Received: 23 September 2009 / Revised: 20 March 2011 / Published online: 15 June 2011 Springer Science+Business Media, LLC 2011

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1 Queueing Syst 2011) 68: DOI /s Linear loss networs Petar Momčilović Mar S. Squillante Received: 23 September 2009 / Revised: 20 March 2011 / Published online: 15 June 2011 Springer Science+Business Media, LLC 2011 Abstract This paper investigates theoretical properties of throughput and cost in linear loss networs. The maximum throughput of the networ with exponential service times is derived and the arrival process that maximizes throughput, given a fixed arrival rate, is established. For general service times, an asymptotically critical loading regime is identified such that the probability of an arbitrary customer being lost is strictly within 0, 1) as the networ size increases. This regime delivers throughput comparable to the maximum at a relatively low networ cost. The paper establishes the asymptotic throughput and networ cost under this critical loading. Keywords Linear loss networs Throughput Critical load Asymptotic properties Mathematics Subject Classification 2000) 60K25 1 Introduction Performance scalability is one of the central problems in designing networ protocols and architectures in practice. It is therefore of critical importance to understand fundamental properties of various performance characteristics as the size of the networ grows while the local resources of each node remain fixed. In this paper we focus P. Momčilović ) Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA petar@ise.ufl.edu M.S. Squillante Mathematical Sciences Department, IBM T.J. Watson Research Center, Yortown Heights, NY 10598, USA mss@watson.ibm.com

2 112 Queueing Syst 2011) 68: on explicit expressions for how various performance characteristics scale in large stochastic loss networs. More specifically, we investigate a linear loss networ in which customers enter the networ at a fixed source node, are relayed from one node to the next node in a fixed sequence of nodes, and exit the networ at a fixed destination node. Our stochastic tandem networ model is motivated by various large-scale networ applications. One specific example is large wireless networs [7] with limited or no cross-traffic interference along the paths from sources to destinations. Another motivating example is optical networs based on emerging trends [15]. Even though the model we consider is basic, it is interesting to understand fundamental performance properties such as how the throughput and cost scale in large networs. The main contributions of this paper are theoretical in nature and include the following new results for stochastic tandem networs. First, we establish that it is feasible to achieve an asymptotic throughput scalability of c/ in a linear -node loss networ as the size of the networ grows. The maximum achievable c in a networ with exponential service times is shown to be equal to the service rate multiplied by 1/ π. In addition, we identify an asymptotically critical loading regime for the linear networ. Under such loading, in the limit as the networ size increases, the probability of an arbitrary customer being lost is strictly within 0, 1). We argue that such a regime is relevant because it delivers high relative throughput at low cost. Explicit expressions for throughput and networ cost are also derived. As the input rate increases, it is further shown that the limiting networ cost per customer delivered) exactly doubles in comparison with the minimum cost possible. Our results have implications on various networ applications. For example, the derived results establish the limits of efficiency of end-to-end transport protocols in a linear loss networ. Achieving asymptotic throughput beyond c/ as the number of hops increase ) requires an implementation of bacpressure control [2] capable of delivering constant throughput in a linear networ [14]. The decrease in throughput as the networ size increases is due to the variability of service times and limited buffer space, not the interference due to cross traffic. The paper is organized as follows. In Sect. 2 we describe a model of a linear loss networ. The main results of the paper are presented in Sect. 3. Concluding remars can be found in Sect. 4. The last section provides proofs of the results presented throughout the paper, together with supporting lemmas. 2 Model We consider a tandem networ consisting of /GI/1/0 nodes indexed by 1, 2,..., see Fig. 1). Customers enter the networ at node 1 and they are relayed between nodes in the networ. Upon service completion a customer at the ith node joins the i + 1)st node. When it reaches the th node and is successfully serviced, the customer exits the networ. In this case we say that a customer reached the destination or is delivered. Customer service times are random with finite positive mean 1/μ and finite variance σ 2 ; service times are independent across nodes and time. The nodes employ a wor-conserving scheduling policy i.e., they service customers whenever feasible). In view of the fact that no buffering queuing) is allowed in the networ, a

3 Queueing Syst 2011) 68: Fig. 1 The linear loss networ consists of bufferless μ-rate servers buffer management customer dropping) policy needs to be specified. It is assumed that customer service times are not available to nodes. In particular, we consider two static policies: i) later arriving priority LAP), and ii) earlier arriving priority EAP). When the service times are exponential, the throughput results presented in this paper are insensitive to the buffer management policy. This follows from the fact that the remaining service time of a customer in service is equal in distribution to the service time of a newly arrived customer. Furthermore, the results on the asymptotically critical regime are insensitive to the buffer management policy even when service times are generally distributed. We note that our model can be viewed as a particle system [12, 13]. The process considered in this paper is complementary to the totally asymmetric exclusion process, which corresponds to a bufferless networ with a bacpressure flow control so that no customers are lost at the expense of blocing [14]. Another complementary relation with particle systems concerns coalescing random wals on the integers [1, 3, 10]. Informally, there is a connection between the asymptotic maximum throughput in a linear loss networ with exponential service times limit 2) in Theorem 1) and the asymptotic dispersion rate for coalescing random wals in [3]. However, it is important to point out that the model considered in this paper differs from the various models considered in the research literature on coalescing random wals. Throughout the paper we assume that the arrival input) process to the networ i.e., the first node of the networ) is stationary and ergodic. Given this assumption, the output process is stationary and ergodic as well. Hence, the throughput θ is defined as the long-term rate at which customers depart from the networ i.e., the th node of the networ). The maximum throughput θ is defined as the throughput achieved under the assumption that there is an infinite buffer in front of the first node; that is, the first node is never idle but no customers are dropped at that node. Note that in this case the departure rate from the first node is equal to μ. Wealso examine the networ under a renewal input process with interarrival times equal in distribution to an almost surely a.s.) positive random variable τ ; the input rate is given by 1/Eτ 0, ). We use the notation θ τ) to indicate the dependency of the throughput on τ the distribution of τ ). 3 Main results This section presents the main results of the paper that characterize properties of throughput and cost in linear -node loss networs. These properties are studied for fixed and as. Our first result describes throughput properties of a linear networ with exponential service times. Namely, it characterizes θ and identifies the throughput maximizing arrival process given a fixed arrival rate, i.e., the optimal customer pacing.

