Queueing Model and Optimization of Packet Dropping in Real-Time Wireless Sensor Networks

Transcription

1 Queueing Moel an Optimization of Packet Dropping in Real-Time Wireless Sensor Networks Marc Aoun, Antonios Argyriou, Philips Research, Einhoven, 66AE, The Netherlans Department of Computer an Communication Engineering, University of Thessaly, Volos, 38, Greece Abstract We consier the problem of moeling the transmission of real-time ata from a single noe of a wireless sensor network to the next hop or access point. Generate packets are place in a single position buffer an are transmitte over the wireless meium to the next hop, or they are iscare an replace when a new ata packet is generate from the sensor. To increase the percentage of packets elivere on-time to the next hop, we introuce a thresholing policy that is supporte by an analytical moel an it is responsible for eciing whether to transmit a ata packet or rop it an transmit the next one. Our moel consiers the behavior at single sensor where ata are generate with a Poisson process an have a fixe ealine associate to them while the impact of the other sensors is moele through their effect on the MAC service time. However, we also evaluate the performance of the packet ropping scheme in a multi-hop sensor chain. The propose policy leas to a higher percentage of on-time elivere packets in both scenarios. Inex Terms Real-time communication, wireless sensor networks, packet ropping, queuing moel. I. INTRODUCTION In wireless sensor networks (WSNs) real-time ata generation an transmission ominate the majority of application scenarios. One of the main characteristics of sensor ata that is generate in real-time is that their value may become obsolete when a new one is generate. In this paper, we investigate the communication of real-time messages by eveloping a simple queueing theory moel an we use it for further optimization. We moel the transmitter of a sensor noe as a single-server queue. The generate ata packets have to be transmitte within a certain ealine. The queueing system is assume to have an overwrite-buffer with space for one packet; when a new packet bearing more recent ata arrives to the queueing system, the buffere packet, if any, is iscare an the new packet is store in the buffer. Incluing the buffer space at the transmitter, the system represents an M/M// queue with ealines. The important contribution of this work to the aforementione moel is that it introuces a thresholing mechanism. Accoring to this mechanism, a buffere ata packet is roppe (remove from the buffer) if the remaining time until its ealine is shorter than a certain thresholθ. The intuition behin this policy is that a packet is not transmitte if there is low probability of being receive before its ealine expires, saving thus transmission time for a newly arriving packet an in aition wireless banwith. By ropping the currently store packet, an waiting on a new packet arrival instea, the percentage of on-time packets is increase because sink Fig. : Multiple wireless sensors generate real-time ata that are elivere to a sink. With multi-hop communication the wireless noes closer to the sink may become the bottleneck since they have to forwar higher portion of the traffic. a newly arriving packet has a higher probability of meeting its ealine. Furthermore, with this proactive approach a packet that woul be characterize lost in the next hop, if it was late, will not be transmitte over the wireless meium an consume banwith. II. RELATED WORKS To the best of our knowlege, the problem of moeling the real-time packet transmission in WSNs has not been aequately aresse. Nevertheless, the secon problem we are intereste in this paper, i.e. the phenomenon of overloaing the core of a WSN that operates uner multiple sources an one sink has been reporte in several works []. Fig. epicts a simple wireless sensor network where problems like the above may occur epening on the locally generate traffic loa at each sensor. Optimization efforts in this type of wireless sensors networks have focuse on reucing the traffic loa an also on esigning meium access control (MAC) protocols that prioritize the noes that are closer to the sink [], [3]. Nevertheless these mechanisms attempt to aress the problem from the perspective of the better utilization of the network without looking at the sources themselves an how to prevent unnecessary ata from being injecte into the network. However, for optimizing the injection of real-time traffic into the network, there is a nee for analytical performance moels that characterize the impact of all possible actions taken by a noe. The works in this area are limite an focus primarily on the queuing moel itself without consiering its applicability in a network setting. Barrer was one of the first to analyze an M/M/ queue with eterministic ealines [4], []. This analysis was extene to systems with stateepenent arrival an service rates, an more general ealines by Brant an Brant [6], [7] an by Movaghar [8]. These

