Abstract: This paper develops a rare-event simulation algorithm for a discrete-time version of the M/G/s loss system and a related Markov-modulated variant of the same loss model. The algorithm is shown to be efficient in the many-server asymptotic regime in which the number of servers and the arrival rate increase to infinity in fixed proportion. A key idea is to study the system as a measure-valued Markov chain and to steer the system to the rare event through a randomization of the time horizon over which the rare event is induced.

Abstract: This paper considers polling systems with an autonomous server that remains at a queue for an exponential amount of time before moving to a next queue incurring a generally distributed switch-over time. The server remains at a queue until the exponential visit time expires, also when the queue becomes empty. If the queue is not empty when the visit time expires, service is preempted upon server departure, and repeated when the server returns to the queue. The paper first presents a necessary and sufficient condition for stability, and subsequently analyzes the joint queue-length distribution via an embedded Markov chain approach. As the autonomous exponential visit times may seem to result in a system that closely resembles a system of independent queues, we explicitly investigate the approximation of our system via a system of independent vacation queues. This approximation is accurate for short visit times only.

Abstract: In IEEE 802.16e (air interface standard for MWiMAX) and IEEE 802.16m (evolution of MWiMAX for IMT-Advanced), power saving is one of the important issues for the battery-powered mobile stations (MSs). According to IEEE 802.16e standard, when an MS switches from awake mode to sleep mode, the MS is required to send a sleep request (MOB-SLP-REQ) message and to receive a sleep response (MOB-SLP-RSP) message. In this paper we propose a new sleep mode scheme, called the power saving mechanism with binary exponential traffic indications. This sleep mode scheme omits MOB-SLP-REQ/RSP messages and has a traffic indication (TRF-IND) interval as a main system parameter, applying the truncated binary exponential increasing method for its length. The proposed scheme in this paper is quite well aligned with the design policy of sleep mode in discussion at IEEE 802.16m in the sense that it tries to minimize the overhead for the state transition between awake mode and sleep mode, and hence it can reduce the delay due to the state transition and enhance the power saving efficiency. We present a mathematical analysis for the proposed scheme and investigate its performance. As performance measures, we provide the sleep interval ratio, the average power consumption, and the mean delay. Using the analytical results, the system parameters such as the initial TRF-IND interval and the maximum binary exponent for TRF-IND intervals can be optimized while satisfying the QoS constraint on the mean delay. The numerical results show that the proposed scheme consumes less energy than the power saving class of type I in the IEEE 802.16e standard.

Abstract: This paper considers a number of structural properties of reflected Levy processes, where both one-sided reflection (at 0) and two-sided reflection (at both 0 and K>0) are examined. With V t being the position of the reflected process at time t, we focus on the analysis of $\zeta(t):=\mathbb{E}V_{t}$ and $\xi(t):=\mathbb{V}\mathrm{ar}V_{t}$ . We prove that for the one- and two-sided reflection, ?(t) is increasing and concave, whereas for the one-sided reflection, ?(t) is increasing. In most proofs we first establish the claim for the discrete-time counterpart (that is, a reflected random walk), and then use a limiting argument. A key step in our proofs for the two-sided reflection is a new representation of the position of the reflected process in terms of the driving Levy process.

Abstract: We consider a (doubly) reflected Levy process where the Levy exponent is controlled by a hysteretic policy consisting of two stages. In each stage there is typically a different service speed, drift parameter, or arrival rate. We determine the steady-state performance, both for systems with finite and infinite capacity. Thereby, we unify and extend many existing results in the literature, focusing on the special cases of M/G/1 queues and Brownian motion.

Abstract: In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service discipline, i.e. only customers standing in front of the gate are served during this visit. In contrast, high priority customers arriving during the visit period of the queue are allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type. Through numerical examples we illustrate that the mixed gated/exhaustive service discipline can significantly decrease waiting times of high priority jobs. In many cases there is a minimal negative impact on the waiting times of low priority customers but, remarkably, it turns out that in polling systems with larger switch-over times there can be even a positive impact on the waiting times of low priority customers.

