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Performance Evaluation of Deadline Monotonic Policy over 802.11 protocol - Ubiquitous Computing and Communication Journal

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Performance Evaluation of Deadline Monotonic Policy over 802.11 protocol - Ubiquitous Computing and Communication Journal Powered By Docstoc
					                Performance Evaluation of Deadline Monotonic Policy
                               over 802.11 protocol

                                  Ines El Korbi and Leila Azouz Saidane
                                       National School of Computer Science
                                       University of Manouba, 2010 Tunisia
                             Emails: ines.korbi@gmail.com Leila.saidane@ensi.rnu.tn



                                                 ABSTRACT
               Real time applications are characterized by their delay bounds. To satisfy the
               Quality of Service (QoS) requirements of such flows over wireless
               communications, we enhance the 802.11 protocol to support the Deadline
               Monotonic (DM) scheduling policy. Then, we propose to evaluate the performance
               of DM in terms of throughput, average medium access delay and medium access
               delay distrbution. To evaluate the performance of the DM policy, we develop a
               Markov chain based analytical model and derive expressions of the throughput,
               average MAC layer service time and service time distribution. Therefore, we
               validate the mathematical model and extend analytial results to a multi-hop
               network by simulation using the ns-2 network simulator.

               Keywords: Deadline Monotonic, 802.11 protocol, Performance evaluation,
               Medium access delay, Throughput, Probabilistic medium access delay bounds.



1   INTRODUCTION                                           priority. To support the DM policy over 802.11, we
                                                           use a distributed scheduling and introduce a new
    Supporting applications with QoS requirements          medium access backoff policy. Therefore, we focus
has become an important challenge for all                  on performance evaluation of the DM policy in terms
communications networks. In wireless LANs, the             of achievable throughput, average MAC layer
IEEE 802.11 protocol [5] has been enhanced and the         service time and MAC layer service time
IEEE 802.11e protocol [6] was proposed to support          distribution. Hence, we follow these steps:
quality of service over wireless communications.                − First, we propose a Markov Chain
    In the absence of a coordination point, the IEEE                 framework modeling the backoff process of
802.11 defines the Distributed Coordination                           n contending stations within the same
Function (DCF) based on the Carrier Sense Multiple                   broadcast region [1].
Access with Collision Avoidance (CSMA/CA)                            Due to the complexity of the mathematical
protocol. The IEEE 802.11e proposes the Enhanced                     model, we restrict the analysis to n
Distributed Channel Access (EDCA) as an extension                    contending stations belonging to two traffic
for DCF. With EDCA, each station maintains four                      categories (each traffic category is
priorities called Access Categories (ACs). The                       characterized by its own delay bound).
quality of service offered to each flow depends on              −     From the analytical model, we derive the
the AC to which it belongs.                                          throughput achieved by each traffic
    Nevertheless, the granularity of service offered                 category.
by 802.11e (4 priorities at most) can not satisfy the           − Then, we use the generalized Z-transforms
real time flows requirements (where each flow is                     [3] to derive expressions of the average
characterized by its own delay bound).                               MAC layer service time and service time
                                                                     distribution.
    Therefore, we propose in this paper a new                   − As the analytical model was restricted to
medium access mechanism based on the Deadline                        two traffic categories, analytical results are
Monotonic (DM) policy [9] to schedule real time                      extended by simulation to different traffic
flows over 802.11. Indeed DM is a real time                          categories.
scheduling policy that assigns static priorities to flow        − Finally, we consider a simple multi-hop
packets according to their deadlines; the packet with                scenario to deduce the behavior of the DM
the shortest deadline being assigned the highest                     policy in a multi hop environment.


                                Ubiquitous Computing- and Communication Journal                              -1 -
                                                          maximum achievable throughput. The native model
     The rest of this paper is organized as follows. In   is also extended in [10] to a non saturated
section 2, we review the state of the art of the IEEE     environment. In [12], the authors derive the average
802.11 DCF, QoS support over 802.11 mainly the            packet service time at a 802.11 node. A new
IEEE 80.211e EDCA and real time scheduling over           generalized Z-transform based framework has been
802.11. In section 3, we present the distributed          proposed in [3] to derive probabilistic bounds on
scheduling and introduce the new medium access            MAC layer service time. Therefore, it would be
backoff policy to support DM over 802.11. In section      possible to provide probabilistic end to end delay
4, we present our mathematical model based on             bounds in a wireless network.
Markov chain analysis. Section 5 and 6 present
respectively throughput and the service time              2.2 Supporting QoS over 802.11
analysis. Analytical results are validated by                2.2.1 Differentiation mechanisms over 802.11
simulation using the ns-2 network simulator [16]. In           Emerging applications like audio and video
section 7, we extend our study by simulation, first to    applications require quality of service guarantees in
take into consideration different traffic categories,     terms of throughput delay, jitter, loss rate, etc.
second, to study the behavior of the DM algorithm in      Transmitting      such     flows     over    wireless
a multi-hop environment where factors like                communications       require    supporting    service
interferences or routing protocols exist. Finally, we     differentiation mechanisms over wireless networks.
conclude in Section 8.
                                                               Many medium access schemes have been
2   LITTERATURE REVIEWS                                   proposed to provide some QoS enhancements over
                                                          the IEEE 802.11 WLAN. Indeed, [4] assigns
2.1 The 802.11 protocol                                   different priorities to the incoming flows. Priority
   2.1.1 Description of the IEEE 802.11 DCF               classes are differentiated according to one of three
     Using DCF, a station shall ensure that the           802.11 parameters: the backoff increase function,
channel is idle when it attempts to transmit. Then it     Inter Frame Spacing (IFS) and the maximum frame
selects a random backoff in the contention window         length. Experiments show that all the three
[0,CW-1], where CW is the current window size and         differentiation schemes offer better guarantees for
varies between the minimum and the maximum                the highest priority flow. But the backoff increase
contention window sizes. If the channel is sensed         function mechanism doesn’t perform well with TCP
busy, the station suspends its backoff until the          flows because ACKs affect the differentiation
channel becomes idle for a Distributed Inter Frame        mechanism.
Space (DIFS) after a successful transmission or an
Extended Inter Frame Space (EIFS) after a collision.          In [7], an algorithm is proposed to provide
The packet is transmitted when the backoff reaches        service differentiation using two parameters of IEEE
zero. A packet is dropped if it collides after            802.11, the backoff interval and the IFS. With this
maximum retransmission attempts.                          scheme high priority stations are more likely to
     The above described two way handshaking              access the medium than low priority ones. The above
packet transmission procedure is called basic access      described researches led to the standardization of a
mechanism. DCF defines a four way handshaking             new protocol that supports QoS over 802.11, the
technique called Request To Send/ Clear To Send           IEEE 802.11e protocol [6].
(RTS/CTS) to prevent the hidden station problem. A
station S j is said to be hidden from S i if S j is         2.2.2 The IEEE 802.11e EDCA
                                                              The IEEE 802.11e proposes a new medium
within the transmission range of the receiver of S i      access mechanism called the Enhanced Distributed
and out of the transmission range of S i .                Channel Access (EDCA), that enhances the IEEE
   2.1.2 Performance evaluation of the 802.11             802.11 DCF. With EDCA, each station maintains
         DCF                                              four priorities called Access Categories (ACs). Each
     Different works have been proposed to evaluate       access category is characterized by a minimum and a
the performance of the 802.11 protocol based on           maximum contention window sizes and an
Bianchi’s work [1]. Indeed, Bianchi proposed a            Arbitration Inter Frame Spacing (AIFS).
Markov chain based analytical model to evaluate the
saturation throughput of the 802.11 protocol. By               Different analytical models have been proposed
saturation conditions, it’s meant that contending have    to evaluate the performance of 802.11e EDCA. In
always packets to transmit.                               [17], Xiao extends Bianchi’s model to the prioritized
     Several works extended the Bianchi model either      schemes provided by 802.11e by introducing
to suit more realistic scenarios or to evaluate other     multiple ACs with distinct minimum and maximum
performance parameters. Indeed, the authors of [2]        contention window sizes. But the AIFS
incorporate the frame retry limits in the Bianchi’s       differentiation parameter is lacking in Xiao’s model.
model and show that Bianchi overestimates the                  Recently Osterbo and Al. have proposed



