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					     End-to-End Delay Measurement for Instant Messaging Relay Nodes

               Muhammad T. Alam, Student Member IEEE, Zheng Da Wu, Member IEEE
                                          School of IT
                                    Bond University, Australia
                                    muhamta123@gmail.com


                                                  ABSTRACT
               In this paper, we provide complete end-to-end delay analyses including the relay
               nodes for instant messages. Message Session Relay Protocol (MSRP) is used to
               provide congestion control for large messages in the Instant Messaging (IM)
               service. Large messages are broken into several chunks. These chunks may
               traverse through a maximum number of two relay nodes before reaching
               destination according to the IETF specification of the MSRP relay extensions. We
               discuss the current solutions of sending large instant messages and introduce a
               proposal to reduce message flows in the IM service. The analysis presented in this
               paper is divided into two parts. At the former part, we consider virtual traffic
               parameter i.e., the relay nodes are stateless non-blocking for scalability purpose.
               This type of relay node is also assumed to have input rate at constant bit rate. The
               later part of the analysis considers relay nodes to be blocking and the input
               parameter to be exponential. The performance analysis with the models introduced
               in this paper is simple and straight forward, which lead to reduced message flows
               in the IM service. Also, using our model analysis a delay based optimization
               problem can be easily deduced.

               Keywords: Instant messaging, MSRP, Stateless, Chunking.


1   INTRODUCTION                                            named MESSAGE. The SIP MESSAGE method
                                                            (RFC 3428 [1]), is able to transport any kind of
     In this paper, we provide complete end-to-end          payload in the body of the message, formatted with
delay analyses including the relay nodes for instant        an appropriate MIME (Multipurpose Internet Mail
messages. Instant messaging (IM) is one of today’s          Extensions) type. 3GPP TS 23.228 [25] already
most popular services. Thus, it is not a surprise that      contains requirements for Application Servers (ASs)
3G IP Multimedia Subsystem (IMS) already has this           to be able to send textual information to an IMS
service well supported in its architecture. IM is the       terminal. 3GPP TS 24.229 [2] introduces support for
service that allows an IMS user to send some content        the MESSAGE method extension. The specification
to another user in near-real time. The content in an        mandates IMS terminals to implement the
instant message is typically a text message, but can        MESSAGE method [1] and to allow implementation
be an HTML page, a picture, a file containing a song,       to be an optional feature in ASs.
a video clip, or any generic file.                               The work over instant messaging [4, 5, 6]
     There are two modes of operation of the instant        observed so far lacks a thorough analysis of the
message service, depending on whether they are              scalable behavior of the nodes involved in providing
stand-alone instant message, not having any relation        the IM service. The messages of IM may be very
with previous or future instant message. This mode          large. Large instant messages have disadvantages
of IM is referred to as “pager mode”. The model is          like service behavior is too slow on low bandwidth
also similar to the SMS (Short Message Service) in          links and more importantly, messages get fragmented
cellular networks. The other model is referred to as        over some transport protocol and then look at SIP
session based instant message that is sent as part of       extension that resolve this issue. Even if messages
an existing session, typically established with a SIP       are compressed, sometimes SIP messages can be too
(Session Initiation Protocol) INVITE request. Both          large. Another problem with SIP is that the fact that
modes have different requirements and constraints,          any proxy can change the transport protocol from
hence the implementation of both models.                    TCP (Transmission Control Protocol) to UDP (User
     The IETF (Internet Engineering Task Force) has         Datagram Protocol) or other transport protocols and
created an extension to SIP that allows a SIP UA            vice versa. The protocols other than TCP and SCTP
(User Agent) to send an instant message to another          (Stream Control Transmission Protocol) are not
UA. The extension consists of a new SIP method              famous for congestion control. If an IMS terminal is


                    Ubiquitous Computing and Communication Journal
sending a large instant message over a transport          the message into chunks and deliver each chunk in a
protocol that does not offer congestion control, the      separate SEND request. The message ID corresponds
network proxies can become congested and stop             to the whole message, so the receiver can also use it
processing other SIP requests like INVITE,                to reassemble the message and tell which chunks
SUBSCRIBE, etc. Even if a terminal sends large SIP        belong with which message.
MESSAGE over a transport protocol that                         Long chunks may be interrupted in mid-
implements end-to-end congestion control e.g., TCP,       transmission to ensure fairness across shared
SCTP, the next proxy can switch to UDP and                transport connections.     This chunking mechanism
congestion may occur.                                     allows a sender to interrupt a chunk part of the way
    To solve the issue of large message passing and       through sending it. The ability to interrupt messages
congestion control in IM, a limit has been placed on      allows multiple sessions to share a TCP connection,
the SIP MESSAGE method such that MESSAGE                  and for large messages to be sent efficiently while
requests cannot exceed the MTU (Maximum                   not blocking other messages that share the same
Transmit Unit) minus 200 bytes. If the MTU is not         connection, or even the same MSRP session. Any
known, this limit is 1300 bytes. Another solution to      chunk that is larger than 2048 octets MUST be
sending SIP MESSAGE requests with large bodies to         interruptible [24].
use the content indirection mechanism [3]. Content             Another characteristic of MSRP is that, MSRP
indirection allows replacing a MIME body part with        messages no not traverse SIP proxies. This is an
an external reference, which is typically an HTTP         advantage, since SIP proxies are not bothered with
(Hyper Text Transfer Protocol) URI (Universal             proxying large instant messages. Also, MSRP does
Resource Identifier).                                     not run over UDP or any other transport protocol that
    Another solution to getting around the size limit     does not offer end-to-end congestion control. It
problem with MESSAGE is to use session-based IM           supports instant messages to traverse zero, one or
mode rather than pager mode. Session-based instant        two MSRP relays (see Figure 1). The relay extension
message mode uses the SIP INVITE method to                of MSRP is defined in [7].
establish a session. An IMS terminal establishes a
session to send and receive instant messages via
Message Session Relay Protocol (MSRP) [24].                                  Relay       Relay
MSRP is a simple text-based protocol whose main            Source            node 1      node 2    Destination
characteristic is that it runs over transport protocols
that offer congestion control. In the IMS, MSRP is                  Flow 1
implemented in the IMS terminals. Analysis is
required to determine the service order of such
                                                                    Flow 2
servers. Our work in this paper is to analyze the
delay bound of the relay nodes that implements the
MSRP to provide instant messaging service. The
benefit of the work lies in the simplicity of the model
derivation.                                                         Flow v
    The rest of the paper is organized as follows.
Section 2 provides a review of MSRP. The SEND                                (β , α )   (β , α )
chunking system, our proposal of scalability over
MSRP relay nodes, delay analysis for both work                Figure 1: IM with maximum 2 relay nodes
conserving non-blocking and blocking situation are
described in Section 3. Finally we conclude the paper
in Section 4.                                                  The default is that SEND messages are
                                                          acknowledged hop-by-hop. Each relay node that
                                                          receives a SEND request acknowledges receipt of the
2   BACKGROUND                                            request before forwarding the content to the next
                                                          relay or the final target. When sending large content,
    There are currently three methods defined in          the client may split up a message into smaller pieces;
MSRP after the INVITE message is sent for an IM           each SEND request might contain only a portion of
session set up: i) SEND: sends an instant message of      the complete message. For example, when Alice
any arbitrary length from one endpoint to another, ii)    sends Bob a 4GB file called "file.mpeg", she sends
VISIT: and endpoint connects to another end point,        several SEND requests each with a portion of the
and iii) REPORT: endpoint or a relay provides             complete message. Relays can repack message
message delivery notifications.                           fragments en-route. As individual parts of the
    MSRP does not impose any restriction on the           complete message arrive at the final destination
size of an instant message. If an IMS user, Alice         client, the receiving client can optionally send
wants to deliver a very large message, she can split      REPORT requests indicating delivery status. MSRP