4 114 Queueing Syst 2011) 68: Theorem 1 The maximum throughput θ in the linear -node networ with exponential service times satisfies ) 2 2 θ = μ 4 +1, 1) 1 implying that, as, θ μ/ π. 2) Furthermore, for a fixed input rate T 1, the throughput of the linear loss networ with exponential service times is maximized when the customer interarrival times are deterministic: max τ:eτ=t θ τ) = θ T ). Proof See Sect. 5. Remar 1 The limit 2) follows from 1) and Stirling s approximation. Hence, μ/ π serves as a relatively accurate approximation for θ even for small values of. Remar 2 The considered networ under Poisson input can be analyzed as a continuous-time Marov process on the finite state space {0, 1} with a well-defined stationary distribution from which the throughput can be derived). An application of standard matrix-analytic methods [11] reveals that this stationary distribution for each has a type of matrix-geometric form in terms of distinct rate matrices which can be computed numerically. We further note that the departure process from a node in the networ has an important dependence structure, and this dependency does not vanish as the size of the networ increases. Theorem 1 provides motivation for an asymptotic regime that we consider next. Note that, for large, the maximum throughput θ is negligible in comparison with the corresponding input rate. This results in an inefficiency since the fraction of customers that are lost approaches 1 as increases indefinitely. The following theorem identifies a limiting regime in which the input and output throughput) rates are proportional, i.e., the probability of an arbitrary customer being lost in the networ is strictly within 0, 1) asymptotically as the networ size increases. Formally, if the renewal arrival process for a -node networ is defined by a interarrival) random variable τ, then a sequence of linear networs indexed by the number of nodes) is in the critical regime if lim θ τ )Eτ 0, 1). Customer loss probabilities of 0 and 1 correspond to the underloaded and overloaded regimes, respectively, and, hence, the reason we refer to the above regime as critical. Despite the fact that the offered load in the critical regime is negligible in comparison with the load that yields the maximum throughput θ, the achieved throughput is a non-negligible fraction of θ. Therefore, the regime is desirable from the operational

5 Queueing Syst 2011) 68: point of view since it balances two conflicting goals: i) achieving high throughput, and ii) maintaining low loss probability. Theorem 2 Consider a -node linear loss networ, where service times are i.i.d. with finite positive mean 1/μ and finite variance σ 2. Let ξ be an a.s. positive random variable. Then the throughput θ satisfies, as, ) 2/π θ ξ ˆθξ) E Eξ ξ/ 2σ 2 0 e x2 /2 dx. Proof See Sect. 5. Remar 3 In the case of deterministic service times σ = 0) the behavior of the maximum throughput θ is fundamentally different since losses can occur only at the first node. The maximum throughput θ is independent of the number of nodes and satisfies θ = θ 1/μ) = μ. Formally, the critical loading for such a networ occurs when the interarrival times are Θ1) rather than Θ ), as. 1 Nevertheless, Theorem 2 holds under deterministic interarrival times as well ˆθξ)= 1/Eξ). Remar 4 The expression for the limiting scaled) throughput ˆθξ) explicitly depends on the service time variance σ 2, but not on the mean 1/μ. For example, the theorem indicates that limiting throughput is the same in a networ with exponential service times with rate μ and a networ with service times uniformly distributed in [0, 2 3/μ] σ = 1/μ in both cases). It is appropriate to represent σ as a product of the mean of the service times and their coefficient of variation. Then, for a fixed mean, the larger the coefficient of variation, the lower the limiting throughput. As a consequence, deterministic service times maximize ˆθξ) among all service distributions with a fixed mean. The preceding theorem allows one to obtain explicit expressions for the limiting scaled throughput for specific renewal processes as illustrated by the following corollary. Corollary 1 Deterministic arrivals) If ξ is a constant, then, as, ) θ ξ 2/ πξ 2) ξ/ 2σ 2 e x2 /2 dx. Poisson arrivals.) If ξ is exponential, then, as, θ ξ ) 2/π Eξ 0 e σ/eξ)2 2σ/Eξ e x2 /2 dx. 1 We use the standard asymptotic notation; see for example [4,Sect.I.3.1].