                               Ubiquitous Computing- and Communication Journal                            -2 -
different works to evaluate the performance of the         information.
IEEE 802.11e EDCA [13], [14], [15]. They propose
a model that takes into consideration all the              3   SUPPORTING DEADLINE MONOTONIC
differentiation parameters of the EDFA especially              (DM) POLICY OVER 802.11
the AIFS one. Moreover different parameters of QoS
have been evaluated such as throughput, average                With DCF all the stations share the same
service time, service time distribution and                transmission medium. Then, the HOL (Head of Line)
probabilistic response time bounds for both saturated      packets of all the stations (highest priority packets)
and non saturated cases.                                   will contend for the channel with the same priority
     Although the IEEE 802.11e EDCA classifies the         even if they have different deadlines.
traffic into four prioritized ACs, there is still no           Introducing DM over 802.11 allows stations
guarantee of real time transmission service. This is       having packets with short deadlines to access the
due to the lack of a satisfactory scheduling method        channel with higher priority than those having
for various delay-sensitive flows. Hence, we need a        packets with long deadlines. Providing such a QoS
scheduling policy dedicated to such delay sensitive        requires distributed scheduling and a new medium
flows.                                                     access policy.

2.3 Real time scheduling over 802.11                       3.1 Distributed Scheduling over 802.11
                                                               To realize a distributed scheduling over 802.11,
    A distributed solution for the support of real-        we introduce a priority broadcast mechanism similar
time sources over IEEE 802.11, called Blackburst, is       to [18]. Indeed each station maintains a local
discussed in [8]. This scheme modifies the MAC             scheduling table with entries for HOL packets of all
protocol to send short transmissions in order to gain      other stations. Each entry in the scheduling table of
priority for real-time service. It is shown that this                                      (       )
                                                           node S i comprises two fields S j , D j where S j is
approach is able to support bounded delays. The
main drawback of this scheme is that it requires           the source node MAC address and D j is the
constant intervals for high priority traffic; otherwise    deadline of the HOL packet of node S j . To
the performance degrades very much.                        broadcast the HOL packet deadlines, we propose to
                                                           use the DATA/ACK access mode.
     In [18], the authors proposed a distributed
priority scheduling over 802.11 to support a class of
                                                               When a node S i transmits a DATA packet, it
dynamic priority schedulers such as Earliest
Deadline First (EDF) or Virtual Clock (VC). Indeed,        piggybacks the deadline of its HOL packet. The
the EDF policy is used to schedule real time flows         nodes hearing the DATA packet add an entry for S i
according to their absolute deadlines, where the           in their local scheduling tables by filling the
absolute deadline is the node arrival time plus the        corresponding fields. The receiver of the DATA
delay bound.                                               packet copies the priority of the HOL packet in ACK
     To realize a distributed scheduling over 802.11,      before sending the ACK frame. All the stations that
the authors of [18] used a priority broadcast              did not hear the DATA packet add an entry for S i
mechanism where each station maintains an entry for        using the information in the ACK packet.
the highest priority packet of all other stations. Thus,
stations can adjust their backoff according to other       3.2 DM medium access backoff policy
stations priorities.
                                                                 Let’s consider two stations S 1 and S 2
     The overhead introduced by the broadcast              transmitting two flows with the same deadline D1 (
priority mechanism is negligible. This is due to the        D1 is expressed as a number of 802.11 slots). The
fact that priorities are exchanged using native DATA       two stations having the same delay bound can access
and ACK packets. Nevertheless, the authors of [18]         the channel with the same priority using the native
propose a generic backoff policy which can be used         802.11 DCF.
by a class dynamic priority schedulers no matter if             Now, we suppose that S 1 and S 2 transmit flows
this scheduler targets delay sensitive flows or rate
sensitive flows.                                           with different delay bounds D1 and D 2 such as
                                                            D1 < D 2 , and generate two packets at time instants
    In this paper, we focus on delay sensitive flows        t 1 and t 2 . If S 2 had the same delay bound as S 1 ,
and propose to support the fixed priority deadline
monotonic scheduler over 802.11 to schedule delay          its packet would have been generated at time t '2 such
sensitive flows. For instance, we use a priority           as t '2 = t 2 + D 21 , where D21 = ( D2 − D1 ) .
broadcast mechanism similar to [5] and propose a
                                                                 At that time, S 1 and S 2 would have the same
new medium access backoff policy where the
backoff value is inferred from the deadline                priority and transmit their packets according to the



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802.11 protocol.                                                      In this section, we propose a mathematical
    Thus, to support DM over 802.11, each station                model to evaluate the performance of the DM policy
uses a new backoff policy where the backoff is given             using Markov chain analysis [1]. We consider the
by:                                                              following assumptions:
  • The random backoff selected in [ 0 , CW − 1]
       according to 802.11 DCF, referred as BAsic                Assumption 1:
       Backoff (BAB).                                            The system under study comprises n contending
  • The DM Shifting Backoff (DMSB):                              stations hearing each other transmissions.
       corresponds to the additional backoff slots that          Assumption 2:
       a station with low priority (the HOL packet               Each station S i transmits a flow Fi with a delay
       having a large deadline) adds to its BAB to               bound Di . The n stations are divided into two traffic
       have the same priority as the station with the
                                                                 categories C1 and C 2 such as:
       highest priority (the HOL packet having the
       shortest deadline).                                            − C1 represents n1 nodes transmitting flows
                                                                           with delay bound D1 .
     Whenever a station S i sends an ACK or hears                     − C 2 represents n 2 nodes transmitting flows
an ACK on the channel its DMSB is revaluated as                            with delay bound D 2 , such as D1 < D 2 ,
follows:
                                                                           D21 = ( D 2 − D1 ) and ( n1 + n 2 ) = n .
    DMSB( S i ) = Deadline( HOL( S i ) ) − DTmin ( S i )   (1)   Assumption 3:
                                                                 We operate in saturation conditions: each station has
                                                                 immediately a packet available for transmission after
    Where DTmin ( S i ) is the minimum of the HOL                the service completion of the previous packet [1].
packet deadlines present in S i scheduling table and             Assumption 4:
Deadline( HOL( S i ) ) is the HOL packet deadline of             A station selects a BAB in a constant contention
node S i .                                                       window [0 ,W − 1] independently of the transmission
                                                                 attempt. This is a simplifying assumption to limit the
    Hence, when S i has to transmit its HOL packet               complexity of the mathematical model.
with a delay bound Di , it selects a BAB in the                  Assumption 5:
contention window [ 0 , CW min − 1] and computes the             We are in stationary conditions, i.e. the n stations
WHole Backoff (WHB) value as follows:                            have already sent one packet at least.