                     Ubiquitous Computing and Communication Journal
nodes can send individual portions of a complete         SEND chunks at the relay nodes if they do not keep
message in multiple SEND requests. As relays             transaction states of a chunk flow for scalability
receive chunks they can reassemble or re-fragment        purpose.
them as long as they resend the resulting chunks in           Long chunks are interrupted in mid-transmission
order.                                                   to ensure fairness across shared transport
     A series of papers [8-10] have studied the          connections. To support this, MSRP uses a
capacity scaling in relay networks. These works          boundary-based framing mechanism. The start line
quantify the impact of large wireless relay networks     of an MSRP request contains a unique identifier that
in terms of signal-to-noise ratio. Most of the work      is also used to indicate the end of the request.
focuses on characterizing one relay node only. The       Included at the end of the end-line, there is a flag that
work of H. Bolcskei et all in [10] demonstrated that     indicates whether this is the last chunk of data for
significance performance gains can be obtained in        this message or whether the message will be
wireless relay networks employing terminals with         continued in a subsequent chunk. There is also a
multiple-input multiple-output (MIMO) capability.        Byte-Range header field in the request that indicates
However, these works do not address the issue of         that the overall position of this chunk inside the
characterizing traffic parameter in relay nodes where    complete message.
the relay nodes do not keep the transaction states. A         This chunking mechanism allows a sender to
signification challenge is to schedule the large         interrupt a chunk part of the way through sending it.
chunks and characterize the traffic parameters under     The ability to interrupt messages allows multiple
delay bounds.                                            sessions to share a TCP connection, and for large
     In    any    IMS      network     the    capacity   messages to be sent efficiently while not blocking
(memory/storage) is large for IM communication.          other messages that share the same connection, or
Large messages have to be broken down into chunks        even the same MSRP session. As mentioned before
to overcome the fixed size limit fact. Real time         that any chunk that is larger than 2048 octets MUST
service of IM is always desirable. However, issues       be interruptible. While MSRP would be simpler to
arise if a) the relay nodes in between source and        implement if each MSRP session used its own TCP
destination IMS terminals possess slow links b)          connection, there are compelling reasons to conserve
traffic order gets distorted before reaching relay       connection. For example, the TCP peer may be a
nodes and c) relay nodes maintains transaction states.   relay device that connects to many other peers. Such
Therefore efficient service discipline of the chunks     a device will scale better if each peer does not create
of IM is necessary. In an IM system, transmission        a large number of connections. The chunking
time typically depends on the number of chunks in        mechanism only applies to the SEND method, as it is
messages. Moreover, service time of the chunks           the only method used to transfer message content
depends on batch size arrivals. The number of            [24]. We call the chunking mechanism i.e., breaking
broken chunks in a large message is not fixed. Thus      one large SEND message into several SEND
analysis of such system is not trivial. In this work,    messages a SEND system.
we explore delay characteristics of instant messages          Proposal: Traditional MSRP [24] may be used
when the messages traverse via relay nodes.              without traditional session set up in IMS to provide
     The end to end delay bound of IMS instant           the congestion control. Also, MSRP relay nodes
messages indeed requires much attention. Although,       should not keep transaction states for the SEND
the study of the fundamental frameworks, namely          chunks.
Integrated and Differentiated services have a long            The benefit of this proposed technique contains
history, defining the flow characteristic of an IMS      reduced message flows in the network as well as
instant message traversing the relay nodes               gaining scalability at the relay nodes. In order to
(maximum number of relay nodes is two for an IMS         comply with this, we propose the following
terminal [7]) under MSRP is not trivial due to the       scheduling. The detail analysis is provided below
arbitrary number of chunks in IM messages. We            that captures the delay bound of the relay nodes.
analyze the end-to-end delay for an IM under work             We provide analysis of one relay node first in
conserving situation. The delay bound is useful to       terms of aggregate flows of two SEND message
formulate the optimization of transmission IM end-       chunk flows and delay bound, which will later be
to-end transmission time.                                used to compute delay bound for SEND systems
                                                         with two relay nodes, source and destination. We
                                                         assume the following for the analysis. One IMS
3   MODELLING                                            source terminal sends multiple large instant
                                                         messages (SEND) to the same destination via two
    The large sized SEND messages in IM, MSRP            relay nodes. Each of the SEND messages is broken
delivers in several SEND messages, where each            into small SEND chunks. It is to be noted that we are
SEND contains one chunk of the overall message.          assuming relay nodes do not keep transaction states
The crucial aspect in this paper is the ordering of      of the chunks. Though, the IETF draft [7] specifies