6 116 Queueing Syst 2011) 68: Fig. 2 The ratio of throughputs in the limit as ) under the deterministic and exponential interarrival processes with the same rate; the random variable ξ is exponentially distributed in this example Jensen s inequality yields ˆθξ) ˆθEξ) implying that deterministic arrivals are throughput optimal in the critical limiting regime for networs with general service times; recall from Theorem 1 that deterministic arrivals are also throughput optimal for finite networs with exponential service times. As can be deduced from Fig. 2,the asymptotic reduction in throughput due to Poisson input compared to the deterministic arrival process) is at most 30% whereas Fig. 2 plots the throughput ratio rather than the throughput reduction). It should be noted that in the case of exponential service times σ = 1/μ) Theorem 2 is consistent with Theorem 1. Namely, ˆθξ/n) is monotonically increasing in n and, as n, ˆθξ/n) 1/ πσ 2, 3) where ξ is such that Eξ 0, ). Next we consider the asymptotic networ cost. Let ψ τ) be the average number of successful service completions per customer delivered as a function of the distribution of interarrival times τ ). Note that if a customer is not lost in the networ then this corresponds to service completions. However, even lost customers account for some successful transmissions: if a customer is lost at the ith node, then i 1) service requirements were completed. In many applications, such as wireless networs, it is appropriate to thin of ψ as an energy cost expended by the networ in delivering a customer. Hence, we shall refer to ψ as the networ cost, the asymptotic properties of which are characterized in the following theorem. Theorem 3 Consider a -node linear loss networ, where service times are i.i.d. with finite positive mean 1/μ and finite variance σ 2. Let ξ be an a.s. positive random variable. Then the networ cost ψ satisfies, as, ) Eξ 2 ξ/ x 2 e 2σ ψ ξ / ˆψξ) x2 /2 dx 2σ 2 E ξ/. 2σ 2 0 e x2 /2 dx Proof See Sect. 5.

7 Queueing Syst 2011) 68: Fig. 3 The limiting scaled networ cost ˆψξ) under deterministic arrivals ξ is a constant) as a function of the parameter σ/ξ In the limiting regime as the input rate increases the scaled cost tends to 2: ˆψξ/n) 2, as n,forξ such that Eξ 0, ). The quantity ˆψ for deterministic arrivals ξ = Eξ) is plotted in Fig. 3 as a function of the parameter σ/ξ. Theorem 3 can be interpreted as follows. In the light-load regime the loss probability is 0 asymptotically), and, hence, the networ cost of delivering a single customer is units the minimum value), i.e., ψ / 1. As the input rate increases so does the loss probability, and, therefore, the networ cost due to lost customers increases the networ cost of delivered customers. As the input rate increases the cost per delivered customer doubles see the preceding limit). Note that ψ depends not only on the customer loss probability but also on where in the networ on average) the customers are lost. Finally, we observe that there exists a tradeoff between throughput and networ cost in the critical regime. That is, higher throughput leads to higher networ cost. In order to quantify this tradeoff, we define ψ ξ)/ψ as the relative networ cost where ψ = is the minimum networ cost) and θ ξ)/θ as the relative throughput motivated by 3) we approximate θ by 1/ πσ 2 for finite ). Theorems 3 and 2 yield ψ ξ)/ψ ˆψξ) [1, 2] and θ ξ)/θ πσ ˆθξ) [0, 1], as, respectively; observe that both ˆψ and πσ ˆθ depend on the input variable ξ in fact, the ratio ξ/σ). By varying the mean of ξ the input rate) it is feasible to numerically evaluate the function Ψ πσ ˆθ)= ˆψ, where Ψ ) quantifies the throughput-cost tradeoff. To illustrate this point we plot the function Ψ πσ ˆθ) for deterministic arrivals in Fig. 4; note that the networ cost is barely above the minimum value for relative throughputs below 1/2 of the maximum. As is readily verified and can be observed from the figure, the function is increasing, strictly convex and continuously differentiable for all πσ ˆθ [0, 1]. Furthermore, assuming a profit function of the form r πσ ˆθξ) ˆψξ) for some r 0, it can be shown by standard Lagrangian methods that the unique optimum of this profit maximization problem satisfies Ψ = r. This general framewor supports a wide variety of profit

8 118 Queueing Syst 2011) 68: Fig. 4 The throughput-cost tradeoff for the limiting system ) under deterministic arrivals, where the quantity πσ ˆθξ) represents the fraction of the maximum throughput while ˆψξ) is the multiplicative increase in networ cost required to deliver a customer. As is apparent, the function is differentiable on 0, 1) with the derivative increasing from 0 lower-left corner) to upper-right corner) functions for which the optimal solution of the throughput-cost tradeoff can be determined through our general approach. 4 Concluding remars Motivated by various large networ applications with limited local resources at each node that do not grow with the size of the networ, we have investigated fundamental properties of performance in linear loss networs and the asymptotic behavior of these properties as the size of the networ grows. For a linear loss networ with exponential service times, we derived an expression for the maximum throughput of the networ and further established the arrival process that maximizes throughput, given a fixed arrival rate. For linear loss networs with generally distributed service times, we introduced a critical regime defined by the fact that the asymptotic loss probability is strictly within 0, 1) as the networ size increases. We established the asymptotic throughput and networ cost under such a critical loading. We conjecture that the maximum throughput θ obeys θ 1/ πσ 2,as. Moreover, we conjecture that polynomial throughput decay occurs in linear loss networs with arbitrary but fixed and finite buffer sizes. 5 Proofs Consider a -node networ with customers arriving at the networ labeled by integers in increasing order according to times of their arrival customer i Z arrives at time t i ). Let L i t) be the index of the node at which customer i is receiving service at time t provided that it is in the networ at time t). In order to properly define right-continuous {L i t), t R} we use the following two conventions: Non-existent nodes 0 and + 1) are used to indicate that a customer has not entered the networ and has been delivered, respectively. That is, L i t) = 0for t<t i, and L i t) = + 1fort d i if d i is the delivery time of customer i.