            WHB( S i ) = DMSB( S i ) + BAB( S i )          (2)
                                                                     Depending on the traffic category to which it
                                                                 belongs, each station S i will be modeled by a
                                                                 Markov Chain representing its whole backoff (WHB)
     The station S i decrements its BAB when it
                                                                 process.
senses an idle slot. Now, we suppose that S i senses
the channel busy. If a successful transmission is                4.1 Markov chain modeling a station of category
heard, then S i  revaluates its DMSB when a correct                       C1
ACK is heard. Then the station S i adds the new                         Figure 1 illustrates the Markov chain modeling a
DMSB value to its current BAB as in equation (2).                station S 1 of category C1 . The states of this Markov
Whereas, if a collision is heard, S i reinitializes its         chain are described by the following quadruplet
DMSB and adds it to its current BAB to allow
                                                                 ( R , i , i − j , D21 ) where:
colliding stations contending with the same priority               •         R : takes two values denoted by C 2 and
as for their first transmission attempt. S i transmits                        ~ C 2 . When R = ~ C 2 , the n 2 stations of
when its WHB reaches 0. If the transmission fails, S i                       category C 2 are decrementing their shifting
doubles its contention window size and repeats the                          backoff (DMSB) during D21 slots and
above procedure until the packet is successfully                            wouldn’t contend for the channel. When
transmitted      or     dropped    after    maximum                           R = C 2 , the D 21 slots had already been
retransmission attempts.
                                                                             elapsed and stations of category C 2 will
                                                                             contend for the channel..
4    MATHEMATICAL MODEL OF THE DM
     POLICY OVER 802.11




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Figure 1: Markov chain modeling a category C1 Station


  •     i : the value of the BAB selected by S 1 in             channel during additional D21 slots. Therefore, S 1
        [0 ,W − 1] .                                            moves      to    the     state ( ~ C 2 , i − j , i − j ,− D 21 ) ,
  •     ( i − j ) : corresponds to the current backoff of       i = 1..W − 1 , j = 0.. min( D21 − 1, i − 1) .
       the station S 1 .
  •     D 21 : corresponds to ( D2 − D1 ) . We choose                  Now, If S 1 is in one of the states
        the negative notation − D 21 for stations of             ( C 2 , i , i − D21 ,− D21 ) , i = ( D21 + 1) ..W − 1 and at
         C1 to express the fact that only stations of          least one of the ( n − 1) remaining stations (either a
        category C 2 have a positive DMSB equal to              category C1 or a category C 2 station) transmits,
         D 21 .                                                 then S 1 moves to one of the states
     Initially S 1 selects a random BAB and is in                ( ~ C 2 , i − D21 , i − D21 ,− D21 ) , i = ( D21 + 1) ..W − 1 .
one of the states ( ~ C2 , i , i ,− D21 ) , i = 0..W − 1 .
During ( D 21 − 1) slots, S 1 decrements its backoff if
                                                                4.2 Markov chain modeling a station of
                                                                          category C2
none of the ( n1 − 1) remaining stations of category                    Figure 2 illustrates the Markov chain modeling
C1 transmits. Indeed, during these slots, the n 2               a station S 2 of category C 2 . Each state of S 2
stations of category C 2 are decrementing their                 Markov chain is represented by the quadruplet
DMSB and wouldn’t contend for the channel.                      ( i , k , D21 − j , D21 ) where:
                                                                    • i : refers to the BAB value selected by S 2 in
     When S 1 is in one of the states                                       [0 ,W − 1] .
( ~ C 2 , i , i − ( D21 − 1) ,− D21 ) , i = D 21 ..W − 1 and        •       k : refers to the current BAB value of S 2 .
                                                          th
senses the channel idle, it decrements its D 21 slot.             •      D21 − j : refers to the current DMSB of S 2 ,
But S 1 knows that henceforth the n 2 stations of                        j ∈ [ 0 , D21 ] .
category C 2 can contend for the channel (the D 21                •      D21 : corresponds to ( D 2 − D1 ) .
slots had been elapsed). Hence, S 1 moves to one of
the states ( C 2 , i , i − D21 ,− D 21 ) , i = D 21 ..W − 1 .      When S 2 selects a BAB, its DMSB equals D21 
                                                                and is in one of the states ( i , i , D 21 , D 21 ) ,
      However, when the station S 1 is in one of the            i = 0..W − 1 . During D21 slots, only the n1
states ( ~ C 2 , i , i − j ,− D 21 ) , i = 1..W − 1 ,           stations of category C1 contend for the channel.
  j = 0.. min( D 21 − 1, i − 1) and at least one of the
 ( n1 − 1) remaining stations of category C1                     If S 2 senses the channel idle during D21 slots, it
transmits, then the stations of category C 2 will               moves to one of the states ( i , i ,0 , D 21 ) , i = 0..W − 1 ,
reinitialize their DMSB and wouldn’t contend for                where it ends its shifting backoff.




                                   Ubiquitous Computing- and Communication Journal                                          -5 -
Figure 2: Markov chain modeling a category C 2 Station


   When S 2 is in one of the states ( i , i ,0 , D 21 ) ,
i = 0..W − 1 , the ( n 2 − 1) other stations of category              •    γ   : the set of states of S 2 , where stations
                                                                               2

C 2 have also decremented their DMSB and can                               of category C 2 contend for the channel
                                                                           (pink states in figure 2).
contend for the channel. Thus, S 2 decrements its
                                                                           γ 2 = { ( i , i ,0 , D21 ) , i = 0..W − 1
BAB and moves to the state ( i , i − 1,0 , D 21 ) ,
                                                                           ∪ ( i , i − 1,0 , D 21 ) , i = 2..W − 1}
i = 2..W − 1 , only if none of ( n − 1) remaining
stations transmits.
                                                                         Therefore, when stations of category C 1 are in
   If S 2 is in one of the states ( i , i − 1,0 , D 21 ) ,          one the states of ξ 1 , stations of category C 2 are in
i = 2..W − 1 , and at least one of the ( n − 1)                     one of the states of ξ 2 . Similarly, when stations of
remaining stations transmits, the n 2 stations of                   category C 1 are is in one of the states of γ 1 ,
category C 2 will reinitialize their DMSB and S 2                   stations of category C 2 are in one of the states of
moves to the state              ( i − 1, i − 1, D21 , D21 ) ,       γ 2.
 i = 2..W − 1 .                                                          Hence, we derive the expressions of S 1
4.3 Blocking probabilities in the Markov chains                     blocking probabilities p11 and p12 shown in
     According to the explanations given in                         figure 1 as follows:
paragraphs 4.1 and 4.2, the states of the Markov
chains modeling stations S 1 and S 2 can be divided                   −     p11 : the probability that S 1 is blocked given
into the following groups:
                                                                           that S 1 is in one of the states of ξ 1 . p11 is
                                                                                                                        '
  •    ξ 1 : the set of states of S 1 where none of the                    the probability that at least a station S 1 of
       n 2 stations of category C 2 contends for the                       the other ( n1 − 1) stations of C 1 transmits
       channel (blue states in figure 1).                                  given that S 1 is in one of the states of ξ 1 .
                                                                                        '

       ξ 1 = { ( ~ C 2 , i , i − j ,− D 21 ) , i = 0..W − 1,
                                                                                                       p 11 = 1 − ( 1 − τ             ) n1 − 1          (3)
        j = 0.. min( max( 0 , i − 1) , D 21 − 1)}
                                                                                                                                11

                                                                           where τ         11
                                                                                                                                     '
                                                                                                 is the probability that a station S 1
                                                                                                        '
  •     γ  : the set of states of S 1 where stations of
            1
                                                                           of C1 transmits given that S 1 is in one of
       category C 2 can contend for the channel                            the states of ξ 1 :
       (pink states in figure 1).                                          τ   11
                                                                                           '
                                                                                             [
                                                                                    = Pr S 1 transmits ξ 1                 ]
       γ 1 = { ( C 2 , i , i − D 21 ,− D 21 ) , i = D 21 ..W − 1}                                             ( ~ C2 ,0 ,0 ,− D21 )
                                                                                                          π   1
                                                                                    =                                                                   (4)
                                                                                           W − 1  min ( max ( 0 ,i − 1) ,D21 − 1)                 
        ξ
  •        : the set of states of S 2 where stations of
            2
       category C 2 do not contend for the channel
                                                                                           ∑
                                                                                           i= 0 
                                                                                                 
                                                                                                                 ∑    π 1~C2 ,i ,i − j ,− D21 )
                                                                                                                          (                        
                                                                                                                                                   
                                                                                                                j= 0                               
       (blue states in figure 2).
       ξ 2 = { ( i , i , D 21 − j , D 21 ) , i = 0..W − 1,                   (            j ,− D21 )
                                                                           π 1R ,i ,i −                is defined as the probability of
        j = 0..( D 21 − 1)}
                                                                           the state ( R , i , i − j ,− D21 ) , in the stationary