                    Ubiquitous Computing and Communication Journal
that the relays may keep transaction states for a very
                                                                                                ⎛ Lmax ⎞ ⎛ Li ⎞
short time, it will be expensive to keep such states      any relay node is no longer than ⎜           ⎟+⎜    ⎟
for the relay nodes if there is huge number of clients                                          ⎝  c ⎠ ⎜αi ⎟
                                                                                                         ⎝    ⎠
being served and traffic flows are massive. We are        [11].
only interested in busy traffic situation for the              We adopt the characteristic of traffic model in
derivation of properties in this research and thus        [12, 13] which has been widely adopted for
server analysis with stateless assumption is more         characterizing network traffic. If the total traffic of a
                                                          flow F (t1 ,t 2 ) arriving in the time interval (t1 , t 2 ]
practical. We assume message chunks are served
according to the order of delivery time tag of the
previous node and all chunks are treated as if they       is bounded by:
belong to a single flow due to the elastic and massive     F (t1 , t 2 ) ≤ σ + ρ (t 2 − t1 )              (1)
flows of SEND chunk messages in a flow. Thus              Then the flow is referred to as conforming to the
performance analysis of an individual flow at a relay
node can be achieved by analyzing the aggregated          traffic parameter   (σ       )
                                                                                   , ρ . Here the assumptions are
flows at this node.                                       under non-overflow condition with a flow injection
                                                          to a leaky bucket with parameters of buffer size, σ
3.1    For Non-blocking Relay Nodes                       and output rate ρ . In other words, ρ is the average
                                                          traffic rate in the long run and σ is the burst bound
     Here we adopt a scheduler which services a job
according to delivery time stamps of the pervious         of the flow   (σ         )
                                                                             , ρ . It is practical to assume that
node. We aim to provide work conserving but               the links of relay nodes will be subject to delay
stateless scheduling of the chunks. Every message         bound in terms of propagation delay. We consider a
chunks has message id that identifies which SEND          chunk to be arrived only after its last bit has arrived
message it belongs. The source node generates the         to a relay node and the delivery time of a chunk at a
chunks and delivers them to a relay node. The             node is the time when the last bit of the chunk leaves
ordering is considered to be the order as the source      the relay node. Note that we are considering the
node generates chunks for the first relay node and        input traffic as the constant bit rate for the relay
then first relay node for the second relay node and so    nodes in this section.
on. During the delivery time these chunks may                  If we consider steady state of the network i.e.,
receive time stamp tags. These chunks may reach /         traffic load less than one then a chunk will only be
propagate to the relay node out of order, and hence       delayed at a node if there is a chunk being served or
the arrival times of the chunks to the relay node may     there are chunks waiting in the buffer with earlier
not always be in order of the order id of the message     delivery time stamps, we assume that the start time
chunks.                                                   of each busy period is initialized at 0. Here, a busy
     Let the propagation delay and link capacity of       period is an interval of time during which the
any link are 0 and c, respectively. The sequence of       transmission queue of the output link is continuously
chunks transmitted by a source to a destination is        backlogged which is consistent with [12].
referred to as a flow (Figure 1). The paths via relay          The previous node’s delivery time stamp tag of
nodes are predetermined as defined in the MSRP            each chunk lags behind its arrival time at any relay
relay extensions [7] (The relay nodes are authorized      node. Note that chunks are served by the order of
by explicitly by the end terminals). Let, at a relay      their previous node’s delivery time stamps which is
node chunk k of a flow i is attached with a time          our assumptions. Thus, the delay for each chunk to
stamp tag according to the delivery time from the         traverse the network remains the same only if the
                        ⎛L ⎞                              time stamps of all chunks are increased by a constant
previous node of Ai + ⎜ i ⎟ where        α j , Li , and   D at the previous node. We can assume that, if the
                   k
                      ⎜α ⎟
                        ⎝    i   ⎠                        burst of each flow is bounded and the capacity of any
                                                          link is no less than the average rate of the flows
 Aik are the input rate, chunk size and the arrival       traversing the link, there exists a worst case delay
time of chunk k of flow i respectively; the delivery      bound in the network, i.e., the worst case delay of a
order time stamp of chunk k of flow i is updated at       flow to traverse any pair of relay nodes is bounded.
the    next   relay   node       with   an   increment         Since we consider that the relay node need not
                                                          keep transaction states of the SEND flows, chunks in
  ⎛ Lmax ⎞ ⎛ Li ⎞
of ⎜     ⎟+⎜
           ⎜
                ⎟ , and chunks are served at the
                ⎟
                                                          the buffer are served by the order of their delivery
  ⎝ c ⎠ ⎝αi ⎠                                             time stamp tags, not their arrival times. There is also
                                                          no distinct relation between the delivery time stamp
increment order of their previous node’s delivery
                                                          of a chunk and its arrival time. Thus, a chunk with an
order time tag, where, Lmax is the maximum size of        earlier delivery time stamp than another chunk,
chunk in all flows. Under these conditions, it is easy    though it arrives later, may be served first. This may
to perceive that the worst case delay of a flow i at      happen due to the well-known traffic distortion




