Scaling laws for delay sensitive traffic in Rayleigh fading

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					  Scaling laws for delay sensitive traffic in Rayleigh
                   fading networks
                                      Nikhil Karamchandani             Massimo Franceschetti
                            California Institute for Telecommunications and Information Technology
                                      Department of Electrical and Computer Engineering
                                               University of California at San Diego
                                                     La Jolla, CA 92093-0407
                                        nikhil@ucsd.edu             massimo@ece.ucsd.edu


   Abstract—The throughput of delay sensitive traffic in a              position of the destination node. This is particularly important
Rayleigh fading network is studied by adopting a scaling limit         in a fading scenario, where the channel strength varies with
approach. The case of study is that of a pair of nodes establishing    time and hence pre-computing routes can be of little help.
a data stream that has routing priority over all the remaining
traffic in the network. For every delay constraint, upper and           Furthermore, since we are interested in the throughput scaling
lower bounds on the achievable information rate between the two        limit for large networks, maintaining a global knowledge
end-points of the stream are obtained as the network size grows.       of connectivity at each node in the network is clearly an
The analysis concerns decentralized schemes, in the sense that all     impossible task.
nodes make next-hop decisions based only on local information,            Intuitively, to achieve a smaller delay, packets of the high
namely their channel strength to other nodes in the network and
the position of the destination node. This is particularly important   priority stream need to be routed along longer hops, which
in a fading scenario, where the channel strength varies with time      in turn restricts the data rate that can be supported, and our
and hence pre-computing routes can be of little help.                  main contribution is to precisely characterize this tradeoff. In
   Natural applications are remote surveillance using sensor           doing so, we also explicitly take into account the time that
networks, and communication in emergency scenarios.                    packets are queued at the intermediate nodes along the multi-
                                                                       hop path, before reaching the destination. Our results show
                       I. I NTRODUCTION
                                                                       that the throughput is proportional to a power law of the
   Wireless networks present a number of challenges unseen             ratio between the delay and the system’s size, whose exponent
in their wired counterparts. First, since the wireless channel is      depends on the channel propagation loss.
a shared medium, simultaneous transmissions may interfere                 One of the main difficulties that we face in our analysis is
with each other and the rate at which any two nodes can                the presence of fading that makes the network connectivity
directly communicate information depends on the amount                 change from time to time. Our approach is to consider the
of interference at the receiver. Further, the channel strength         network as a sequence of random graphs, each corresponding
depends on the loss due to propagation of the signal in                to a given time slot and defined by the set of links that
space, and also varies in time due to multipath fading. A              can support a given information rate in that slot. We then
practical strategy for communication in this scenario is multi-        consider the problem of navigating through this sequence of
hop routing, wherein the nodes co-operate by buffering and             random graphs using only local knowledge at each node of the
forwarding each others’ data packets. Channel estimation is            graph topology. We then use some graph properties that arise
performed at each hop along the route and packets are sent at          with high probability as the network size grows, to derive the
a certain rate which depends on the current channel conditions.        corresponding throughput scaling laws for the high priority
   In this context, we formulate the following problem. Con-           traffic stream.
sider a wireless fading network in which a set of sources is              We now want to spend a few words on the related literature.
communicating to their respective destinations. Now suppose            On the one hand, our work is reminiscent of that of navigation
a single source-destination pair wants to exchange some delay          in small-world graphs and long-range percolation models.
sensitive information. Hence, data exchanged between this              Typically, in these models the network links are dynamically
pair of nodes is given priority over other network traffic.             added at random to each node visited during the navigation
For example, such a situation can arise whenever a “critical”          process and the objective is to reach a destination node with
event requires a real time data link between two nodes in the          the fewest number of hops possible, see [1] [2] [3] [4]. On
network. We ask: what is the maximum rate of communication             the other hand, however, our wireless communication context
that such a pair can achieve for any given delay constraint?           is very different from these works. More directly related are
   In addressing the above question, we only consider decen-           the works that characterize the trade-off between throughput
tralized routing schemes, which means that all nodes make              and delay in wireless networks. Most of these were inspired
next-hop decisions based only on local information, namely             by the results in [5], and are concerned with mobility of the
their channel strength to other nodes in the network and the           nodes in the network, see [6] [7] [8] [9] [10]. Furthermore,
they all consider dense networks, in which a growing number               •   Coherence time ≫ 1 slot. This is the slow fading sce-
of nodes are distributed over a fixed area and there is no                     nario.