                                     Ubiquitous Computing- and Communication Journal                                                                   -6 -
       conditions and Π                               1    = π    {   ( R ,i ,i −
                                                                      1
                                                                                      j ,− D21 )
                                                                                                    }    is the          −       p 22 : the probability that S 2 is blocked

       probability vector of a category C 1 station.                                                                            given that S 2 is in one of the states of γ 2 .
                                                                                                                                       p 22 = 1 − ( 1 − τ    12   ) n1 ( 1 − τ 22 ) n2 − 1          (9)
  −     p12 : the probability that S 1 is blocked given
       that S 1 is in one of the states of γ 1 . p12 is                                                                     The blocking probabilities described above
                                                   '                                                                   allow deducing the transition state probabilities and
       the probability that at least a station S 1 of
                                                                                                                       having the transition probability matrix Pi , for a
       the other ( n1 − 1) stations of C 1 transmits
                                                                                                                       station of traffic category C i .
       given that S 1 is in one of the states of γ 1 or
                    '
                                                                                                                            Therefore, we can evaluate the state
                            '
       at least a station S 2 of the n 2 stations of                                                                   probabilities by solving the following system [11]:
                                  '
       C 2 transmits given that S 2 is in one of the
                                                                                                                                             Π i Pi = Π i
       states of γ 2 .                                                                                                                      

                                       p 12 = 1 − ( 1 − τ                         ) n1 − 1 ( 1 − τ 22 ) n2
                                                                                                                                            
                                                                                                                                             ∑
                                                                                                                                             j
                                                                                                                                                  π ij = 1                                      (10)
                                                                          12                                                                
                                                                                                             (5)
                                                                                                                       4.4 Transition probability matrices
       where τ 12 is the probability that a station S 1
                                                      '                                                                  4.4.1 Transition probability matrix of a
                                     '                                                                                        category C1 station
       of C 1 transmits given that S 1 is in one of
                                                                                                                           Let P1 be the transition probability matrix of
       the states of γ 1 .                                                                                             the station S 1 of category C1 . P1 { i , j} is the
                           τ    12
                                              '
                                                  [
                                       = Pr S 1 transmits γ                                 1   ]                      probability to transit from state i to state j . We
                                                        (
                                                      π 1C2 ,D21 ,0 ,− D21 )                                           have:
                                      =         W−1
                                                                                                             (6)
                                                                                                                       P1 { ( ~ C 2 , i , i − j ,− D 21 ) , ( ~ C 2 , i , i − ( j + 1) ,− D 21 )}
                                                 ∑        π   1
                                                               ( C2 ,i ,i − D21 ,− D21 )
                                               i = D21                                                                 = 1 − p11 , i = 2..W − 1, j = 0.. min( i − 2 , D 21 − 2 )
                                                                                                                                                                                               (11)
       and τ        22
                                                            '
                           the probability that a station S 2 of                                                       P1 { ( ~ C 2 , i ,1,− D 21 ) , ( ~ C 2 ,0 ,0 ,− D 21 )} = 1 − p11 ,
                                  '
       C 2 transmits given that S 2 is in one of the                                                                   i = 1.. min(W − 1, D 21 − 1)
       states of γ 2 .                                                                                                                                                                         (12)
                                                                                                                       P1 { ( ~ C 2 , i , i − D 21 + 1,− D 21 ) , ( C 2 , i , i − D 21 ,− D 21 )}
           τ   12   = Pr          [   S '2    transmits γ                 2   ]                                        = 1 − p11 , i = D 21 ..W − 1
                                                              ( 0 ,0 ,0 ,D21 )                                                                                                                  (13)
                                                       π
                    =                                         2
                                                                                                             (7)        P1{ ( ~ C2 , i , i − j ,− D21 ) , ( ~ C2 , i − j , i − j ,− D21 )}
                                W−1                                   W−1                                                                                                                       (14)
                                                                                                                        = p11 , i = 2..W − 1, j = 1.. min( i − 1, D21 − 1)
                                ∑i= 0
                                          π
                                              ( i ,i ,0 ,D21 )
                                              2                   +   ∑
                                                                      i= 2
                                                                              π
                                                                                      ( i ,i − 1,0 ,D21 )
                                                                                      2

                                                                                                                       P { ( ~ C2 , i , i ,− D21 ) , ( ~ C2 , i , i ,− D21 )} = p11 ,
                                                                                                                        1
           ( i ,k ,D21 −       j ,D21 )
                                                                                                                                                                                               (15)
       π   2                                  is defined as the probability                                            i = 1..W − 1
       of the state                              ( i , k , D21 −            j , D 21 ) ,                in the
                                                                                                                       P { ( C2 , i , i − D21 ,− D21 ) , ( ~ C2 , i − D21 , i − D21 ,− D21 )}
       stationary condition. Π                                        2   = π  {        ( i ,k ,D21 −
                                                                                        2
                                                                                                        j ,D21 )
                                                                                                                   }    1
                                                                                                                       = p12 , i = ( D21 + 1) ..W − 1
       is the probability vector of a category C 2                                                                                                                                              (16)
       station.
                                                                                                                       P1{ ( C2 ,i ,i − D21 ,− D21 ) ,( C2 ,( i − 1) ,( i − 1 − D21 ) ,− D21 )}
     In the same way, we evaluate p 21 and p 22 the                                                                    = 1 − p12 ,i = ( D21 + 1) ..W − 1
blocking probabilities of station S 2 as shown in                                                                                                                                               (17)
figure 2:
   −     p 21 : the probability that S 2 is blocked                                                                                                                              1
                                                                                                                       P { ( ~ C2 ,0 ,0 ,− D21 ) , ( ~ C2 , i , i ,− D21 )} =
                                                                                                                        1                                                          ,
        given that S 2 is in one of the states of ξ 2 .                                                                                                                          W              (18)
                                                                                                                       i = 0..W − 1
                                        p 21 = 1 − ( 1 − τ                11      )   n1
                                                                                                             (8)
                                                                                                                         If ( D 21 < W ) then:




                                                                  Ubiquitous Computing- and Communication Journal                                                                              -7 -
 P { ( C2 , D21 ,0 ,− D21 ) , ( ~ C2 , i , i ,− D21 )} =
                                                               1
                                                                 ,                                 τ 11 = f (τ 11 ,τ 12 ,τ 22 )
  1                                                                                               
                                                                                                   τ 12 = f (τ 11 ,τ 12 ,τ 22 )
                                                               W                      (19)
i = 0..W − 1                                                                                      
                                                                                                   τ 22 = f (τ 11 ,τ 12 ,τ 22 )
                                                                                                   under the constraint
    By replacing p11 and p 12 by their values in                                                  
equations (3) and (5) and by replacing P1 and Π 1                                                  τ 11 > 0 ,τ 12 > 0 ,τ 22 > 0 ,τ
                                                                                                                                            11    < 1,τ     12   < 1,τ   22   < 1
in (10) and solving the resulting system, we can                                                                                                                               (28)
          ( R ,i ,i − j ,− D21 )
express π 1                      as a function of τ 11 , τ 12 and
                                                                                                 Solving the above system (28), allows deducing
τ 22 given respectively by equations (4), (6) and                                            the expressions of τ 11 , τ 12 and τ 22 , and deriving
(7).                                                                                         the state probabilities of Markov chains modeling
                                                                                             category C 1 and category C 2 stations.
    4.4.2  Transition probability matrix of a
         category C2 station
     Let P2 be the transition probability matrix of                                          5    THROUGHPUT ANALYSIS
the station S 2 belonging to the traffic category C 2 .
The transition probabilities of S 2 are:                                                          In this section, we propose to evaluate Bi , the
                                                                                             normalized throughput achieved by a station of
 P2 { ( i , i , D21 − j , D21 ) , ( i , i , D21 − ( j + 1) , D21 )}                          traffic category C i [1]. Hence, we define:
                                                                                      (20)
 = 1 − p21 , i = 0..W − 1, j = 0..( D21 − 1)
                                                                                             −      Pi ,s : the probability that a station S i belonging
P2 { ( i , i , D21 − j , D21 ) , ( i , i , D21 , D21 )} = p21 ,                                    to the traffic category C i transmits a packet
                                                                                      (21)
i = 0..W − 1, j = 0..( D21 − 1)                                                                    successfully. Let S 1 and S 2 be two stations
                                                                                                   belonging respectively to traffic categories C 1
P2 { ( i , i ,0 , D21 ) , ( i , i − 1,0 , D21 )} = 1 − p22 ,                                       and C 2 . We have:
                                                                                      (22)
i = 2..W − 1
                                                                                             P1,s = Pr [ S1 transmits successfully ξ 1 ] Pr [ξ 1 ]
P2 { ( 1,1,0 , D21 ) , ( 0 ,0 ,0 , D21 )} = 1 − p22                               (23)       + Pr [ S1 transmits successfully γ 1 ] Pr [γ 1 ]
                                                                                             = τ 11 ( 1 − p11 ) Pr [ξ 1 ] + τ 12 ( 1 − p12 ) Pr [γ 1 ]
P2 { ( i , i ,0 , D21 ) , ( i , i , D21 , D21 )} = p22 ,
                                                                                      (24)                                                                                     (29)
i = 1..W − 1
                                                                                             P2 ,s = Pr [ S 2 transmits successfully ξ                         ] Pr[ξ 2 ]
P2 { ( i , i − 1,0 , D21 ) , ( i − 1, i − 1, D21 , D21 )} = p22 ,                                                                                           2