9 Queueing Syst 2011) 68: In the case that customer i is displaced from the networ by customer j at time l i, we set L i t) = L j t) for t l i; note that this necessarily implies L n t) = L j t) for t l i and min{i, j} n max{i, j}. Finally, assuming that customer 0 enters the networ at time t = 0 we define a stopping time: γ = inf { t 0 : L υ t) = L 0 t)}, 4) where υ = 1 and υ = 1 for the LAP and EAP policy, respectively. 5.1 Proof of Theorem 1 First we present a preliminary lemma. Lemma 1 Consider a -node linear networ with exponential service times. For an arbitrary sequence of arrival times we have, for 0 n<and t 0, P [ L i t) = n + 1 t i = 0 ] = e μt μt) n /n!. Remar 5 Observe that under the LAP policy if no customer enters the networ after customer i then P[L i t) = n + 1 t i = 0,t i+1 = ]=e μt μt) n /n!, since n service requirements need to be completed for the customer to reach the n + 1)st node; similarly for the EAP policy we have P[L i t) = n + 1 t i = 0,t i 1 = ]= e μt μt) n /n!. Effectively, the lemma states that in order to determine whether customer i is dropped, it is sufficient to consider only customer i + 1) for the LAP policy or customer i 1) for the EAP policy. That is, although customer i can be displaced by any customer with an index higher lower) than i, one can consider only customer i + 1) customer i 1)) and disregard all other customers. Proof of Lemma 1 The proof is based on the memoryless property of the exponential distribution. Consider the LAP policy the argument for the EAP policy is almost identical. Suppose that customer j displaces customer i at time l i from node n. The service time of customer j at node n is equal in distribution to the remaining beyond time l i ) service time of customer i at the same node. Hence, the evolution of L i t) for t l i is probabilistically the same regardless of whether either customer i or customer j is displaced. Proof of Theorem 1 Recall that when service times are exponential then the two considered static buffer management policies are equivalent as far as throughput results are concerned. Without loss of generality we assume the LAP policy. Define p i,, 2 i, to be the probability that a given customer, which at a fixed time is at the ith node while there is a customer at node 1 and there are no customers at nodes 2,...,i 1, reaches the destination upon completing service at node. For notational convenience we set p i, 1, for i>, and p 1, 0, for 1. Two properties of {p i, } i, are relevant to our analysis: i) conditioning on whether the customer at node i or node 1 first completes service renders the recursion p i, = 1 2 p i+1, p i 1, 1, 5)

10 120 Queueing Syst 2011) 68: which follows from the fact that both service times are exponential random variables with the same mean; ii) the sequence of {p i, } for fixed i is non-increasing, i.e., p i, 1 p i,. 6) Proof for the maximum throughput The proof consists of two parts. First, the problem of finding the maximum throughput is recast as a problem of determining a quantity related to a symmetric random wal on the integers. Second, the random-wal problem is addressed. The throughput of the G/M/1/0 system is monotonic in the arrival sequence [8]it is essential that the service times are exponential). Let {A 1) i,i N} and {A 2) i,i N} be two sequences of arrival times, and θ 1) and θ 2) be the corresponding throughputs. In particular, if the system is initially empty, then {A 1) i,i N} {A 2) i,i N} implies θ 1) θ 2). Hence, for the purposes of finding the maximum throughput θ, it is sufficient to assume that the first node in the networ is saturated, i.e., there is always a customer at the first node. In this case the departure process from the first node is a rate-μ Poisson process. Then the maximum throughput satisfies θ = μp 2,, 7) i.e., it is equal to the rate at which eventually delivered customers arrive to the second node in the networ. An expression for p 2, can be obtained by combinatorial arguments. To this end, consider a customer that just arrived to the second node, say at t = 0, without loss of generality; term this customer as the first customer. Since the first node is saturated there is a customer at the first node as well; term this as the second customer. In view of Lemma 1 see also Remar 5) it is sufficient to consider only these two customers, i.e., we can assume that no new customers are input to the networ. Then p 2, is the probability that the second customer does not displace the first customer in the networ. Now, consider two infinite linear networs = ) the first customer is in the first networ at node 2 at t = 0, the second customer is in the second networ at node 1att = 0, and there are no other customers in either networ. The service times of the customers in the two new networs are identical to those in the original networ. Let Yt) be the difference of occupied node indices between the first and second networs at time t minus 1, and thus initially Y0) = 0. In addition, define {c i } i 1 as an increasing sequence of times at which either customer completes service at some node. Then p 2, = P [ Yc i ) 0, i 2 2 ] 8) since by time c 2 2 the first customer is either lost in the original networ or has completed service at the th node. Observing that {Yc i )} i 1 is a symmetric random wal due to the exponential nature of the service times) together with 8) result in [5, p. 77] ) 2 2 p 2, = The statement 1) follows from 7) and the preceding equality. The limit 2) isdue to 1) and Stirling s approximation for the factorial function.