                      Ubiquitous Computing and Communication Journal
problem [12]. Therefore in this regard, it is more                                                                  }
                                                                 β1 + α1 (Ri − max{Ri −1 , min {Ai , Ai ,...., Ai ,} )
reasonable to evaluate a chunk’s delay with reference                                        n                       1               1                2           n

to its previous delivery node’s time stamp, rather                      in

than its arrival time at the current node. We need to            ≥ ∑ Ls
characterize traffic of this kind of scheduling for the                s =ii

relay nodes. We define a parameter (β , α ) such that                                                                                                     (2)
the total traffic of the flow of chunks, whose time                                  (   p
                                                                                                             {
                                                                 β2 +α2 Rj − max Rj −1, min Aj , Aj ,....,Aj ,   1
                                                                                                                             {   1           2             p
                                                                                                                                                                }})
stamps are in the range of (t1 , t 2 ] , is no larger                  jp

than β + α (t 2 − t1 ) , which is similar to the (σ , ρ )        ≥ ∑Ls
                                                                   s= j1
traffic model. We assume that chunks are ordered by
their previous node’s delivery time stamps as                                                                                                             (3)
P1 , P2 ,..., Pk ,....( Ri ≥ R j , if i > j , where Ri is        Since,
the previous node’s delivery time tag of chunk Pi).
For     any    two     chunks     Pm     and   Pk,
                                                                                 {
                                                                 max Rin , R j p = Rk ,                  }
                                          k
(k ≥ m ), β + α (Rk − Rm ) ≥ ∑ Li , where Li is                        ⎪                             {
                                                                       ⎧min Ai1 , Ai2 ,...., Ain , , ⎫ ⎪                         }
                                         i =m
the size of Pi. A chunk may receive service as long
                                                                   min ⎨
                                                                       ⎩
                                                                                                       ⎬
                                                                       ⎪min A j1 , A j2 ,...., A j p , ⎪
                                                                                                       ⎭ {                           }
as there is no chunk in the buffer when it arrives.
                                                                 = min {Am , Am +1 ,...., Ak } and
Thus, it is necessary to take into account the arrival
time of a chunk to characterize traffic in a relay node.
                                                                                 min {Ri1 −1 , R j1 −1 } = Rm −1 ,
Therefore, we define the traffic parameter for any
two chunks of a flow as follows: for any two chunks              We have,
Pk        and        Pm        of         a        flow
                                                                 (β1 + β2 ) + (α1 +α2 )
(k ≥ m ≥ 1),
                                                                   (                             {
                                                                 × Rk − max Rm−1, min{Am, Am+1,...,Ak }                                          })
β + α(Rk − max {Rm−1, min {Am , Am+1,...,Ak }}) ≥ ∑Li
                                                           k


                                                          i =m
                                                                             [                       (                   {
                                                                       ≥ β1 +α1 × Rin − max Ri1−1, min Ai1 , Ai2 ,....,Ain               {                            }})]
 where Ai is the arrival time of chunk Pi, i=1,2,…; we
refer to F (t1 , t 2 ) = β + α (t 2 − t1 ) in the time                       [                       (                   {                   {                             }})]≥ ∑L
                                                                                                                                                                               k
                                                                       + β2 +α2 × Rjp − max Rj1−1, min Aj1 , Aj2 ,....,Ajp                                                          s
                                                                                                                                                                              s=m
interval (t1 , t 2 ] as the traffic function of this flow
                                                                                                                (4)
with the traffic parameter         (β , α ) . We apply the            Application of Theorem 1: If the function of all
additive property of        (σ , ρ ) traffic model [12] to       traffic flows are known, the virtual traffic aggregated
                                                                 function can be derived by Theorem 1.
obtain the following:
                                                                 However, the chunk pattern may be distorted at a
    Proposition 1: Given two flows with traffic
                                                                 relay node. In such case, we can provide the
parameters (β 1 , α 1 ) and (β 2 , α 2 ) the traffic             following relation for a flow in terms of worst case
parameter of the aggregated traffic of the two flows             delay of the outgoing traffic.
is   (β   1                   )
              + β 2 , α1 + α 2 .                                      Proposition 2: Assume that the traffic parameter
                                                                 of the input traffic of a SEND chunk flow at a relay
    Proof: Assume that chunks are ordered by their
delivery order. Given any two chunks Pk and
                                                                                     (                   )
                                                                 node is β , α and the worst case delay to traverse
                                                                 a relay node is D (let the mean service time of a
Pm (k ≥ m ) of the aggregated flow, assuming                     chunk at this current node is d). We can characterize
chunks                     Pi1 , Pi2 ,......          and        the output traffic of this flow as β ′,                         (               α ) where the
              (
Pin , i1 < i2 < .... < in and n ≤ (k − m + 1) bel    )           buffer                   requirement
                                                                                                 {
                                                                 β ′ = max 0, α (D − d ) + Lmax + β .                                }
                                                                                                                                                                      is

ong to flow 1, and the rest of the chunks
                                                                     Proof: Assume that chunks are ordered by their
Pj1 , Pj2 ,......                                     and        delivery times at this current node, i.e., for chunks
              (
Pj p , j1 < j2 < .... < j p and p ≤ (k − m + 1)       )                                          (
                                                                 Pk and Pm k ≥ m, Tk ≥ Tm where the delivery                 )
belong to flow 2. Thus for the virtual traffic                   order time tag of chunk Pi , i = 1,2,..., is Ti and is
parameter, we have
                                                                 also the arrival time of Pi of the output traffic. As




                          Ubiquitous Computing and Communication Journal
the worst case delay of a chunk is D, we have the                                                           v
following relation:                                                         this node is    c, c ≥ ∑ α i . Under these assumptions,
Ti ≤ Ri + D                              (5)                                                               i =1
                                                                            the wost case delay bound at a current relay node is:
Again, since the delivery order of each chunk is
                                                                             1⎡ v                        ⎤
                                                                               ⎢∑ (β i − α iθ i ) + Lmax ⎥
delayed           by             d           and                                                                         (9)
β ′ = max{0, α (D − d ) + Lmax }+ β . ,                for any               c ⎣ i =1                    ⎦
two chunks k and m (k ≥ m ≥ 1) , we get                                          Proof: For any chunk Pk if we assume m to be
β ′ + α [Rk + d − max{Rm−1 + d , Tm }]                                      the biggest integer k > m > 0 such that Rk < Rm