notion of priority traffic, or fading. In the present work, we           We now describe how information is communicated across the
address the problem of priority traffic when the channel is              network. If a source s wants to transmit a message to a target
subject to random fading, and we focus on extended networks             t, it does so by using a multi-hop scheme. The message is
where a fixed density of nodes is placed over a growing                  divided into packets, all of the same size and in each slot a
area and the nodes do not move. We remark that even if                  node can transmit a single packet to a chosen destination. The
the nodes are fixed, fading can still arise because of mobile            coding is point to point i.e., in any slot a sender transmits
objects in the environment. Furthermore, for the sake of ease           information to only one receiver, and similarly any receiver
of exposition, we will limit our analysis to the case in which          while decoding messages from a sender considers all other
nodes are regularly placed on a grid. The case in which the             signals as noise. The transmission of a packet from a node
nodes are randomly distributed in space according to a Poisson          i to j in a slot v is successful if and only if the number of
point process is amenable to a similar analysis (subject to             information bits in the packet is less than R(SINRv ), where
                                                                                                                              ij
technical complications) and is discussed in [11]. Finally, we          R(·) denotes the maximum number of bits that two nodes can
wish to mention the works [12] [13], where the problem of               directly communicate to each other in a slot of unit duration,
characterizing the information-theoretic capacity of wireless           for a given SINR. Its precise form may depend on the specific
fading networks has been addressed.                                     coding-decoding strategy, modulation and bit error rate (BER)
   The remainder of the paper is organized as follows. In               requirement. For example, if the slot duration is long enough,
Section. II, we introduce our network model. We illustrate              there exist codes for transmission over the AWGN channel
the problem and define some notation in Section. III. The                which perform very close to capacity and R(SINR) is of the
main results are presented and discussed in Section. IV. A              form log(1+SINR). Further, we assume that the packet size is
short sketch of the proofs is given in Section V. Finally, in           kept constant across transmissions and the relays only forward
Section. VI we draw conclusions.                                        the packets on links which can support the required rate, i.e.
                     II. N ETWORK M ODEL                                the SINR at the corresponding receiver is high enough.
                                                                           Now we discuss how nodes decide their next hops while
   Consider an n × n grid, with nodes at {(x, y) : x ∈
                                                                        routing packets from the source to the target. We assume
{1, 2, . . . , n}, y ∈ {1, 2, . . . , n}}. Define the distance between
                                                                        that all nodes are aware of the underlying grid structure of
any two nodes a, b as the Manhattan distance d(a, b) =
                                                                        the network. Hence, the routing schemes that we study will
|ax − bx | + |ay − by |. Assume that time is divided into slots
                                                                        be geographic in nature. Further, we restrict our attention to
of unit duration, and the nodes communicate over a wireless
                                                                        decentralized schemes. We define a decentralized scheme as
channel of unit bandwidth. For medium access, consider a
                                                                        one where any node, when it gets the message, decides its
decentralized strategy where in every slot, each node indepen-
                                                                        next hop only on the basis of local information, namely the
dently and with probability p decides to transmit a packet to
                                                                        channel strength to other nodes in the network and the position
its intended destination. All nodes transmit at power P . If a
                                                                        of the target t. In the schemes that we propose, we assume
node i transmits to node j in slot v, the instantaneous signal
                                                                        that this kind of local channel state information is available to
to interference plus noise ratio (SINR) at j is given by
                                                                        each node, and explicitly take into account the time required
                               P l(i, j)h(i, j)v                        by a node to find a suitable relay.
                   SINRv =
                       ij                 v      ,
                                   N + Ij
where l(i, j) = d(i, j)−α , α > 2 is the path loss between i                             III. P ROBLEM S TATEMENT
and j, N is the thermal noise power (assumed to be the same                Randomly pick a source s and a target t amongst the nodes
                     v
for all receivers), Ij is the interference at node j in slot v          in the grid. Assume that the source generates a traffic stream
           v
and h(i, j) is the (random) fading coefficient between nodes             which is delay sensitive and hence is given priority over the
i and j in slot v. We consider a Rayleigh fading model such             packets of other traffic in the network. More precisely, we
that in any time slot v, the probability density function (pdf)         assume that at any intermediate node, a queued packet of
of h(i, j)v is given by                                                 this high priority stream is scheduled for transmission at least
                                                                                     ¯                ¯
                                                                        once every K slots, where K is some constant. Note that this
    fh(i,j)v (x) = e−x , x ∈ (0, ∞), and E[h(i, j)v ] = 1.