11 Queueing Syst 2011) 68: Proof for the optimal pacing The throughput θ τ) can be expressed as a product of the probability p that an arbitrary customer is not lost in the -node networ and the rate T 1 at which the customers arrive to the first node. The probability p can be established by considering how far in the networ a given customer reaches by the time a new arrival occurs at the first node. To this end, a given customer reaches node i = 1,..., at the time of the following customer s arrival to the first node with probability E[π i 1 τ)], where π i x) := e μx μx) i /i!. Indeed, E[π i 1 τ)] is the probability that i 1) service completions occur between two arrivals, in which case a customer is at node i when the next customer arrives at node 1. For example, the customer of interest is lost at the first node with probability E[π 0 τ)]= Ee μτ, i.e., the service time at the first node exceeds the interarrival time between the customers. On the other hand, a customer leaves the last th) node in the networ before the next customer enters the networ with probability E i= π i τ). This line of reasoning leads to [ θ τ) = p /T = 1 1 ] T E p i+1, π i τ) + π i τ). 9) i=1 Combining E i=0 π i τ) = 1 with 9) renders the following alternative expression for the throughput: i= θ τ) = 1 T 1 T 1 1 p i+1, )E [ π i τ) ] ; i=0 recall that p 1, 0, for 1. Next, consider θ ) :[0, ) [0, ) as a function of a real-valued argument. Since the function is twice differentiable on [0, ), it is straightforward to obtain its second derivative: d 2 θ x) dx 2 = μ2 e μx T 1 i=0 μx) i q i, 10) i! where 1 p,, i = 1, q i = 2p, p 1, 1, i = 2, p i+3, + 2p i+2, p i+1,, 0 i 3. Utilizing 5), the coefficients {q i } i can be rewritten as 1 p,, i = 1, q i = p 1, 1 p 1,, i = 2, p i+1, 1 p i+1,, 0 i 3, and it then follows from 6) that all q i,1 i<, are nonnegative: q i = p i+1, 1 p i+1, 0.

12 122 Queueing Syst 2011) 68: In view of the preceding inequality and 10), we can conclude that θ ) is concave. Therefore, Jensen s inequality is applicable, and combining it with 9) yields θ τ) θ Eτ)= θ T ). This completes the proof of the theorem. 5.2 Proofs of Theorems 2 and 3 Before providing the proofs of the two theorems we state and prove a number of preliminary results. Let {s i } i 1 be an i.i.d. sequence of nonnegative random variables with Es i = 1/μ, Vars i ) = σ 2, and define a right-continuous counting process {Xt), t 0} based on this sequence: Xt) = inf{i 0 : s 1 + +s i >t}. Furthermore, define {Yν t), t 0} as Y ν t) = {Ẋt ν), t ν +, 0, 0 t< ν +, where {Ẋt),t 0} is an independent copy of {Xt),t 0}, ν is an a.s. finite random variable independent of the two processes, and ) + denotes the positive-part operator. Note that Yν t) = Ẋt ν), t 0, on the event {ν 0}. Two stopping times are associated with the defined processes: T = inf{t 0 : Xt) + 1} and Tν = inf{t 0 : Xt) Yν t)}. The following lemma characterizes these two stopping times in the limit as, where denotes the minimum operator. Lemma 2 We have, as, and P [ T ν T ] 2/πE EX T ν T ) / 2/πE ν + / 2σ 2 0 ν + / 2σ πσ 2) 1 E ν + ) 2 e x2 /2 dx e x2 /2 dx ν + / 2σ 2 x 2 e x2 /2 dx. Proof First, observe that event {ν >0} implies {Yν t) = 0, 0 t< ν + }. Since Xt) 1, t 0, it follows that { T ν T } { { = inf Xt) Y ν t) } } > 0 ν + t<t { { } = inf Xt) Ỹ 0 t<t ν t)} > 0, 11)

13 Queueing Syst 2011) 68: where Ỹν t) = Y ν t), fort ν +, and Ỹν t) = μ ν + μt, for0 t< ν + ; observe that Ỹν t) Y ν t) = 0, for 0 t< ν +. Second, introducing scaled processes { ˆX t) = Xt) μt)/, t 0} and {Ŷν t) = Ỹ ν t) μt)/, t 0} leads to { inf Xt) Ỹ 0 t<t ν t)} = { inf ˆX t) Ŷ 0 t< T ν t)}, 12) where T = T / 1/μ 13) a.s., as, due to the strong law of large numbers. The process {Ỹν t), t 0} is defined in such a way that Ŷν t) = μν,for0 t<ν+ /. Third, the functional central limit theorem for counting processes for example, see [16, p. 236]) yields { ˆX t), t 0} {μ 3/2 σ Bt), t 0} and {Ŷν t), t 0} {μ3/2 σ Ḃt) μν, t 0}, as,ind[0, ), J 1 ), where {Bt),t 0} and {Ḃt),t 0} are independent standard Brownian motions. These two limits, the independence of { ˆX t), t 0} and {Ŷν t), t 0}, the continuity of Brownian paths, the continuity of the inf operator and the continuous mapping theorem further imply, as, { inf 0 s t { ˆX s) Ŷ ν s)},t 0} { inf 0 s t { 2μ 3 σ 2 Bs) + μν },t 0}. 14) Fourth, combining 12), 13) and 14) renders, as, P [ { inf Xt) Ỹ 0 t<t ν t)} > 0 ] [ ] P inf Bt) > b, 15) 0 t<1/μ where b = ν/ 2μσ 2. Now, the reflection principle and the strong Marov property of Brownian motion [9, p. 80] result in [ ] P inf Bt) > b = b + μ 2/πE e x2 /2 dx. 16) 0 t<1/μ Finally, 11), 15) and 16) imply the first statement of the lemma. Next, we consider the second statement of the lemma. To this end, let X t) = Xt)/, T ν = T ν / and observe that EX Tν T ) / = E X T ν T ). 17) Due to the functional strong law of large numbers for counting processes, we have X t) μt 0 18) sup 0 t T a.s., as, for any fixed T. On the other hand, 14) implies T ν τ,as, where τ = inf{t 0 : Bt) b}. This wea limit and 13) further yield E T ν T ) Eτ μ 1 ),as, which combined with 17) and 18) results in, as, 0 EX T ν T ) / Eμτ 1). 19)