2 Fig. : A schematic representation of the moel for a sensor noe that aopts proactive packet ropping. stuies all eal with non-preemptive systems with ealines to service beginnings, but in [9], Movaghar consiers ealines to service enings with preemption. Movaghar an Kargahi [], [] have evise an approximation for an M/M/ queue with the earliest-ealine-first (EDF) iscipline, which is known to stochastically maximise the fraction of packets serve before their ealine (see, e.g., []). Lehoczky [3] analyze an M/M/ queue with ealines to service enings an the EDFpolicy in heavy traffic. He argue that, since the ealines of all store packets have to be taken into account, this queue gives rise to a Markov process on a statespace of infinite imension. Lehoczky shows that the Markov process collapses to a tractable one-imensional process in heavy traffic. Lehozcky later use these results to analyze control policies in [4], an extene his analysis to Jackson networks in []. Doytchinov et al. [6] extene this analysis to a GI/G/ queue with ealines, an Kruk et al. [7] to networks of such queues. A. Sensor System Moel III. PERFORMANCE ANALYSIS We now provie further etails for the performance analysis of the propose system moel. Packets are generate at each sensor accoring to a Poisson process with parameter. The packets have exponentially istribute service times with parameter that correspon to the behavior of the MAC protocol an the impact of channel contention from other noes. A packet transmission is consiere successful if it is complete within time units after the arrival of the packet to the queue, i.e., if a packet arrives at time t, its transmission must finish before its global ealine t +, with being a fixe relative ealine. Packet transmission at the physical layer (PHY) is non-preemptive, meaning that if a packet starts being transmitte it will finish regarless of whether another packet arrives. We efine the packet elivery ratio (PDR) of the system as the fraction of packets that are transmitte successfully, i.e. within the ealine requirement. B. PDR Calculation In this section, we compute the PDR of the system in the presence of packet ropping enote by γ(θ). The parameter θ enotes the packet ropping threshol given in secons, an it will be iscusse in more etail later. We enote by L the number of packets present in the system (incluing the transmitter) immeiately before a packet arrival. A packet arriving to the system will fin the latter in one of the following three states: Empty (L = ): The transmitter is available, an there is no packet occupying the buffer spot. Busy Transmitter (L = ): The transmitter is transmitting a packet but the buffer is empty. Full, (L = ): The transmitter is busy an the buffer also contains a packet. In that case, an arriving packet overwrites the buffere one. The PDR γ(θ) is by efinition equal to the probability that the transmission of an arbitrary packet is successful. To compute this probability, we conition on whether a packet arrives at an empty system or not. Denoting by B the service time that the arriving packet woul experience, We have: γ(θ) = P(L = )P(B ) θ +( P(L = )) e t P(B t)e t t () The rationale behin Eq. () is as follows: If an arbitrary packet P arrives at an empty system, it is transmitte successfully only if the require transmission time B is less than. This explains the term P(L = )P(B ). If P arrives at a non-empty system (L = or L = ), we conition on the length of the remaining (resiual) transmission time t of the packet that is currently being transmitte. The resiual service time is exponentially istribute, so its ensity is e t. Moreover, given that the resiual service time is t, the service of P is successful if the following three conitions are all met: First, the time until the ealine is larger than θ when the resiual service ens, i.e., t θ. This is taken into account in the integration region of Eq. (). Secon, the service time is less than t, which explains the factor P(B t ). Thir, there were no other arrivals uring t time units, which explains the term e t. C. Probability of an Empty System Next, we calculate P(L = ), i.e. the probability that an arbitrary packet arrives at an empty system. For this purpose we efine the cycle time C which is the time between two consecutive time instants at which the system becomes empty an it is epicte in Fig. 3. The cycle time is separate into two perios: A first ile perio uring which the system is empty, an a busy perio uring which the transmitter is occupie until it becomes empty again. The busy perio spreas throughout the time that the system is not in a state L =. It consists of a collection of consecutive states characterize by L = or L =. We shall refer to the case where L = uring a busy perio as clearance perio (CP ), which is the amount of time the buffer is continuously occupie. A clearance perio ens when the buffer is cleare, i.e. when either a ispatch to the transmitter occurs, or when the store packet is enie service an remove from the system following the thresholing proceure. The ile time lasts until a packet arrives, so the mean ile time is /. If the busy perio is enote by BP, we have: E[C] = +E[BP]. () From the PASTA-property [8], we have that the probability that a packet arrives at an empty system, P(L = ),

3 Packet arrivals First arrival, begins a clearance perio L = L = L = L = L = time L = L = clearance perio (CP) CP Ile time Ile time busy perio cycle new cycle begins Fig. 3: A cycle consisting of ile time an busy perio. it is equal to the probability that the system is empty at an arbitrary time. To erive this probability we ivie the mean ile time by the mean cycle time as follows: P(L = ) = / E[C] = / /+E[BP]. (3) Now we erive an expression for the busy perio. The busy perio starts when a packet arrives to an empty system. After this, the expecte time until the first event (an arrival or service completion) is /(+). Furthermore, with probability /(+ ), the first event is a service completion, an the busy perio ens after / time units. With probability /(+ ), the first event is an arrival. In this case, the arriving packet is store in the buffer an the buffer has to be cleare, which takes in expectation E[T] time