                                                                        model differs from a relay network as described in [14], where
The fading coefficients are assumed to be independent across             all nodes only cooperate to facilitate communication between
both space and time. This means that in every slot v, h(i, j)v          a single source and destination. In our case, other source-
and h(k, l)v are independent for all i = k or j = l; and                destination pairs (of lower priority) may also simultaneously
for all i, j, h(i, j)v remains constant for a certain number of         communicate and potentially cause interference to the high
time slots, called the coherence time of the channel, and then          priority stream, see Fig. 1. Similarly, we do not consider
changes in an independent fashion. The following two cases              schemes that flood the network with packets of the high
are considered:                                                         priority stream, since it aversely affects the communication
   • Coherence time = 1 slot. This is the fast fading scenario.         of other data streams in the network.
                                                                             where K1 is a constant and α is the path loss exponent.
                                                                             Theorem 2. In both the fast fading and slow fading scenarios,
                                                                             there exists a decentralized scheme Π with average packet
                                                                             delay DΠ (n) (where DΠ (n) is O(nγ ), 0 < γ < 1) and
                                                                             achievable throughput TΠ (n), such that for n large enough
                                                                             we have
                                                                                                                  K3
                                                                                            TΠ (n) ≥ K2 R                 ,
                                                                                                              ( D n )α
                                                                                                                   (n)
                                                                                                                   Π

                                                                             where K2 , K3 are constants, and α is the path loss exponent.
                                                                                While the upper bound in Theorem 1 holds for any decen-
                                                                             tralized multi-hop scheme, the lower bound in Theorem 2 is
                                                                             shown constructively, by presenting a specific routing scheme
                                                                             that achieves the bound using only local channel state infor-
                                                                             mation. Upper and lower bounds are almost tight, as they
                                                                             differ only by a logarithmic factor in the argument of the R(·)
                                                                             function.
                                                                                In the sequel, we discuss our approach to prove the above
                                                                             results. We do not report all the details here because of space
                                                                             constraints and will only provide an outline. The complete
Fig. 1. Source s communicates high priority data to target t. There can be   proofs are available in [11]. First, notice that the results are
other co-existing traffic (of lower priority) in the network.
                                                                             intuitive: the average distance between the source and the
                                                                             target is of order n and if we cover it using H hops, then
   Next, we define the throughput and delay for any routing                   the delay will be at least H time slots and the average hop
                                                                                                                          n
scheme in the above setting.                                                 length along the path will be of order H . Due to the path
   The throughput of a given routing scheme Π, used to convey                loss, the received signal power at each hop will be of order
                                                                               n                                                        n
high priority data from s to t in a grid of size n×n is denoted              ( H )−α . A bound on the throughput of the type R(( H )−α )
by TΠ (n). Denoting the (random) number of bits successfully                 then follows. Despite the simplicity of this “back of the
transferred by Π from s to t in M slots by BΠ (M, n), a                      envelope” calculation, we need quite a bit of work to make it
throughput TΠ (n) is achievable iff                                          rigorous. First, in addition to the path loss, we have to account
                                                                             for both the interference from other nodes in the network and
                                         1
         lim Pr TΠ (n) ≤ lim inf           BΠ (M, n)        = 1.             the random fading process. Second, in the computation of the
        n→∞                     M →∞     M                                   delay, we have to consider not only the number of hops it takes
The average packet delay for scheme Π is denoted by DΠ (n).                  for the packet to reach the destination, but also the (random)
This is measured by the average number of time slots that                    amount of time spent in the queues at the intermediate nodes,
a packet takes to reach the destination after it leaves the                  and this time again depends on the fading process. Finally, in
source. This includes the number of hops before it arrives                   the case of Theorem 2, we have to come up with a constructive
at the destination and the time it is queued at the intermediate             strategy that discovers the route to the target using only local
nodes. Accordingly, denoting the (random) delay of packet j                  channel state information at each node.
      j
by DΠ (n), we define                                                             To overcome the above technical difficulties, we proceed as
                                                                           follows. We characterize the fading network in terms of a time
                                        k
                                     1      j                                sequence of random graphs, whose topology depends on the
             DΠ (n) = E lim sup          DΠ (n) .
                             k→∞ k j=1                                       channel conditions. We compute bounds on the probability of
                                                                             existence of edges in these graphs and then use some of their
In this work, we focus on the scaling limit of the above                     structural properties to derive upper and lower bounds on the
quantities as the grid size n increases. We now present our                  achievable throughput for any given delay constraint.
main results and discuss them briefly.
