Document Sample

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 R EFERENCES 2n 4a 1 [1] M. Franceschetti and R. Meester, “Navigation in small world networks : S = a=1 1 + M a−α A scale-free continuum model,” Journal of Applied Probability, vol. 43, no. 4, pp. 1173–1180, Dec. 2006. 2n [2] A. J. Ganesh and M. Draief, “Efﬁcient routing in poisson small-world log S = −4 a log(1 + M a−α ). networks,” Journal of Applied Probability, vol. 43, no. 3, pp. 678–686, a=1 2006. [3] J. Kleinberg, “The small-world phenomenon: An algorithmic perspec- Next, we have tive,” in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000. 2n [4] G. Sharma and R. Mazumdar, “Hybrid sensor networks: A small world,” a log(1 + M a−α ) in Proceedings of the ACM International Symposium on Mobile Ad Hoc a=2 Networking and Computing (MobiHoc), 2005. 2n [5] M. Grossglauser and D. Tse, “Mobility increases the capacity of ad-hoc wireless networks,” IEEE/ACM Transactions on Networking, vol. 10, ≤ x log(1 + M x−α ) dx 1 no. 4, pp. 477–486, Aug. 2002. 2n 2n [6] A. E. Gamal, J. Mammen, B. Prabhakar, and D. Shah, “Optimal x2 αM x1−α throughput-delay scaling in wireless networks - part i : The ﬂuid model,” = log(1 + M x−α ) + dx IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2568– 2 x=1 2 1 1 + M x−α 2592, Jun. 2006. M 1 αM 2n 1−α [7] ——, “Optimal throughput-delay scaling in wireless networks - part ii ≤ 2n2 log 1+ − log(1+M )+ x dx : Constant-size packets,” IEEE Transactions on Information Theory, (2n)α 2 2 1 vol. 52, no. 11, pp. 5111–5116, Nov. 2006. M 1 αM [8] M. J. Neely and E. Modiano, “Capacity and delay tradeoffs for ad-hoc ≤ 2n2 log 1+ − log(1 + M ) + . mobile networks,” IEEE Transactions on Information Theory, vol. 51, (2n)α 2 2(α − 2) no. 6, pp. 1917–1937, Jun. 2005. [9] G. Sharma, R. Mazumdar, and N. Shroff, “Delay and capacity tradeoffs Thus, we have in mobile ad-hoc networks: A global perspective,” in Proceedings of the d(i, j) α IEEE Conference on Computer Communications (INFOCOM), 2006. log S ≥ −2 log(1 + β(d(i, j))α )−8n2 log(1 + β( ) ) [10] L. Ying, S. Yang, R. Srikant, and G. Dullerud, “Coding achieves 2n the optimal delay-throughput tradeoffs in mobile ad-hoc networks,” in 2αβ(d(i, j))α Proceedings of the IEEE International Symposium on Modeling and − Optimization in Mobile, Ad-Hoc and Wireless Networks (WiOpt), 2007. (α − 2) [11] N. Karamchandani and M. Franceschetti, “Scaling laws for delay = −δβ (d(i, j)). sensitive trafﬁc in rayleigh fading networks,” 2007, in preparation. [Online]. Available: http://ans.ucsd.edu/ans/images/pdf/fading-draft.pdf. Then from (6), we have [12] A. Jovicic, P. Viswanath, and S. R. Kulkarni, “Upper bounds to transport βd(i,j)α N − +δβ (d(i,j)) capacity of wireless networks,” IEEE Transactions on Information Pr(1v i,j,β = 1) ≥ e P . Theory, vol. 50, no. 11, pp. 2555–2565, Nov. 2004. [13] F. Xue, L. L. Xie, and P.R.Kumar, “The transport capacity of wireless networks over fading channels,” IEEE Transactions on Information Theory, vol. 51, no. 3, pp. 834–847, Mar. 2005. [14] O. Dousse, M. Franceschetti, and P. Thiran, “On the throughput scaling of wireless relay networks,” IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2756–2761, Jun. 2006.

DOCUMENT INFO

Shared By:

Categories:

Tags:
wireless networks, wireless sensor networks, ad hoc, the network, base station, data rate, hoc networks, power control, sensor networks, university of waterloo, tsinghua university, rayleigh fading, cognitive radio, channel estimation, power allocation

Stats:

views: | 10 |

posted: | 2/6/2010 |

language: | English |

pages: | 5 |

OTHER DOCS BY zbm17245

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.