units. Once the buffer has been cleare, the expecte time before the system becomes empty is again equal to E[BP] ue to the memorylessness of service times. From the above we have that the mean busy perio is E[BP] = + + (E[T]+E[BP]), (4) + where E[T] is the mean buffer clearance perio. We will erive E[T] next. With algebraic manipulations (4) yiels E[BP] = + E[T]. () Now, it remains to etermine E[T], the expecte length of the buffer clearance perio. Without loss of generality, we assume that the buffer clearance perio starts at time. We conition on t the time at which the next event (arrival or service completion) occurs. Because the time until the next event is exponentially istribute with parameter +, we obtain: E[T] = + + θ θ θ (+)e (+)t ( θ)t + (+)e (+)t tt + (+)e (+)t (t+e[t])t. The three integrals (from left to right) cover the following three possibilities: First, if there are no events before time θ, Γ Θ Fig. 4: The PDR as a function of θ, with = = =. the buffer is cleare at time θ because the ealine of the buffere packet becomes smaller than θ. Secon, if the first event occurs at time t < θ, an the first event is a service completion, the buffer is cleare at time t because the buffere packet enters service. Thir, if the first event occurs at time t < θ an is a packet arrival, the arriving packet overwrites the buffere packet, an a new buffer clearance perio begins. After rewriting an evaluating the integrals (the rightmost two using partial integration), we obtain: E[T] = e (+)( θ) +e (+)( θ). (6) Combining Equations (3), (), an (6) we have that P(L = ) = + +e (+)( θ) + + Finally, by substituting (7) into () an evaluating the integrals we have that the PDR γ(θ) is given by γ(θ) = P(L = )( e )+ [ ( P(L = )) + ( e (+)( θ)) e (7) ( e ( θ))]. (8) Note that if, ealines become irrelevant an two special cases occur for specific values of θ. If θ =, packets in the buffer are always transmitte, regarless of the time until their ealine. The PDR of the system is thus equal to that in an M/M// queue (see e.g., [9, Section.7]. If θ =, packets in the buffer are never transmitte. Only packets arriving to an empty system are. In this case, the PDR is equal to that in an M/M// queue. By substituting an θ, these values follow from Equation (8). IV. NUMERICAL RESULTS FOR A SINGLE SENSOR In this section, we stuy the PDR numerically. In Fig. 4, we present γ(θ) for various values of θ, with = = =. We clearly see that the PDR of the system is inee increase by the threshol θ, as long as θ is chosen appropriately. Having establishe that an appropriately chosen threshol increases PDR, we stuy how much the PDR can be increase by this threshol. Note that γ() is the PDR of the system

4 Gain % Λ 3 Λ 4 Λ 6 Gain % Λ Fig. 6: PDR gain for values equal to 3, 4 an 6. Gain % (a). 4 Λ 6 (b). 8. Fig. : Maximal PDR improvement.. without thresholing. We efine θ as the threshol that maximizes the relative increase in PDR, i.e., we efine { } γ(θ) γ() θ = arg max : θ. γ() We efine the maximal PDR improvement as the maximal relative increase in PDR, i.e., as (γ(θ ) γ())/γ(). In the sequel, we set =. We can make this assumption without loss of generality; all parameters are relative to each other, so for any set of parameters, we can scale time in such a way that = without changing the PDR of the system. In Fig. (a) we isplay the maximal PDR improvement for between an, an between an. Fig (a) implies that, in this parameter region, we can obtain an increase of up to % in PDR by setting the threshol to its optimal value. Furthermore, we see that the relative increase in PDR is maximal if.3, but even for larger values of there can be a PDR enhancement. In aition to this, the maximal PDR improvement grows as grows, so the PDR policy is especially beneficial if the system is overloae. Likewise, in Fig. (b) we show the maximal PDR improve- ment for up to. In this region, the increase in PDR can be as much as 3%. Furthermore, the value of with the largest relative increase in PDR, as well as the range of for which gain is achieve become smaller. This is conveye by Fig. 6, which plots the gain for three increasing values of, equal to 3, 4 an 6. The narrower range of where gain is achieve is explaine by a shorter sojourn time of buffere packets that later enter service, ue to more frequent buffer overwriting for an increasing arrival rate. The shorter sojourn time reflects itself in a larger lea-time when the buffere packet is assesse, by the packet ropping mechanism, for ispatch to the transmitter. This in turn presents itself in a ecrease of the upper value of for which packet ropping still results in transmission enials for some packets, an as such, still results in PDR gain. We also present values of the improvement for specific values of an in Table I for a single sensor. We clearly see that, for this parameter region, the maximal PDR increase grows as becomes larger. Furthermore, given a fixe ealine, the maximal improvement increases up to a certain value of, an ecreases beyon that value. V. NUMERICAL RESULTS FOR A SENSOR CHAIN Accoring to our setup, each noe in the sensor chain generates locally a loa an also forwars the successfully receive ata packets from the previous noe. Here, we assume that this total loa follows again an exponential istribution. This assumption is necessary since if aopt a generalize istribution for the arrival rate at a noe, this will make the problem intractable. The close form analysis analysis was only possible for a single queue as we have seen in the previous sections. n= TABLE I: The maximal PDR improvement (%) for specific values of an.