                                                                                         V. N ETWORK AS A RANDOM GRAPH
                    IV. M AIN R ESULTS
                                                                             A. Dealing with dependencies
Theorem 1. In both the fast fading and slow fading scenarios,
for any decentralized scheme Π with average packet delay                        In this section, we model the fading network as a sequence
DΠ (n) (where DΠ (n) is O(nγ ), 0 < γ < 1), we have that for                 of random graphs. Define 1v to be the indicator random
                                                                                                            k
n large enough, any achievable throughput is bounded by                      variable for whether node k transmits in slot v. Further, let 1v
                                                                                                                                            i,j
                                             n                               be the indicator random variable corresponding to the event
                                      log( DΠ (n) )
                 TΠ (n) ≤ R K1                         ,                     that in slot v, node i can successfully communicate a packet
                                           n
                                       ( DΠ (n) )α                           of size R(β) bits to node j, i.e., SINRv > β. Consider the
                                                                                                                      ij
sequence of random graphs Gv constructed with the nodes of                  where
the network as vertices and with edges ev between all i, j for
                                        ij
                                                                            δβ (d(i, j)) =
which 1v = 1. Thus, 1v is the indicator random variable for
        i,j             i,j
the edge ev being present in Gv . We have,                                                                        βd(i, j)α αβd(i, j)α
            ij                                                              2 log(1 + βd(i, j)α )+4n2 log(1+               )+          .
                                                                                                                   (2n)α      α−2
   Pr(1v = 1) = Pr(SINRv > β)
       i,j              ij                                                  The derivation of (5) can be found in the Appendix.
                                     v
                              β(N + Ij )d(i, j)α                               Thus, we have computed upper and lower bounds on the
              = Pr h(i, j)v >                                       , (1)
                                     P                                      probability of existence of edges in the random graph Gv cor-
                                                                            responding to the fading network. We now construct two other
where                                                                       graphs Gu and Gl with the same set of vertices, but with edges
              v
                                                                            drawn independently and with probability corresponding to the
             Ij   =           P d(k, j)−α h(k, j)v 1v .
                                                    k                (2)    upper and lower bounds derived in (3) and (5) respectively. It
                      k=i,j                                                 is clear that while edges in Gv are not independent, Gv is
                                                                            stochastically dominated by the graph Gu and dominates the
   It is important to notice at this point that the edges of Gv
                                                                            graph Gl in which edges are indeed independent. It follows
are not independent. Knowing if edge ev is present in Gv
                                            i,j
                                                                            that if we can find a path in Gl , we can also find it in Gv ,
gives information on the amount of interference at the nodes
                                                                            while if a path cannot be found by any decentralized algorithm
in the neighborhood of j, and in turn on the connections of
                                                                            in Gu , then it cannot be found in Gv .
these nodes. To take care of these unwanted dependencies,
                                                                               Since, in order to transport a packet containing R(β) bits of
starting from Gv we construct two other graphs in which the
                                                                            information across the network, one needs to discover paths
states of the edges are independent, and which dominate and
                                                                            between the source and the destination on Gv , we can now
are dominated respectively by the edges of the original graph
                                                                            exploit the independence structure of Gu and Gl to find bounds
Gv . We will then use these two graphs to derive bounds on
                                                                            on the existence of such paths. In the next section, we briefly
our quantities of interest, namely the throughput and the delay.
                                                                            outline the procedure to complete the proofs of the theorems.
   We start by evaluating some bounds on the probability
                                                          v                 Please refer to [11] for details.
in (1). An upper bound trivially follows by substituting Ij = 0,
yielding                                                                    B. Proofs outline
                                                                              •   Theorem 1 : The distance between the randomly chosen
                                                   βN d(i, j)α
        Pr(1v = 1)
            i,j       ≤ Pr h(i, j)v >                                             source and target is of order n. Then, to cover this
                                                       P                          distance in H hops, a packet needs to make at least one
                                  βd(i,j)α N
                                                                                                     n
                      = e−            P        .                     (3)          hop of length Ω( H ). Further, the packet delay is then at
                                                                                  least H. We compute the probability that a decentralized
On the other hand, we can find a corresponding lower bound                         algorithm can find such a long hop in Gu , within the first
                                   v
by bounding the interference Ij from above. Notice that                           H hops of its operation. Since this probability decreases
any node j in the grid has at most 4 × a other nodes at                           with the size of the packet, it leads to an upper bound on
(Manhattan) distance a from itself, for all a ∈ {1, 2, . . . 2n}.                 the achievable throughput for any decentralized scheme
To upper bound the interference, add fictitious nodes to the                       with a certain average packet delay.