i = 2..W − 1
                                                                                 (25)        + Pr [ S 2 transmits successfully γ                   2     ] Pr[γ 2 ]
                                                                                             =τ   22 ( 1 −   p 22 ) Pr [γ     2]
 P2 { ( i , i − 1,0 , D21 ) , ( i − 1, i − 2 ,0 , D21 )} = 1 − p22 ,                                                                                (30)
                                                                                      (26)
 i = 3..W − 1                                                                                −      Pidle : the probability that the channel is idle.
                                                     1
 P2 { ( 0 ,0 ,0 , D21 ) , ( i , i , D21 , D21 )} =     , i = 0..W − 1 (27)
                                                     W                                           The channel is idle if the n1 stations of
                                                                                             category C 1 don’t transmit given that these stations
    By replacing p 21 and p22 by their values in
                                                                                             are in one of the states of ξ 1 or if the n stations
equations (8) and (9) and by replacing P2 and Π 2
                                                                                             (both category C 1 and category C2 stations) don’t
in (10) and solving the resulting system, we can
           ( i ,k ,D − j ,D21 )                                                              transmit given that stations of category C 1 are in
express π 2 21                  as a function of τ 11 , τ 12
                                                                                             one of the states of γ 1 . Thus:
and τ       22      given respectively by equations (4), (6)
                                                           ( R ,i ,i −   j ,− D21 )
and (7). Moreover, by replacing π                          1                          and    Pidle = ( 1 − τ    11   ) n1   Pr [ξ 1 ] + ( 1 − τ   12   ) n1 ( 1 − τ 22 ) n2    Pr [γ 1 ]
    ( i ,k ,D21 −   j ,D21 )                                                                                                                                                   (31)
π   2                          by their values, in equations (4), (6)
and (7), we obtain a system of non linear equations                                              Hence, the expression of the throughput of a
as follows:
                                                                                             category C i station is given by:




                                                     Ubiquitous Computing- and Communication Journal                                                                          -8 -
                                 Pi ,s T P
    Bi =
                                                 2              
             PIdle Te + Ps Ts +  1 − PIdle −
                                               ∑       ni Pi ,s  Tc
                                                                 
                                                                          For all the scenarios, we consider that we are in
                                                                                                                    n
                                                i= 1                    presence of n contending stations with       stations
                                                                   (32)                                             2
                                                                          for each traffic category. In figure 3, n is fixed to
     Where Te denotes the duration of an empty                            8 and we depict the throughput achieved by the
                                                                          different stations present in the network as a
slot, Ts and Tc denote respectively the duration of
                                                                          function of the contention window size W ,
a successful transmission and a collision.                                 ( D21 = 1) . We notice that the throughput achieved
              2            
 1 − PIdle −
             ∑    ni Pi ,s 
                             corresponds to    the                       by category C1 stations (stations numbered from
             i= 1                                                        S 11 to S 14 ) is greater than the one achieved by
probability of collision. Finally T p denotes the                         category C 2 stations (stations numbered from S 21
average time required to transmit the packet data                         to S 24 ).
payload. We have:

     (
Ts = T PHY + TMAC + T p + T D + SIFS +  )
                                                                 (33)
( TPHY    + T ACK + T D ) + DIFS

      (
Tc = TPHY + TMAC + T p + TD + EIFS  )                            (34)

     Where T PHY , TMAC and T ACK are the
durations of the PHY header, the MAC header and
the ACK packet [1], [13]. T D is the time required to
transmit the two bytes deadline information.
Stations hearing a collision wait during EIFS before
resuming their backoff.

     For numerical results stations transmit 512                          Figure 3: Normalized throughput as a function of
bytes data packets using 802.11.b MAC and PHY                             the contention window size ( D 21 = 1, n = 8 )
layers parameters (given in table 1) with a data rate
equal to 11Mbps. For simulation scenarios, the                                 Analytically, stations belonging to the same
propagation model is a two ray ground model. The                          traffic category have the same throughput given by
transmission range of each node is 250m. The                              equation (31). Simulation results validate analytical
distance between two neighbors is 5m. The EIFS                            results and show that stations belonging to the same
parameter is set to ACKTimeout as in ns-2, where:                         traffic category (either category C1 or category C 2
                                                                          ) have nearly the same throughput. Thus, we
ACKTimeout = DIFS + ( T PHY + T ACK + T D ) + SIFS
                                                                          conclude the fairness of DM between stations of the
                                               (35)                       same category.
Table 1: 802.11 b parameters.                                                 For subsequent throughput scenarios, we focus
                                                                          on one representative station of each traffic
                                                                          category. Figure 4, compares category C1 and
                                                                          category C 2 stations throughputs to the one
                                                                          obtained with 802.11.

                                                                              Curves are represented as a function of W and
        Data Rate                11 Mb/s                                  for different values of D21 . Indeed as D21
            Slot                   20 µs
                                                                          increases, the category C1 station throughput
           SIFS                    10 µs
           DIFS                    50 µs                                  increases, whereas the category C 2 station
       PHY Header                 192 µs                                  throughput decreases. Moreover as W increases,
       MAC Header                 272 µs                                  the difference between stations throughputs is
           ACK                   112 µs                                   reduced. This is due to the fact that the shifting
     Short Retry Limit               7                                    backoff becomes negligible compared to the
                                                                          contention window size.


                                            Ubiquitous Computing- and Communication Journal                               -9 -
                                                             We propose to evaluate the Z-Transform of the
    Finally, we notice that the category C1 station     MAC layer service time [3], [14], [15] to derive an
obtains better throughput with DM than with             expression of the average service time. The service
802.11, but the opposite scenario happens to the        time depends on the duration of an idle slot Te , the
category C 2 station.                                   duration of a successful transmission Ts and the
                                                        duration of a collision Tc [1], [3],[14]. As Te is the
                                                        smallest duration event, the duration of all events
                                                                           Tevent 
                                                        will be given by          .
                                                                           Te 

                                                        6.1 Z-Transform of the MAC layer service time

                                                          6.1.1 Service time Z-transform of a category
                                                                C1 station:
                                                             Let TS 1 ( Z ) be the service time Z-transform of
                                                        a station S1 belonging to traffic category C 1 . We
                                                        define:

Figure 4: Normalized throughput as a function of               H 1( R ,i ,i −      j ,− D21 )   (Z) :
                                                                                            The Z-transform of the
the contention window size (different D21 values)
                                                        time already elapsed from the instant S 1 selects a
     In figure 5, we generalize the results for         basic backoff in [ 0 ,W − 1] (i.e. being in one of the
different numbers of contending stations and fix the    states ( ~ C 2 , i , i ,− D 21 ) ) to the time it is found in the
contention window size W to 32.
                                                        state ( R ,i ,i − j ,− D21 ) .
                                                             Moreover, we define:

                                                                      11
                                                          •         Psuc : the probability that S 1 observes a
                                                                   successful transmission on the channel,
                                                                   while S 1 is in one of the states of ξ 1 .
                                                                                     Psuc = ( n1 − 1)τ 11 ( 1 − τ 11 ) n1 − 2
                                                                                       11
                                                                                                                                                (36)

                                                                      12
                                                          •         Psuc : the probability that S 1 observes a
                                                                   successful transmission on the channel,
                                                                   while S 1 is in one of the states of γ 1 .
                                                                       Psuc = ( n1 − 1)τ 12 ( 1 − τ 12 ) n1 − 2 ( 1 − τ 22 ) n2
                                                                         12
                                                                                                                                                (37)
Figure 5: Normalized throughput as a function of                              + n2τ 22 ( 1 − τ 22 ) n2 − 1 ( 1 − τ 12 ) n1 − 1
the number of contending stations
                                                            We evaluate H 1( R ,i ,i − j ,− D21 ) ( Z ) for each state
     All the curves show that DM performs service
differentiation over 802.11 and offers better           of S1 Markov chain as follows:
throughput for category C1 stations independently
                                                                                                           Ts 
of the number of contending stations.                                                            1  11  Te 
                                                          H 1( ~ C2 ,i ,i ,− D21 ) ( Z ) =        + Psuc Z        +
                                                                                                 W 
                                                                                                    
                                                                                                    
6   SERVICE TIME ANALYSIS                                                           Tc  
                                                                                     min ( i + D21 − 1,W − 1)

     In this section, we evaluate the average MAC         (p  11   −     11
                                                                       Psuc   )      T 
                                                                                  Z e             ∑   H 1( ~ C 2 ,k ,i ,− D21 )   (Z)
                                                                                          
                                                                                                 k = i+ 1
layer service time of category C 1 and category C 2
stations using the DM policy. The service time is                                                       Ts                  Tc                
                                                                                                12  Te 
                                                                                                                         (                 )       
                                                                                                                               
                                                          + H ( C 2 ,i + D21 ,i ,− D21 ) ( Z )  Psuc Z
                                                            ˆ1                                                                  T
the time interval from the time instant that a packet                                                                    12
                                                                                                               + p11 − Psuc Z  e                 
becomes at the head of the queue and starts to                                                                                                    
contend for transmission to the time instant that                                                                                                 
either the packet is acknowledged for a successful                                                                                              (38)
transmission or dropped.                                       Where:


                               Ubiquitous Computing- and Communication Journal                                                                 - 10 -
                                                                                                                                   Ts 

                                                                                                                                           (( 1 −
                                                                                                                                   
              H 1( C ,i + D ,i ,− D ) ( Z ) = H 1( C ,i + D ,i ,− D ) ( Z )
                ˆ                                                                                              TS1 ( Z ) =                          p11 ) H 1( ~ C 2 ,0 ,0 ,− D21 ) ( Z )
                                                                                                                                    T
                                                                                                                                 Z e 
             
             
                      2      21        21              2    21      21

              if ( i + D 21 ) ≤ W − 1                                                                                                                             Tc 
                                                                                                                                                                   Te 
                                                                                                                                                                    )∑                  (
                                                                                                                                                                      m
              ˆ                                                                                               + ( 1 − p12 ) H 1( C2 ,D21 ,0 ,− D21 ) ( Z )                 p11 H 1( ~ C2 ,0 ,0 ,− D21 ) ( Z )
              H 1( C2 ,i + D21 ,i ,− D21 ) ( Z ) = 0 Otherwise
             
                                                                                                                                                                 Z
                                                                                                                                                            i= 0 
                                                                       (39)                                                                                      
                                                                                                               + p12 H 1( C2 ,D21 ,0 ,− D21 ) ( Z )        ))   i

       We also have:                                                                                                                                                                                             m+ 1
                                                                                                                   Tc                                                                                   
                                                                                                                  T                                                                                         
                                           ( (1 −    p11 ) Z ) j H 1( ~ C2 ,i ,i ,− D21 ) ( Z )
                                                                                                                             (
                                                                                                               +  Z  e  p11H 1( ~ C2 ,0 ,0 ,− D21 ) ( Z ) + p12 H 1( C 2 ,D21 ,0 ,− D21 ) ( Z )       )   
                     j ,− D21 ) ( Z ) =
 H 1( ~ C2 ,i ,i −                                                                                                                                                                                          
                                                             Ts                                      Tc                                                                                                 
                                                                         (                   )
                                                                                                     
                                                11
                                          1 − Psuc Z 
                                                              Te              11
                                                                     − p11 − Psuc Z 
                                                                                                        Te                                                                                      (44)
 i = 2..W − 1, j = 1..min( i − 1, D21 − 1)
                                                                                                                  6.1.2 Service time Z-transform of a category
                                                                                                      (40)                C2 station:
                                                                                                                    In the same way, let TS2 (Z) be the service
                                           ( (1 −    p11 ) Z ) D21 H 1( ~ C2 ,i ,i ,− D21 ) ( Z )              time Z-transform of a station S 2 of category C 2 .
H 1( C2 ,i ,i − D21 ,− D21 ) ( Z ) =
                                                         Ts                                          Tc    We define:
                                                                             (                   )
                                                                                                     
                                           1−      11  Te 
                                                 Psuc Z                  − p11 −          11
                                                                                        Psuc
                                                                                                        T
                                                                                                     Z e           H 2( i ,k ,D21 − j ,D21 ) ( Z ) : The Z-transform of the
+ ( 1 − p12 ) ZH 1( C 2 ,i + 1,i + 1− D21 ,− D21 ) ( Z ) ,i = D21 ..W − 2                                      time already elapsed from the instant S 2 selects a
                                                                                                      (41)     basic backoff in [0 ,W − 1] (i.e. being in one of the
                                                                                                               states ( i , i , D21 , D 21 ) ) to the time it is found in the
H 1( C 2 ,W − 1,W − 1− D21 ,− D21 ) ( Z )
                                                                                                               state ( i , k , D21 − j , D 21 ) .
    ( (1 −    p11 ) Z ) D21 H 1( ~ C2 ,W − 1,W − 1,− D21 ) ( Z )                                                    Moreover, we define:
=
                        Ts                                  Tc 

                                     (                  )
                                                            
                  11  Te                         11          T                                                            21
         1−     Psuc Z            − p11 −        Psuc       Z e                                                  •      Psuc : the probability that S 2 observes a
+ ( 1 − p12 ) ZH 1( C 2 ,i + 1,i + 1− D21 ,− D21 ) ( Z ) ,i = D21 ..W − 2                                                successful transmission on the channel,
                                                                                                      (42)               while S 2 is in one of the states of ξ 2 .
                                                                                                                                        Psuc = ( n1 − 1)τ 12 ( 1 − τ 12 ) n1 − 1
                                                                                                                                          11
                                                                                                                                                                                                 (45)
                                          (1 −      p11 ) ZH 1( ~ C2 ,1,1,− D21 ) ( Z )
H 1( ~ C 2 ,0 ,0 ,− D21 ) ( Z ) =
                                                     Ts                                         Tc 
                                                                                                                            22
                                                                     (                      )Z                     •      Psuc : the probability that S 2 observes a
                                                                                                
                                               11  Te                                11         Te 
                                      1−     Psuc Z             − p11 −              Psuc
                                                                                                                         successful transmission on the channel,
                      min ( W − 1,D21 − 1)
                                                                                 1                                       while S 2 is in one of the states of γ 2 .
+ ( 1 − p11 ) Z               ∑      H 1( ~ C2 ,i ,1,− D21 ) ( Z ) +
                                                                                 W
                              i= 2                                                                                                Psuc = n1τ 12 ( 1 − τ 12 ) n1 − 1 ( 1 − τ 22 ) n2 − 1
                                                                                                                                    22