5 n= noes n= noes TABLE II: The maximal PDR improvement (%) for specific values of an an for sensor chains of n= an n= noes. We now present values of the improvement for specific values of an in Table II for a chain of an sensors. Regaring the results, we see that as the number of noes in the chain is increase, the loa that must be forware naturally increases. Therefore, contrary to the case of low an n = (seen in Table I) where the PDR improvement of the propose scheme is not that important, in the case of n = it is. The reason is that each noe has to forwar an increase traffic loa an not only the low that is generate locally. As the number of noes becomes even higher, an in our case n becomes, the PDR improvement becomes even higher. Recall that the results are relative to the case that oes not apply proactive packet ropping. The same tren follows for higher where the significant performance benefits are preserve. The reason for the reuce importance of is ue to the fact that the ominates the system behavior. VI. CONCLUSIONS In this paper, we evelope an analytical moel that captures the impact of proactive packet ropping on the transmission of real-time packets from a single sensor noe. We showe that with the use of the analytical moel, an optimal ropping policy can be ientifie, leaing to PDR improvement of up to 3% for a single hop. When this moel is applie in the case of a multi-hop chain network of sensors, the PDR improvement is more significant. This is preominantly ue the ability of our moel to preict the elaye arrival of a packet an subsequently rop it before transmission. We plan to exten our moel to a system that exhibits ifferent service time istributions an larger buffer size. Nevertheless, as it seems from our current analysis these extensions are not trivial. REFERENCES [] Y. Liu, Y. He, M. Li, J. Wang, K. Liu, an L. Mo, Does wireless sensor network scale? a measurement stuy on greenorbs, in IEEE INFOCOM,. [] S. Olariu an I. Stojmenovic, Design guielines for maximizing lifetime an avoiing energy holes in sensor networks with uniform istribution an uniform reporting, in IEEE INFOCOM, 6. [3] X. Wu, G. Chen, an S. K. Das, Avoiing energy holes in wireless sensor networks with nonuniform noe istribution, IEEE TPDS, 8. [4] D. Barrer, Queueing with impatient customers an inifferent clerks, Operations Research, vol., no., pp , 97. [], Queueing with impatient customers an orere service, Operations Research, vol., no., pp. 6 66, 97. [6] A. Brant an M. Brant, On the m(n)/m(n)/s queue with impatient calls, Performance Evaluation, vol. 3, no. -, pp. 8, 999. [7], Asymptotic results an a Markovian approximation for the m(n)/m(n)/s+gi system, Queueing Systems, vol. 4, no. -, pp ,. [8] A. Movaghar, On queueing with customer impatience until the beginning of service, Queueing Systems, vol. 9, no. 4, pp , 998. [9], On queueing with customer impatience until the en of service, Stochastic Moels, vol., no., pp , 6. [] M. Kargahi an A. Movaghar, Non-preemptive earliest-ealine-first scheuling policy: A performance stuy, in IEEE International Symposium on Moeling, Analysis, an Simulation of Computer an Telecommunication Systems, Georgia, ATL, USA,, p.. [], A metho for performance analysis of earliest-ealine-first scheuling policy, Journal of Supercomputing, vol. 37, no., pp. 97, 6. [] S. Panwar, D. Towsley, an J. Wolf, Optimal scheuling policies for a class of queues with customer ealines to the beginning of service, Journal of the ACM, vol. 3, no. 4, pp , 988. [3] J. Lehoczky, Real-time queueing theory, in IEEE Real-Time Systems Symposium, Washington, DC, USA, 996, p [4], Using real-time queueing theory to control lateness in real-time systems, Performance Evaluation Review, vol., no., pp. 8 68, 997. [], Real-time queueing network theory, in IEEE Real-time Systems Symposium, San Francisco, CA, USA, 997, p [6] B. Doytchinov, J. Lehoczky, an S. Shreve, Real-time queues in heavy traffic with earliest-ealine-first queue iscipline, The Annals of Applie Probability, vol., no., pp ,. [7] L. Kruk, J. Lehoczky, S. Shreve, an S.-N. Yeung, Earliest-ealinefirst service in heavy-traffic acyclic networks, The Annals of Applie Probability, vol. 4, no. 3, pp. 36 3, 4. [8] R. Wolff, Poisson arrivals see time averages, Operations Research, vol. 3, no., pp. 3 3, 98. [9], Introuction to Queueing Theory. Prentice-Hall, Inc., 989.