             [                    {
   ≥ β ′ + α Rk + d − max Rm−1 + d , Rm + D                  }]             and Tk > Tm where Ri and Ti are the previous
                                                                            node’s delivery time tag and the delivery time of Pi
        {
≥ min β + α (Rk − Rm−1 ), β + Lmax + α (Rk − Rm )                       }   at current node. Thus
                                           (6)                              Rm > Rk ≥ Ri ,                        for    all     m<i<k
And
                                                                                                                          (10)
β +α[Rk − max {Rm−1, min [Am , Am+1,....,Ak ]}]
                                                                  k
                                                             ≥ ∑Li          Tk > Ti ≥ Tm ,                        for    all   m<i<k
                                                                  i=m                                                     (11)
                       k
i..e., α(Rk −Rm−1 ) ≥ ∑Li
                                                                            In other words,            Pm is transmitted before chunks
                       i=m                                                   Pm +1 ,..., Pk ; however, its previous node’s delivery
                                        (7)
                                                                            time tag is greater than that of chunks Pm +1 ,...Pk .
Now let the previous node’s delivery order of a
                                                                            Thus
chunk   Pi , i = 1,2,..., at the outgoing link of the
                                                                                                                    Lm
                              ⎛L ⎞                                          min{Am +1 ,..., Ak } > Tm −                                  (12)
relay node is: Ri′ = Ri + D + ⎜ max ⎟ . Thus                                                                         c
                              ⎝ α ⎠                                                                                  ⎛L ⎞
from Eq. (6) and (7) we have:                                               Since,  Pm +1 ,..., Pk arrive after Tm − ⎜ m ⎟ and
β + α[Rk − max{Rm−1 , min[Tm ,Tm+1 ,...Tk ]}]
       ′        ′                                                                                                    ⎝ c ⎠
                                                                            depart before Pk at the current relay node, we have
      ≥ min{β + α(Rk − Rm−1 ), β + Lmax + α(Rk − Rm )}                                          k


              ⎧k     k
                                ⎫ k
                                                                                             ∑L        i

         ≥ min⎨∑Li , ∑Li + Lmax ⎬ ≥ ∑Li                                     Tk = Tm +       i = m +1
                                                                                                                                         (13)
              ⎩i=m i=m+1        ⎭ i=m                                                    c
                 (8)                                                        Note that Ri ≥ Ai for all i = 1,2,..., and thus
Thus the characteristic of traffic parameter for worst
case Delay D           (           )
               is β ′, α and proposition 2 is                               Rk ≥ Ri ≥ Ai ≥ Tm − ⎜ m ⎟
                                                                                                      ⎛L ⎞
                                                                                                                               for
proved.                                                                                               ⎝ c ⎠
Next we analyse the worst case delay bound of a                             i = m + 1,..., k − 1 . Furthermore we have the traffic
SEND chunk flow to traverse a relay node.                                   function,
    Proposition 3: Let,      Pki be the kth chunk of flow i
and assume that the chunks are ordered according to
                                                                                      {i{       {i i
                                                                            θi = maxminRm−1 − minAm, Am+1,...,Aki }},0
                                                                                        k≥m>1
                                                                                                                                 }
their current node’s delivery order time tag. Define                                    {i i
                                                                                    = minAm , Am+1,...,Ak }+θ
                                                                                                        i

θi = max{mink ≥m>1{R       i
                           m−1         {
                                 − min A , A ,..., A ,0
                                           i
                                           m
                                                 i
                                                 m+1
                                                         i
                                                         k   }} }
                                                                                        {i {i i
                                                                                    ≤ maxRm−1, minAm , Am+1,...,Ak }}
         i         i
where   R and A are the delivery time tag from
         m         m
                                                                                                                 i

previous node and the arrival time at current node of
 i                                                                          i.e.,
Pm ; Lmax be the maximum size of a chunk. Assume
                                                                               βi +αi [Rki − (minAm, Am+1,...,Aki }+θi )] ≥ ∑Lij
                                                                                                {i i
                                                                                                                                     k
that the input traffic of a relay node consists of flows
1,2,…, v , whose traffic parameters are (β i , α i )                                                                             j=1
respectively and the capacity of the output link of                                                             (14)
                                                                            Since, chunks Pm +1 ,..., Pk comprise the chunks of
                                                                            flows 1,2,… v , we have



                       Ubiquitous Computing and Communication Journal
                                                                                    are updated / serviced by an increment d at the relay
      ⎧
                       [                             ⎫
∑⎨βi + αi Rk − (min{Am , Am+1,...,Ak }+θi ) ≥ i=∑1Li ⎬                ]
 v                                              k
           i         i    i        i
                                                                                    node, then input traffic parameter for the next relay
 i =1 ⎩                                         m+   ⎭                              node           is (β ′, α ) where buffer requirement
i.e.,                                                                               is β ′ =        max{0, α (D + δ − d ) + L }+ β .              max
  k                v
                        ⎛     ⎞⎡     ⎛    L ⎞⎤       v
                                                                                    The delay bound of proposition 3 can further be
  ∑1Li ≤ ∑(βi −αiθi ) + ⎜ ∑αi ⎟⎢Rk − ⎜Tm − cm ⎟⎥
                        ⎝ i=1 ⎠⎣     ⎝        ⎠⎦                                                                                  ⎛    v
                                                                                                                                             αi ⎞
i =m+    i =1
                                                                                    tightened. For instance, if ⎜
                                                                                                                                  ⎝
                                                                                                                                    ∑           ⎟ → 0, then the
                                                                                                                                              c ⎠
                                                                      (15)                                                            i =1