                                                                                                      (43)                                                                                       (46)
                                                                                                                                 + ( n2 − 1)τ 22 ( 1 − τ 22 ) n2 − 2 ( 1 − τ 12 ) n1
       If S 1 transmission state is ( ~ C 2 ,0 ,0 ,− D 21 ) ,
the transmission will be successful only if none of                                                                 We evaluate H 2( i ,i ,D21 − j ,− D21 ) ( Z ) for each state
the ( n1 − 1) remaining stations of C 1 transmits.                                                             of S1 Markov chain as follows:
                                                                                                                                                    1
Whereas when the station S 1 transmission state is                                                               H 2( i ,i ,D21 − j ,D21 ) ( Z ) =     , i = 0 and i = W − 1 (47)
 ( C 2 , D21 ,0 ,− D21 ) , the transmission occurs                                                                                                 W
successfully only if none of ( n − 1) remaining                                                                                                               Ts 
                                                                                                                                                1  22  Te 
                                                                                                               H 2( i ,i ,D21 ,D21 ) ( Z ) =       +  Psuc Z        +
stations (either a category C 1 or a category C 2                                                                                              W 
station) transmits.                                                                                                                                  
                                                                                                                                      Tc  

                                                                                                               (                    )
                                                                                                                                       
    If the transmission fails, S 1 tries another                                                                p 22 − Psuc Z  e   H 2( i + 1,i ,0 ,D21 ) ( Z ) , i = 1..W − 2
                                                                                                                             22        T

transmission. After m retransmissions, if the                                                                                                
packet is not acknowledged, it will be dropped.                                                                                              
                                                                                                                                                                                  (48)

                                                                                                                       To compute H 2( i ,i ,D21 −                   j ,D21 )   (Z) ,       we define
       Thus:
                                                                                                               Tdec ( Z ) , such as:
                                                                                                                 j




                                                             Ubiquitous Computing- and Communication Journal                                                                                    - 11 -
                                                                                                                                                                         m+ 1
                                                                                                                           Tc                           
 0
Tdec   (Z) =   1                                                                         (49)                                                           
                                                                                                     TS 2 ( Z ) =  p 22 Z  e  H 2( 0 ,0 ,0 ,D21 ) ( Z ) 
                                                                                                                             T
                                                                                                                                                                                +
                                                                                                                                                          
                                                 (1 −       p 21 ) Z                                                                                      
Tdec ( Z ) =
  j
                                                                                                                                                                                                                  i
                                       Ts                                  Tc                                     Ts                                                Tc                             
                                                                                                                                                           m
                                                                                                                                                                                                           
                 21
                                                 (                     )               j− 1
                                                                                                                                                            ∑
                                                                            
           1 −  Psuc Z                 Te               21
                                                + p 21 − Psuc Z               Te 
                                                                                       Tdec ( Z )   (1 −   p 22 ) Z    Te    H 2( 0 ,0 ,0 ,D21 ) ( Z )           p 22 Z  Te  H 2
                                                                                                                                                                                       ( 0 ,0 ,0 ,D21 ) ( Z ) 
                                                                                                                                                          i= 0                                             
                                                                                                                                                                                                           
for j = 1..D 21                                                                                                                                                                        (54)
                                                                                          (50)
                                                                                                     6.2 Average Service Time
       So:                                                                                                From equations (43) (respectively equation
                                                                                                     (54)), we derive the average service time of a
H 2( i ,i ,D21 −   j ,D21 )   (Z) =     H 2( i ,i ,D21 −    j + 1,D21 )    ( Z )Tdec ( Z ) ,
                                                                                  j
                                                                                                     category C 1 station ( respectively a category C 2
i = 0..W − 1, j = 1..D 21 , ( i , j ) ≠ ( 0 , D 21 )                                                 station). The average service time of a category C i
                                                                                          (51)       station is given by:
       And:                                                                                                               X i = TS i( 1) ( 1)        (55)

H 2( i ,i − 1,0 ,D21 ) ( Z ) = ( 1 − p 22 ) ZH 2( i + 1,i ,0 ,D21 ) ( Z )
                     (1 −     p 22 ) ZH 2( i ,i ,0 ,D21 ) ( Z )                                          Where TS i( 1) ( Z ) , is the derivate of the service
+
                Ts                   Tc                                                        time Z-transform of station S i [11].
        22  Te 
                                  (                   )               D21
                                        
                                                                      Tdec ( Z )
                                  22     T
   1 −  Psuc Z        + p 22 − Psuc Z  e 
                                                                                                        By considering the same configuration as in
                                                                                                   figure 3, we depict in figure 5, the average service
i = 2..W − 2                                                                                         time of category C 1 and category C2 stations as a
                                                                                          (51)
                                                                                                     function of W . As for the throughput analysis,
                                                                                                     stations belonging to the same traffic category have
H 2( W − 1,W − 2 ,0 ,D21 ) ( Z )
                                                                                                     nearly the same average service value. Simulation
               (1 −     p 22 ) ZH 2( W − 1,W − 1,0 ,D21 ) ( Z )                                      service time values coincide with analytical values
=                                                                                                    given by equation (55). These results confirm the
                 Ts                   Tc                                      (52)
         22  Te 
                                   (                    )               D21                         fairness of DM in serving stations of the same
                                         
                                                                        Tdec ( Z )
                                   22     T
    1 −  Psuc Z        + p 22 − Psuc Z  e                                                         category.
                                                                      
                                                                      


    According to figure 2 and using equations (44),
we have:

H 2( 0 ,0 ,0 ,D21 ) ( Z ) = H 2( 0 ,1,0 ,D21 ) ( Z )Tdec ( Z )
                                                         21      D


                   ( 1 − p 22 ) ZH 2( 1,1,0 ,D21 ) ( Z )
+
                     Ts                          Tc             (53)
         22  Te 
                                  (                     )
                                                     
  1 −  Psuc Z                            22
                            + p 22 − Psuc Z         Te  T D21 ( Z )
                                                           dec
                                                         
                                                         

    Therefore, we can derive an expression of S 2
Z-transform service time as follows:                                                                 Figure 6: Average service time as a function of the
                                                                                                     contention window size (D21=1, n=8)

                                                                                                         In figure 8, we show that category C 1 stations
                                                                                                     obtain better average service time than the one
                                                                                                     obtained with 802.11 protocol. Whereas, the
                                                                                                     opposite scenario happens for category C 2 stations



                                                        Ubiquitous Computing- and Communication Journal                                                                                - 12 -
independently of n , the number of contending           D 21 = 4 , the probability that      S 1 service time
stations within the network.
                                                        exceeds 0.005s equals 0.28%. Whereas, station S 2
                                                        service time exceeds 0.005s with the probability
                                                        5.67%. Thus, DM offers better service time
                                                        guarantees for the stations with the highest priority.

                                                            In figure 9, we double the size of the contention
                                                        window size and set it to 64. We notice that
                                                        category C 1 and category C 2 stations service time
                                                        curves become closer. Indeed, when W becomes
                                                        large, the BAB values increase and the (DMSB)
                                                        becomes negligible compared to the basic backoff.
                                                        The whole backoff values of S 1 and S 2 become
                                                        near and their service time accordingly.


Figure 7: Average service time as a function of the
number of contending stations

6.3 Service Time Distribution

    Service time distribution is obtained by
inverting the service time Z transforms given by
equations (43) and (54). But we are most interested
in probabilistic service time bounds derived by
inverting the complementary service time Z
transform given by [11]:

                ~        1 − TS i ( Z )
                Xi (Z) =                        (55)    Figure 9: Complementary service time distribution
                            1− Z                        for different values of D21 (W=64)

     In figure 8, we depict analytical and simulation       In figure 10, we depict the complementary
values of the complementary service time                service time distribution for both category C 1 and
distribution of both category C 1 and category C 2      category C 2 stations and for values of n , the
station (W = 32 ) .                                     number of contending nodes.