network so that for each a, j has exactly 4a total number                     •   Theorem 2 : In this case, we define a routing strategy that
of nodes at distance a. Denote these nodes by {ka,b }4a ,    b=1                  looks for a path connecting the source to the destination in
and the corresponding fading coefficients to node j in slot v                      Gl . We divide the network into smaller cells, each of area
by {h(ka,b , j)v }4a . For fictitious nodes, we generate fading
                  b=1
                                                                                      n
                                                                                  O( D ). The straight line joining the source and the target
coefficients independently at random, according to the same                        passes through at most O(D) such cells. We consider
distribution as the coefficients for the real nodes. We consider                   schemes that route packets by forwarding them from cell
the case when all the nodes (real and fictitious) interfere with                   to cell along this line. Now, when an intermediate node
the communication between i and j in slot v. It then follows                      receives a packet of the stream, it attempts to find a
from (2) that                                                                     suitable relay in the next cell. In the fast fading case, it
                                                                                  can pick a node arbitrarily and just wait until the channel
                       2n    4a
             v                                                                    to this node becomes strong enough to successfully
            Ij < P                                     v
                                   a−α h(ka,b , j)v ≡ Ij .           (4)
                                                                                  transfer the packet. On the other hand, in the slow fading
                      a=1 b=1
                                                                                  case, it has to actively search for a suitable relay inside
Substituting (4) into (1) , we have                                               the next cell by polling different nodes in each slot. The
                                                                                  probability that a suitable relay is found increases as the
                                                      v
                                               β(N + Ij )d(i, j)α                 packet size decreases, and we find a bound on the packet
   Pr(1v = 1) > Pr h(i, j)v >
       i,j                                                                        size such that this probability tends to one as n → ∞.
                                                      P
                            βd(i,j)α N
                                                                                  Thus, the packet is routed into the next cell. The number
                       −                 +δβ (d(i,j))
                  ≥ e           P
                                                        ,            (5)          of hops that a packet takes before reaching the target is
      O(D). Additionally, using the independence structure of                                                             A PPENDIX
      Gl , we are able to evaluate the average queuing delay                     A. Proof of Eqn. (5)
      at each hop and hence a lower bound on the achievable
      throughput for any given delay constraint follows.
                                                                                 Pr(1v
                                                                                     i,j,β = 1)
             VI. C ONCLUSION AND F UTURE W ORK
   In this work, we derived scaling laws for decentralized
                                                                                                                   v
                                                                                                            β(N + Ij )d(i, j)α
                                                                                 > Pr h(i, j)v >
schemes which route delay sensitive data in extended Rayleigh                                                      P
fading networks. We first derived an upper bound on the                                                                              2n   4a
                                                                                                            βd(i, j)α N                                d(i, j) α
throughput achievable by any decentralized scheme under a                        = Pr h(i, j)v >                        +                      β(             ) h(ka,b , j)v
given delay constraint, and then proposed specific schemes                                                       P         a=1
                                                                                                                                                         a
                                                                                                                                         b=1
which are almost order optimal. Future work includes eval-                                                  2n   4a       ∞
                                                                                 (a)      βd(i,j)α N                            −
                                                                                                                                      d(i,j)
                                                                                                                                    β( a )α +1              z
                                                                                        −
uating similar bounds for more general scenarios such as                         = e          P                               e                                 dz
                                                                                                                      0
networks with multiple source destination pairs and uniform                                                a=1 b=1
                                                                                                           2n                                 4a
or non-uniform traffic requirements.                                                        βd(i,j)α N                      1
                                                                                       −
                                                                                 =e            P                                                   ,                         (6)
                         ACKNOWLEDGMENT                                                                   a=1    β( d(i,j) )α + 1
                                                                                                                      a
   The authors are grateful to Prof. Rene Cruz for some helpful                  where (a) results from the fading coefficients, {h(ka,b , j)}, b ∈
discussions. This work was partially supported by the National                   {1 . . . 4a}, a ∈ {1 . . . 2n}, being independent unit exponential
Science Foundation, under CAREER award CNS-0546235 and                           random variables.
award CCF 0635048.                                                               Now, let M = β(d(i, j))α . Consider
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