From Eq. (13) and Eq. (15) we have                                                  worst               case       delay              bound             would      be
                       k
                                                                                    ⎛     v
                                                                                                   ⎞
                   ∑L           m
                                                                                    ⎜ ∑ β i + Lmax ⎟
Tk = Tm + i=m+1                                                                     ⎝ i =1         ⎠                          . On the other hand, if
              c                                                                                                    c
                                                                                    θ = min i {θ i } ,             and the delivery time tag at the
            ⎛ v ⎞⎡     ⎛    L ⎞⎤ v
            ⎜∑αi ⎟⎢Rk −⎜Tm − m ⎟⎥ + ∑(βi −αiθi )                                    pervious node of all chunks are decreased by θ ,
            ⎝ i=1 ⎠⎣   ⎝     c ⎠⎦ i=1                                               then the traffic functions of all flows remain the
    ≤ Tm +
                            c                                                       same and the actual worst case delay bound from
                                                                                                     ⎛ v            ⎞
                                                                                                     ⎜ ∑ β i + Lmax ⎟
                                        v
                         (β −α θ )
               Lmax ∑ i i i                                                                          ⎝ i =1         ⎠ − θ . Therefore, it
        ≤ Rk +     + i=1                                                            proposition 3 is
                c           c                                                                                c
                                                                      (16)          is possible to tighten the worst case delay as well in
                                                                                    this instance. If all chunks’ delivery time stamps at
If there does not exist such m, then                           P1 ,..., Pk −1 all
                                                                                    the previous node are increased or decreased by a
leave the node before                           Pk and thus have                    constant at the entrance to a relay node, their
                                                                                    delivery time remains unchanged. If all chunks’
            ⎛ k
                     ⎞       v          v
                 α i ⎟ Rk + ∑ (β i − α iθ i )
      ∑ Li ⎜ ∑ ⎠ i=1                                                                previous node’s delivery time tag decreased by θ ,
Tk = i =1 ≤ ⎝ =1
              i                                                                     applying proposition 3, for any chunk Pk we have the
                                                                                    following:
         c                 c                                                                                           v
i.e.,                                                                                                              ∑ [β       i   − α i (θ i − θ )] + Lmax
                            v                                                       Tk − (Rk − θ ) ≤                i −1

                           ∑ (β         i   − α iθ i )                                                                                        c
Tk − Rk ≤                  i =1
                                                                                    i.e.,
                                            c                                                              v

Thus               the                  delay            is   bounded
                                                                     (17)
                                                                          by                              ∑ [β     i   − α i (θ i − θ )] + Lmax
                                                                                    Tk − Rk ≤             i −1
                                                                                                                                                            −θ
∑ (β                                )
  v

          i       − α iθ i                                                                                                        c
                                            Lmax                                                                                                   (18)
 i =1
                                        +            and proposition 3 is           i.e.,      the       worst         case       delay           is bounded      by
              c                              c                                        v
proved.
    Application of proposition 2 and proposition 3:                                 ∑ [β       i   − α i (θ i − θ )] + Lmax
The proposed propositions are straight forward for
                                                                                     i −1
                                                                                                                                       −θ
performance analysis. From the above relation, we                                                              c
can also characterize the outgoing traffic parameter                                Now if we take the propagation delay into account,
of a relay node for a given propagation delay, δ .                                  the increment for flow n, 1 ≤ n ≤ v , should be
Let δ be the propagation delay of a chunk of a flow                                   v

(β , α ) i.e, the propagation delay of a chunk from a                               ∑ [β       i    − α i (θ i − θ )] + Lmax
relay node to the next relay node. Then the worst
                                                                                     i −1
                                                                                                                                             − θ + δ n ,i       where
                                                                                                               c
case delay of a flow is D + δ if this is the first relay
node i.e., there is no update at the previous node of
                                                                                    δ n,i     is the propagation delay of flow n to traverse
this flow. Here we assume that this is the first relay                              the link between relay node i and its next adjacent
node and the flows arrive from the source directly to                               relay node.
this node. In this case if all of the chunks of the flow                                Example: In order to further analyze the