Figure 8: Complementary service time distribution       Figure 10: Complementary service time
                                                        distribution for different values of the contending
for different values of D21 , (W = 32 )                 stations
     All the curves drop gradually to 0 as the delay        Analytical and simulation results show that
increases. Category C 1 stations curves drop to 0       complementary service time curves drop faster
faster than category C 2 curves. Indeed, when           when the number of contending stations is small for
                                                        both category C 1 and category C 2 stations. This


                                  Ubiquitous Computing- and Communication Journal                      - 13 -
means that all stations service time increases as the       CW max < 1024 and K =1.
number of contending nodes increases.                          Analytical and simulation results show that
7 EXTENTIONS OF THE ANAYTICAL                              throughput values increase with stations priority.
   RESULTS BY SIMULATION                                   Indeed, the station with the lowest delay bound has
                                                           the maximum throughput.
     The mathematical analysis undertaken above
show that DM performs service differentiation over               Moreover, figure 12 shows that stations
802.11 protocol and offers better QoS guarantees           belonging to the same traffic category have the
for highest priority stations                              same throughput. For instance, when n is set to 15
     Nevertheless, the analysis was restricted to two      (i.e. m = 3 ), the three stations of the same traffic
traffic categories. In this section, we first generalize   category have almost the same throughput.
the results by simulation for different traffic
categories. Therefore, we consider a simple multi-
hop and evaluate the performance of the DM policy
when the stations belong to different broadcast
regions.

7.1 Extension of the analytical results
     In this section, we consider n stations
contending for the channel in the same broadcast
region. The n stations belong to 5 traffic categories
where n = 5 m and m is the number of stations of
the same traffic category. A traffic category C i is
characterized by a delay bound Di , and
 Dij = Di − D j is the difference between the
deadline values of category C i and category C j           Figure 12: Normalized throughput: different
stations. We have:                                         stations belonging to the same traffic category
                        Dij = ( i − j ) K    (53)
                                                                 In figure 13, we depict the average service
     Where K is the deadline multiplicity factor           time of the different traffic category stations as a
and is given by:                                           function of K , the deadline multiplicity factor. We
                 Di + 1,i = Di + 1 − Di = K  (53)          notice that the highest priority station average
                                                           service time decreases as the deadline multiplicity
      Indeed, when K varies, the deadline values of        factor increases. Whereas, the lowest priority
all other stations also vary. Stations belonging to        station average service time increases with K .
the traffic category C i are numbered from S i1 to
 S im .




                                                           Figure 13: Average service time as a function of
Figure 11: Normalized throughput for different             the deadline multiplicity factor K
traffic category stations
                                                                In the same way, the probabilistic service time
    In figure 11, we depict the throughput achieved        bounds offered to S 11 (the highest priority station)
by different traffic categories stations as a function     are better than those offered to station S 51 (the
of the minimum contention window size CW min               lowest priority station). Indeed, the probability that
such as CW min is always smaller than CW max ,              S 11 service time exceeds 0.01s=0.3%. But, station



                                Ubiquitous Computing- and Communication Journal                           - 14 -
S 51 service time exceeds 0.01s with the probability   and D21 = D 2 − D1 =5 slots. Flows F3 and F4 are
of 36%.                                                transmitted respectively by S 12 and S 4 and have
                                                       the same delay bound. Finally, F5 and F6 are
                                                       transmitted respectively by S 5 and S 6 with delay
                                                       bounds D1 and D2 and D 2 ,1 = D2 − D1 = 5 slots.

                                                          Figure 16 shows that the throughput achieved
                                                       by F1 is smaller than the one achieved by F2 .




Figure 14: Complementary             service   time
distribution (W=32, n=8)

    The above results generalize the analytical
model results and show once again that DM
performs service differentiation over 802.11 and
offer better guarantees in terms of throughput,
average service time and probabilistic service time
bounds for flows with short deadlines.
                                                       Figure 16: Normalized throughput using DM
                                                       policy
7.2 Simple Multi hop scenario
    In the above study, we considered that                 Indeed, both flows cross nodes 6 and 7, where
contending stations belong to the same broadcast        F1 got a higher priority to access the medium than
region. In reality, stations may not be within one      F2 when the DM policy is used. We obtain the
hop from each other. Thus a packet can go through
                                                       same results for flows F5 and F6 . Flows F3 and
several hops before reaching its destination. Hence,
factors like routing protocols or interferences may     F4 have almost the same throughput since they
preclude the DM policy from working correctly.         have equal deadlines.
                                                           Figure 17 show that the complementary service
    In the following paragraph, we evaluate the        time distribution curves drop to 0 faster for flow F1
performance of the DM policy in a multi-hop            than for flow F2 .
environment. Hence, we consider a 13 node simple
mtlti-hop scenario described in figure 15.




                                                       Figure 17: End to end complementary service time
                                                       distribution
Figure 15: Simple multi hop scenario
Six flows are transmitted over the network. Flows      The same behavior is obtained for flow F5 and F6,
packets are routed using the AODV protocol.            where F5 has the shortest delay bound.
Flows F1 and F2 are respectively transmitted by
stations S 1 and S 2 with delay bounds D1 and D2           Hence, we conclude that even in a multi-hop



                              Ubiquitous Computing- and Communication Journal                        - 15 -
environment, the DM policy performs service            [6] IEEE 802.11 WG, ”Draft Supplement to Part
differentiation over 802.11 and provides better QoS        11: Wireless Medium Access Control (MAC)
guarantees for flows with short deadlines.                 and physical layer (PHY) specifications:
                                                           Medium Access Control (MAC) Enhancements
8    CONCLUSION                                            for Quality of Service (QoS)”, IEEE
     In this paper we first proposed to support the        802.11e/D13.0, (January 2005).
DM policy over 802.11 protocol. Therefore, we          [7] J. Deng, R. S. Chang: A priority Scheme for
used a distributed backoff scheduling algorithm and        IEEE 802.11 DCF Access Method, IEICE
introduced a new medium access backoff policy.             Transactions in Communications, vol. 82-B,
Then we proposed a mathematical model to                   no. 1, (January 1999).
evaluate the performance of the DM policy. Indeed,     [8] J.L. Sobrinho, A.S. Krishnakumar: Real-time
we considered n contending stations belonging to           traffic over the IEEE 802.11 medium access
two traffic categories characterized by different          control layer, Bell Labs Technical Journal, pp.
delay bounds. Analytical and simulation results            172-187, (1996).
show that DM performs service differentiation over     [9] J. Y. T. Leung, J. Whitehead: On the
802.11 and offers better guarantees in terms of            Complexity of Fixed-Priority Scheduling of
throughput, average service time and probabilistic         Periodic, Real-Time Tasks, Performance
service time bounds for the flows having small             Evaluation (Netherlands), pp. 237-250, (1982).
deadlines. Moreover, DM achieves fairness              [10]K. Duffy, D. Malone, D. J. Leith: Modeling
between stations belonging to the same traffic             the 802.11 Distributed Coordination Function
category.                                                  in Non-saturated Conditions, IEEE/ACM
                                                           Transactions     on     Networking      (TON),
     Then, we extended by simulation the analytical        Vol. 15 , pp. 159-172 (February 2007)
results obtained for two traffic categories to         [11]L. Kleinrock: Queuing Systems,Vol. 1: Theory,
different traffic categories. Simulation results           Wiley Interscience, 1976.
showed that even if contending stations belong to      [12]P. Chatzimisios, V. Vitsas, A. C. Boucouvalas:
 K traffic categories, K > 2 , the DM policy offers        Throughput and delay analysis of IEEE 802.11
better QoS guarantees for highest priority stations.       protocol, in Proceedings of 2002 IEEE 5th
Finally, we considered a simple multi-hop scenario         International Workshop on            Networked
and concluded that factors like routing messages or        Appliances, (2002).
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policy and DM still provides better QoS guarantees         Throughput Analysis of IEEE 802.11e EDCA
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                                                           Networks , LCN’05 (2005).
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                              Ubiquitous Computing- and Communication Journal                       - 16 -

				
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