14 124 Queueing Syst 2011) 68: Since the distribution of τ is nown for example, see [9, p. 80]), it is straightforward to obtain Eμτ 1) = b + μ 2/πE e x2 /2 dx + 2/πμE b +) 2 x 2 e x2 /2 dx. 0 b + μ Finally, the statement follows from the preceding equality and 19). Now, for a finite sequence of nonnegative reals {x i } i=1 define a right-continuous counting process {W{x} i=1 )t), t R} by W 0, t <0, {x i } ) i=1 t) = i, i 1 j=1 x j t< i j=1 x j, + 1, t j=1 x. The following lemma provides a comparison of two processes in terms of underlying sequences. Lemma 3 Let {x i } i=1 and {ẋ i} i=1 be sequences of nonnegative reals such that x i ẋ i x for i S {1,...,} and x i =ẋ i for i S. Then, for t R, W {ẋ i } i=1) t x S ) W {xi } i=1) t) W {ẋi } i=1) t + x S ). 20) Proof The proof is by induction on the size of the set S. The base case is provided by the instance S =0; then x i =ẋ i for i {1,...,} and the statement holds trivially. Assume that 20) holds with S =j for some j<. Now, consider S =j + 1 and let i = max{i : i S}. Define a third sequence {ẍ i } i=1 by ẍ i =ẋ i for i i and ẍ i = x i for i = i. Then the inductive assumption yields, for t R, W {ẍ i } i=1) t xj ) W {xi } i=1) t) W {ẍi } i=1) t + xj ). However, the inductive assumption also renders, for t R, W {ẋ i } i=1) t x) W {ẍi } i=1) t) W {ẋi } i=1) t + x). Combining the preceding two equalities results in W {ẋ i } i=1) t xj + 1) ) W {xi } i=1) t) W {ẋi } i=1) t + xj + 1) ), which completes the proof. Before stating the last preliminary result we introduce some additional notation. Consider the -node networ. Let ξ i be the interarrival time between the customers i 1) and i, fori 1, and i and i + 1), fori<0. For notational simplicity let ν i = i j=1 ξ sgni) j, i N. Without loss of generality, assume that customer 0 enters the networ at time t = 0. In this case, sgni) ν i can be interpreted as the time at which customer i enters the networ, where sgn0) = 0. Next, let s i,j, i Z, j 1,

15 Queueing Syst 2011) 68: be the service time of customer i at node j if customer i had priority over all other customers or, equivalently, if customer i were the only customer to enter the networ). Note that it is feasible to define s i,j even though customer i might not receive service at node j due to losses. When customer i is lost at node j, then s i,n, n<j, represents the actual service time of customer i at node n. We define the process {Zi t), t R} by setting Z i t) to be the potential location node index) of customer i at time t in the -node networ if customer i had priority over all other customers: Z i t) = inf {j 0 : j s i,n >t sgni) ν i }. n=1 Here, events {Zi t) = 0} and {Z i t) > } indicate that the customer has not entered the networ at time t and has exited the -node networ by time t, respectively. It is convenient to define the following two stopping times: τ = inf { t 0 : Z 0 t) > } 21) and τi = inf { t 0 : Zi t) = Z 0 t)}. 22) Service times {s i,j } i,j are based on customers, i.e., servers process customers at unit rates and customer demands are random. Equivalently, one can define service times based on nodes, i.e., customers have unit service demands and service rates are random. In particular, we set ṡ i,j, i 0, to be the uninterrupted service time at node j of the customer with the ith lowest highest) positive negative) index when i is positive negative); here, uninterrupted service time means the service time of the customer if that customer had priority over all other customers after its arrival at node j. For example, suppose that customers 0, 3, 5 and 9 arrive consecutively at node j at times 10, 11, 12 and 18 with corresponding service requirements times) 2, 4, 4 and 10, respectively; customers 1, 2, 4, 6, 7 and 8 are lost in the networ prior to reaching node j. Then, assuming the LAP policy, customers 0 and 3 are lost at node j after being in service one time unit each), and s 0,j = 2, s 3,j = s 5,j = 4, s 9,j = 10 and ṡ 1,j =ṡ 2,j = 4, ṡ 3,j = 10. Note that the sequences {ṡ 1,j } j=1 and {ṡ 1,j } j=1 are i.i.d., independent of {s 0,j } j=1 due to independence of service times across time and nodes. In the proofs of Theorems 2 and 3 we use the fact that the quantity ṡ i, = i 1) max satisfies for every δ>0, as, To this end, since 1 n i, 1 j ṡsgni) n,j P [ ṡ i, >δ ] 0. 23) P [ ṡ i, >δ ] = 1 1 P [ ṡ sgni),1 >δ / i 1 )]) i, for 23) to hold it is sufficient to have P[ṡ sgni),1 >δ ] 0, as, for any δ>0. However, this limit follows from Eṡsgni),1 2 < for example, see [6,p.150]).