                                            Ubiquitous Computing and Communication Journal
proposed propositions, consider two cases.                relay nodes as:
Let two flows; flow 1 and flow 2 are contending for             ⎡ v                          ⎤
                                         2L
the bandwidth of a link with a capacity of  . The
                                                                ⎢ ∑ (β ri − α riθ i ) + Lmax ⎥ v
                                          c                ∑2⎢ i=1
                                                          r =1, ⎢            c
                                                                                             ⎥ + ∑ α 2i δ 2,e
                                                                                             ⎥ i =1
                                               L                ⎢                            ⎥
reserved bandwidths of the two flows are both    ,              ⎣                            ⎦
                                               c
                                                                                                     (19)
and all chunks are of size L. However, the inter-
arrival times of two consecutive chunks of flows 1        Where, r represents the index of relay nodes, α 2i is
and 2 are c and c/2, respectively. Assume that the        the traffic rate of flow i reaching the end destination
first chunks of both flows arrive at time 0, and the
                                                          terminal e from the second relay node, and δ 2,e is the
                                            Aik ,
arrival time of the kth chunk of flow i, i=1,2, is
                                                          propagation delay for a chunk of a flow to reach
                                      (k − 1)c            from the second relay node to the destination end
where Ai = ( k − 1)c if i=1, and Ai =
        k                          k
                                               if         terminal. Eq. (19) achieves the goal of our work in
                                         2
                                                          stateless work conserving situation. Note that the
i=2. The previous node’s time tag attached to the kth
                                                          design and analysis of the above work are consistent
chunk of flow i is, however, kc, which is
                                                          with [18-20] with the traffic parameter behaving as
independent of i and will make each flow attain its       virtual clock arrivals as shown in [21]. Li and
reserved bandwidth. Therefore, it can be observed         Knightly [22, 23] provided analysis for multihop
that the worst case delay of flow 1 is c, and it is       stateless scheduling, but the simplicity of our
infinity for flow 2. However, if the previous node’s      analysis is perhaps preferred to be deployed
time tag of the kth chunk of flow i, i=1,2, is set to     regarding virtual traffic flows.
Aik + c , then the worst case delays of both flows             So we far we developed scheduling model with
become infinity. We can observe such characteristic       consideration that the relay nodes are in steady state
from the propositions we derived. The delivery order      that is the traffic load is less than one. However, this
at the previous node attached to the kth chunk of         situation may not hold and the chunks may be
                                                          blocked in overload condition. Losses may occur
flow 2 are i) kc and ii)   Aik + c respectively. In the   because chunks are rejected when they arrive at a full
first case the traffic parameters of the two flows are    (arrival) buffer. In this case a full retransmission is
⎛   L⎞     ⎛   L⎞                                         initiated by the sender IMS terminal after a timeout,
⎜ 0, ⎟ and ⎜ 0, ⎟ i.e., same. By the aggregate            a significant increase in the end-to-end delay.
⎝    c⎠    ⎝    c⎠                                        Chunks may also be corrupted as they arrive a relay
property from proposition 1, we have the traffic          node. There is a large series of recent work on the
                                       ⎛  2L ⎞            asymptotic analysis of loss messages with different
parameter of the aggregate flow as ⎜ 0,      ⎟ and        server characteristics. The characteristic of loss
                                       ⎝   c ⎠            probability for corrupted messages can be located in
                                                 c        Abramov’s PhD work in [15]. Abramov showed the
by proposition 3 the delay bound of any chunk is          effect of adding redundant chunks to a large message.
                                                 2        Summarizing from [14, 15] to our context, adding
since θ 1 = θ 1 = 0 . Therefore, since the delivery       redundant chunks when the load at relay node is
order at the previous node of a chunk lags behind its     slightly greater than 1, will decrease message loss
arrival time, bounded by c and infinity in flows 1 and    probability with the rate of geometric progression.
2 respectively, and then the worst case delay of the      But, adding redundant chunk is not profitable if the
           3c                                             load is much greater than 1, i.e., the loss probability
flows are       and infinity respectively. In the later   will increase for this case.
            2
                                           ⎛      L⎞
case, the traffic parameter of flow 2 is ⎜ ∞,       ⎟ .   3.2      For Overloaded Relay Nodes
                                           ⎝      c⎠
                                   ⎛      2L ⎞                In this section of the paper, we analyze chunks
Thus the aggregate traffic flow is ⎜ ∞,       ⎟ and the   being blocked or rejected due to full buffer. In the
                                   ⎝       c ⎠            previous section we presented delay analysis for one
worst case delay is infinity. Thus, we see that the       source and destination with two intermediate relay
worst case delays of both flows become infinity           nodes. In practical, there may be times chunks arrive
according to our analysis.                                in batches at the relay nodes from the same IMS
    From the above analysis using proposition 3, we       terminal [7]. Alternatively, chunks from the different
can find the end-to-end delay bound for an IMS            sources can traverse the same two relay nodes in a
source to an IMS destination terminal using two           network. We focus on analyzing the response time




                     Ubiquitous Computing and Communication Journal
and mean transmission times of chunks in batch              given probability.
arrival conditions in this section. Let the batches of      The value of the time-out period is set to a multiple k
chunks arrive as Poisson process with mean arrival          of the estimated mean transmission time, for some
rate λ0 (number of batch arrivals per unit time) and         k ≥ 1 set by the user. Let, the rate of successful
mean batch size b. We provide thorough derivation           transmissions be λ ′′ . This makes the probability that
of mean transmission time from a source IMS                 a chunk is successfully transmitted to the
terminal to a destination IMS terminal including two                   λ ′′
relay nodes in between them. Here we focus on               receiver          . Thus the probability that a chunk is
analyzing the response time and mean transmission
                                                                       λ
                                                            rejected at either of the relay nodes due to a full
times of SEND systems.
                                                            buffer is,
In a batch-Poisson stream, we can assume that
successive batches arrive after intervals which are                       λ ′′
                                                            p f = 1−                                       (22)
independent and exponentially distributed. Let the                        λ
service rates (service times being independent of           An IMS terminal may also define the probability of
each other and of the arrival process), are µ1 and          retransmission p r . For p r = 0 , there will be no
µ2    for relay node 1 and 2, respectively. The             retransmissions. This will impact with higher losses
departure process from the first relay node is              of chunks but lower congestion. For p r = 1 , there is
approximated as Poisson with unit batches. We see           always a retransmission attempt and so there are no
that both the relay nodes behave as M/GI/1 servers          losses and the number of retransmissions is unlimited.
under the above assumptions though a deeper                 This causes extra load on the network which might
analysis to correlate the arrival process of the second     result in congestion and hence significantly longer
relay node with the departure process at the first          delays.
relay node is required. Any chunk which has not             It is desirable for every system that losses do not
transferred successfully is lost. We are interested of
                                                            exceed a specified maximum loss rate Lr . This leads
the losses due to the full buffer in this section. By
full buffer we mean that the relay nodes are                to the relation with the chunk arrival rates as follows:
considered to have finite buffer only.                       (1 − p r )(λ − λ ′′) ≤ bλ0 Lr                  (23)
We assume the effect on the sender and receiver is          We now provide the expressions for relay node
the same as a full-buffer loss. Obviously, losses due       utilization below.
to a full buffer at each node, cause an additional          For relay node 1, the utilization is
transmission delay by a timeout of duration much
                                                                  λ′
greater than the per-link transmission delay and            U1 =                                          (24)
typically a multiple of the estimated end-to-end delay.           µ1
Let, the retransmission rate i.e., the net rate at which    Where λ ′ is the throughput from the first relay node.
chunks are resent to the sender node due to losses of
                                                            And for the second relay node, the utilization is:
all types is λ r . The total rate of chunk arrivals,               λ ′′
including       retransmissions,       is       therefore   U2 =                                           (25)
                                                                   µ2
 λ = bλ0 + λ r                                   (20)
                                                                 Note that we assumed λ the batch arrival rates
Let, T1 , T2 , T3 be the constant transmission delays
                                                            at the first relay node, λ ′ the batch arrival rates at
for a chunk passing over the links between the sender       the second relay node which is the throughput of the
and first relay node, first relay node and second relay     first relay node and λ ′′ the throughput of the second
node, and second relay node and the receiver IMS
                                                            relay node. Here we find that the mean rate of
terminal respectively. Hence the transmission delay
                                                            retransmission in the system is,
for successful chunks, excluding the time spent in the
nodes, is                                                    λ r = p r (λ − λ ′′)                         (26)
T = T1 + T2 + T3                               (21)              With these analyses, it is plain to compute the
                                                            two throughputs of the relay nodes. For M/M/1
If a transmission has been successful, then the             machines performance evaluation can be performed
receiver sends a REPORT message for the                     from the following expressions:
successfully delivered chunk to the sender node
through the route, according to the MSRP. When the                ρ1 (1 − ρ1A ) µ1   1