16 126 Queueing Syst 2011) 68: Now, for υ { 1, 1}, we define {Vυ t), t R} by V υ t) = inf {j 0 : j ṡ υ,n >t υ ξ υ }. n=1 Recall the definitions of γ, τ and τi in 4), 21) and 22), respectively. In addition, stopping times τυ s) = inf{ t 0 : Vυ t + s) = Z 0 t)} 24) and { ζi s) = inf t 0 : Vsgni) t + s) = sup will be used to state the last preparatory lemma. sgni)j> i Z j t) } Lemma 4 For j>0, the following relationships hold: { } γ <τ inf τ i { τ ) υ υṡ j, γ <τ } 25) υi> j and { ) ) τ υ υṡ j, <ζ υj υṡ j, τ } { γ τυ ) υṡ j, <τ }, 26) where υ = 1 and υ = 1 for the LAP and EAP policy, respectively. Proof Consider 25) first. For notational simplicity let E 1 ={γ <τ inf υi> j τi }. It is sufficient to show that, on the event E 1, L υ t) V υ ) t + υṡ j, 27) for all 0 t γ, since 27) implies {τυ υṡ j, ) γ } by definitions 4) and 24). For every realization of service and arrival times there exists γ such that Vυ t) = W {ṡ υ,i } ) ) γ i=1 t υ ξυ 28) for 0 t γ ; on the event {τ <γ } variable γ can be interpreted as the node where customer 0 is displaced from the networ. Furthermore, there exists a sequence { s υ,i } γ i=1 such that, for 0 t γ, L υ t) = W { s υ,i } i=1) t υ ξυ ). 29) Under the LAP policy, s 1,i represents the length of the time interval between the arrival of the first customer with a positive index at node i and the time when node i becomes idle for the first time thereafter; a similar interpretation exists for the EAP policy. For example, assume LAP, ṡ 1,i =ṡ 2,i = 2 and ṡ 3,i = 4 with the corresponding customers arriving at node i at times 10, 11 and 14; then, s 1,i = 3. Note that, on the event E 1, customer 0 is displaced from the networ by customer υi with 1

17 Queueing Syst 2011) 68: i j this follows from an observation that if customer 0 is displaced by customer υi then at least {τυi <τ }. This fact implies that S ={i :ṡ υ,i s υ,i } is such that S j; moreover, ṡ υ,i s υ,i ṡj, for i S since s υ,i j n=1 ṡυn,i if customer 0 is displaced from the networ by a customer υj, then before customer 0 is lost, service of the customer with the lowest highest) positive negative) index under LAP EAP) is interrupted by a new arrival at most j 1) times). Applying Lemma 3 yields 27) and, thus, 25) holds. The proof of 26) proceeds along similar lines. For notational simplicity let E 2 = {τυ υṡ j, )<ζ υj υṡ j, ) τ }. It is sufficient to show that, on the event E 2, V υ t υṡ j, ) L υ t) 30) for all 0 t γ, since 30) implies {γ τυ υṡ j, )}. Next we argue that, on E 2, customer 0 is displaced by a customer υi for some 1 i j. To this end, let ωυj be the earliest time when all customers with indices υ,2υ,...,jυ are lost; if at least one of these customers is delivered then we set ωυj =. Lemma 4 then implies Vυ ) t υṡ j, L υ t) Vυ ) t + υṡ j, for 0 t ωυj τ. Thus, on {τυ υṡ j, )<ζ υj υṡ j, )} E 2 one has ωυj τ, i.e., 30) holds. Next, for 0 t γ, L υ t) and V υ t) can be represented as 28) and 29) with S ={i :ṡ υ,i s υ,i }, S j and ṡ υ,i s υ,i ṡj, for i S. The conclusion follows from Lemma 3. Next we present the proof of Theorem 2. Proof of Theorem 2 Consider customer 0 and observe that P[γ τ ] is the probability that it is delivered not lost) in a -node networ since it can be displaced only by a customer with positive negative) indices under the LAP EAP) policy). Hence, the throughput satisfies θ ξ ) = P [ γ τ ] /Eξ. 31) Next we bound from above and below P[γ <τ ]=1 P[γ τ ]. We consider an upper bound first. Note that the union bound and Lemma 4 yield, for j 1 and δ>0, P [ γ <τ ] P [ τυ ) υṡ j, <τ ] [ + P inf τ υi> j i <τ ] P [ τυ ) υδ <τ ] + P [ ṡj, >δ ] [ + P inf τ υi> j i <τ ], 32) where the second inequality follows from the monotonicity of τυ ). Combining the union bound with Lemma 2 results in [ lim P inf τ i <τ ] 2/π E υi> j ν i / e x2 /2 dx. 33) 2σ 2 i=j+1

18 128 Queueing Syst 2011) 68: Since ξ>0 a.s., it follows that Ee ξ/ 2σ 2 < 1 and i=j+1 E ν i / e x2 /2 dx 2 2σ 2 i=j+1 as j. Combining 33) and 34) renders [ lim lim P inf j Ee ν i/ 2σ 2 = 2 Ee ξ/ 2σ 2 ) j+1 0, 34) 1 Ee ξ/ 2σ 2 υi> j τ i <τ ] = 0. 35) Taing lim as on both sides of 32), applying Lemma 2 and 23), then letting j and using the preceding limit result in lim P[ γ <τ ] 2/πE ξ δ) + / e x2 /2 dx. 2σ 2 An upper bound follows by letting δ 0 in the preceding inequality: lim P[ γ <τ ] 2/πE ξ/ e x2 /2 dx. 36) 2σ 2 A similar line of reasoning results in a lower bound. Namely, Lemma 4 and the union bound yield, for j 1 and δ>0, P [ γ <τ ] P [ τυ ) ) υṡ j, <ζ υj υṡ j, τ ] P [ τυ ) υṡ j, <τ ] P [ τυ ) )] υṡ j, ζ υj υṡ j, P [ τυ ) υδ <τ ] P [ τυ ) )] υδ ζ υj υδ 2P [ ṡj, >δ ], 37) where the last two inequalities are also due to the monotonicity of τ υ ) and ζ υ ). Next, invoing the union bound and Lemma 2 result in lim P[ τ ) )] υ υδ ζ υj υδ 2/π E i=j+1 ν i ξ 1 δ) + / e x2 /2 dx 2σ 2 = 2/π E i=j ν i δ) + / 2σ 2 e x2 /2 dx, where the equality is due to the i.i.d. nature of the sequence {ξ i } i Z. The preceding inequality and the same argument used in 34) yield lim lim P[ τ ) )] υ υδ ζ υj υδ = 0. 38) j

19 Queueing Syst 2011) 68: Taing lim as on both sides of 37), applying Lemma 2 and 23), then letting j and using 38) result in lim P [ γ <τ ] 2/πE ξ+δ)/ e x2 /2 dx. 2σ 2 Letting δ 0 in the preceding inequality yields the desired lower bound: lim P [ γ <τ ] 2/πE Finally, combining 36) and 39) renders lim P[ γ <τ ] = 2/πE ξ/ 2σ 2 e x2 /2 dx. 39) ξ/ 2σ 2 e x2 /2 dx. The statement of the theorem follows from the preceding limit and 31). We conclude the section with the proof of Theorem 3. Proof of Theorem 3 Without loss of generality consider customer 0 and note that the networ cost ψ satisfies ) EZ ψ ξ = 0 γ τ ) 1 Eξθ. 40) ξ) Now, Lemma 4 and the monotonicity of {Z0 t), t R} imply, for arbitrary j 1, EZ0 γ τ ) E [ Z0 γ τ ) ] 1 {γ <τ inf υi> j τi } E [ Z0 ) τ υ υṡ j, τ ) 1 {γ <τ inf υi> j τ EZ0 ) τ υ υṡ j, τ ) [ + 1)P inf τ i <τ ], υi> j where the last inequality is due to Z0 γ τ ) + 1). By restricting the set of values for ṡj, in the preceding inequality, one obtains the following lower bound for any δ>0: EZ0 γ τ ) E [ Z0 ) τ υ υṡ j, τ ) ] [ 1 {ṡ j, δ } + 1)P inf τ i <τ ] υi> j EZ0 ) τ υ υδ τ ) [ + 1) P inf τ i <τ ] υi> j + P [ ṡ j, >δ ]). 41) On the other hand, Lemma 4, the monotonicity of {Z 0 t), t R}, and Z 0 γ τ ) + 1) also imply, for arbitrary j 1 and δ>0, i } ]

20 130 Queueing Syst 2011) 68: EZ0 γ τ ) E [ Z0 γ τ ) ] 1 {τ υ υṡ j, )<ζ υj υṡ j, ) τ } + + 1)P [ τυ ) )] υṡ j, ζ υj υṡ j, EZ0 ) τ υ υṡ j, τ ) + + 1)P [ τυ ) )] υṡ j, ζ υj υṡ j, EZ0 ) τ υ υδ τ ) + + 1) P [ ṡj, >δ ] + P [ τυ ) )]) υṡ j, ζ υj υṡ j,. 42) Finally, combining 41) and 42), letting first and then j, invoing Lemma 2, 23), 35), 38), and then passing δ 0 yield lim EZ 0 τ γ ) / = 2/πE ξ/ 2σ πσ 2) 1 Eξ 2 e x2 /2 dx ξ/ 2σ 2 x 2 e x2 /2 dx. The statement of the theorem follows from the preceding limit, Theorem 2 and 40). Acnowledgements The authors than Devavrat Shah for pointing out reference [15]. We also than anonymous referees for helpful comments, in particular for bringing references [1, 3, 10] to our attention. P.M. thans Predrag Jelenović for discussions. P.M. was supported by the National Science Foundation under Grant CNS References 1. Arratia, R.: Limiting point processes for rescalings of coalescing and annihilating random wals on Z d. Ann. Probab. 96), ) 2. Bertseas, D., Gallager, R.: Data Networs, 2nd edn. Prentice-Hall, Englewood Cliffs 1992) 3. Bramson, M., Griffeath, D.: Clustering and dispersion rates for some interacting particle systems on Z 1. Ann. Probab. 82), ) 4. Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge 2001) 5. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. I. Wiley, New Yor 1968) 6. Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd edn., vol. II. Wiley, New Yor 1971) 7. Gupta, P., Kumar, P.R.: The capacity of wireless networs. IEEE Trans. Inf. Theory 462), ) 8. Jelenović, P., Momčilović, P., Squillante, M.: Scalability of wireless networs. IEEE/ACM Trans. Netw. 152), ) 9. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New Yor 1991) 10. Krapivsy, P., Redner, S., Ben-Naim, E.: A Kinetic View of Statistical Physics. Cambridge University Press, Cambridge 2010) 11. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM, Philadelphia 1999) 12. Liggett, T.: Interacting Particle Systems. Springer, New Yor 1985) 13. Liggett, T.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin, 1999)

21 Queueing Syst 2011) 68: Martin, J.: Large tandem queueing networs with blocing. Queueing Syst. Theory Appl ), ) 15. Saleh, A., Simmons, J.: Evolution toward the next-generation core optical networ. J. Lightwave Technol. 249), ) 16. Whitt, W.: Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New Yor 2002)