                                                            λ′ =                                           (27)
arrival buffer at a node is full, the rejection of an                1 − ρ1A +1  1

arriving chunk will result in no 200OK being sent.
The value of the sender’s timer will reach the pre-               ρ (1 − ρ 2A ) µ 2      2


specified time-out value, the chunk will be deemed
                                                            λ ′′ = 2                                       (28)
                                                                      1 − ρ 2A +1    2

lost and a retransmission will be attempted with a




                     Ubiquitous Computing and Communication Journal
           ρ1 , ρ 2 are                                                  U1(2−U1)(M2B −b) +(λ′) b3M2S U1(1−U1b)
                                                                                                    2
Where,                    load at the first relay node and           ′
                                                                 W (0) =
                                                                     *
                                                                                                     +
the second relay node respectively; and A1 , A2 are                              2λ′b2(U1 −1)          λ′b(U1 −1)
the buffer sizes of the first relay node and the second                                                          (34)
relay node respectively.
Let the service time random variable be S, with mean
                                                                 M 2 B and M 2 S are the second moments of batch
       1                                                         size and service time distributions, respectively.
s=         , the batch size random variable be B with                 The total expected response time at relay node 2,
       µ                                                         W2 can be computed as the M/M/1 response time
mean b, Yn be the queue length immediately after                 approximation as:


                                                                                              ) (1− (A +1)ρ                   )
the last chunk in the nth batch departs and G X (z ) be                           1
                                                                 W2 =                                     A2
                                                                                                               + A2 ρ2A2 +1
                                                                          µ2 (1− ρ2 )(1− ρ
                                                                                         A2         2     2
the generating function of the probability mass                                          2
function of discrete random variable X. Let                                                                       (35)
U n denotes the number of chunks that arrive during                   The total expected response time W, i.e., the
the service of (all the chunks in) the nth batch. Let            combined time spent in the two relay nodes on a
π n (k ) = P(Yn = k ) for n ≥ 1, k ≥ 0, so that                  successful transmission attempt, is the sum of the
                                                                 expected response times at each node, i.e.,
               ∞
GYn ( z ) = ∑ π n (k ) z k . At equilibrium, assuming            W = W1 + W2                                      (36)
              k =0                                                    Thus, the mean transmission time (MTT) for a
                                                                 chunk that is successful on its first attempt is
this        exists,        let    π n (.) → π (.)          and
                                                                  MTT = W + T                                     (37)
GYn (.) → GY (.) as n → ∞ . The random variable                  Eq. (37) achieves the goal of out model. Failed
B denotes a generic batch size random variable                   chunks, due to full buffer, retry a number of times
Bn and we use V similarly to denote a generic                    given by the retransmission probability p r . Because
                                                                 each retry is made independently of previous
instance of Vn .                                                 attempts, this number of attempts is a geometric
     Let the sojourn time, or waiting time, in the               random variable with parameter p r . The overhead
queue of the last chunk in a batch – i.e. the sum of
the time it spends waiting to start service and its              incurred by a failed transmission, i.e. the elapsed
service time – be W.                                             time between the start of an attempt that
The Laplace-Stieltjes transform of the response time             subsequently fails and the start of the next attempt,
distribution in such an M/GI/1/∞ queue with batch                consists of the time-out delay of k*MTT for chunks
arrivals can then be shown to be given by:                       lost due to a full buffer (k mean successful
[1 − GB (H )]W * (θ ) = (1 − ρ1′)[GB (S * (θ )) − GB (H )]
                                                                 transmission times). We express this overhead, L as
                                                                 follows:
                                                    (29)          L = k * MTT * p f                               (38)
                     θ
                     −1
Where      z = G (1 − )                             (30)
                     λ′
                     B


θ = λ ′(1 − GB ( z ))                               (31)         4       CONCLUSIONS
and
                                                                     Providing instant messaging in real time is
H = z −1 S * (θ ).                          (32)                 indeed open challenge today. Previous works on
W is the average response time. * here denotes data              relay nodes are centered on one node only. We have
indexed by relay node i = 1,2.                                   shown a complete en-to-end delay evaluation that
                                                                 includes two relay nodes (maximum number that a
     The mean waiting time (and arbitrary higher
                                                                 source MRSP terminal can select) for both buffer
moments) at relay node 1 can be approximated by
                                                                 blocking and non-blocking situation. In the former
applying the recurrence formula of the Laplace-
                                                                 analysis constant bit rate was considered to be
Stieltjes transform of the response time distribution
                                                                 consistent with Cruz’s [12, 13] famous work
                                                                                                               (σ        )
of the M/GI/1/∞ queue with batch arrivals. The
computation of higher moments of waiting time of                 conserving virtual traffic parameter model         , ρ .
this type can be located in [16, 17].                            With our model, the performance evaluation of end-
We have,                                                         to-end delay for large instant messages becomes
W1 = −W *′ ( x) | x =0                      (33)                 straight forward. The later analysis was provided for
Where                                                            buffer overloaded situation. The analysis presented
                                                                 leads to the common optimization problem of
                                                                 minimizing Eq. (37) with the respect to the relay




                          Ubiquitous Computing and Communication Journal
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                                                         IEEE Transactions on, Vol: 37 (1), (1991), pp: 114 –
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About UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.