Abstract: Stochastic loss networks are often very effective models for studying the random dynamics of systems requiring simultaneous resource possession. Given a stochastic network and a multi-class customer workload, the classical Erlang model renders the stationary probability that a customer will be lost due to insufficient capacity for at least one required resource type. Recently a novel family of slice methods has been proposed by Jung et al. (Proceedings of ACM SIGMETRICS conference on measurement and modeling of computer systems, pp. 407---418, 2008) to approximate the stationary loss probabilities in the Erlang model, and has been shown to provide better performance than the classical Erlang fixed point approximation in many regimes of interest. In this paper, we propose some new methods for loss probability calculation. We propose a refinement of the 3-point slice method of Jung et al. (Proceedings of ACM SIGMETRICS conference on measurement and modeling of computer systems, pp. 407---418, 2008) which exhibits improved accuracy, especially when heavily loaded networks are considered, at comparable computational cost. Next we exploit the structure of the stationary distribution to propose randomized algorithms to approximate both the stationary distribution and the loss probabilities. Whereas our refined slice method is exact in a certain scaling regime and is therefore ideally suited to the asymptotic analysis of large networks, the latter algorithms borrow from volume computation methods for convex polytopes to provide approximations for the unscaled network with error bounds as a function of the computational costs.

Abstract: We analyze the behavior of Generalized Processor Sharing (GPS) queues with heavy-tailed service times. We compute the exact tail asymptotics of the stationary workload of an individual class and give new conditions for reduced-load equivalence and induced burstiness to hold. We also show that both phenomena can occur simultaneously. Our proofs rely on the single big event theorem and new fluid limits obtained for the GPS system that can be of interest by themselves.

Abstract: Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are classical circuit switch telephone networks (loss networks) and present-day wireless mobile networks. Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to upper bounds for loss probabilities and analytic error bounds for the accuracy of the approximation for various performance measures. The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications: pure loss networks as under (i) GSM networks with fixed channel allocation as under (ii). The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning.

Abstract: We consider piecewise-deterministic Markov processes that occur as scaling limits of discrete-time Markov chains that describe the Transmission Control Protocol (TCP). The class of processes allows for general increase and decrease profiles. Our key observation is that stationary results for the general class follow directly from the stationary results for the idealized TCP process. The latter is a Markov process that increases linearly and experiences downward jumps at times governed by a Poisson process. To establish this connection, we apply space---time transformations that preserve the properties of the class of Markov processes.

Abstract: Consider two servers of equal service capacity, one serving in a first-come first-served order (FCFS), and the other serving its queue in random order. Customers arrive as a Poisson process and each arriving customer observes the length of the two queues and then chooses to join the queue that minimizes its expected queueing time. Assuming exponentially distributed service times, we numerically compute a Nash equilibrium in this system, and investigate the question of which server attracts the greater share of customers. If customers who arrive to find both queues empty independently choose to join each queue with probability 0.5, then we show that the server with FCFS discipline obtains a slightly greater share of the market. However, if such customers always join the same queue (say of the server with FCFS discipline) then that server attracts the greater share of customers.

Abstract: This paper analyzes transient characteristics of Gaussian queues. More specifically, we determine the logarithmic asymptotics of P(Q0 >p B, Q TB >q B), where Qt denotes the workload at time t. For any pair (p, q), three regimes can be distinguished: (A) For small values of T , one of the events {Q0 >p B} and {Q TB >q B} will essentially imply the other. (B) Then there is an intermediate range of values of T for which it is to be expected that both {Q0 >p B} and {Q TB >q B} are tight (in that none of them essentially implies the other), but that the time epochs 0 and T lie in the same busy period with overwhelming probability. (C) Finally, for large T , still both events are tight, but now they occur in different busy periods with overwhelming probability. For the short-range dependent case, explicit calcula-

Abstract: We study in this paper an M/M/1 queue whose server rate depends upon the state of an independent Ornstein---Uhlenbeck diffusion process (X(t)) so that its value at time t is μ ?(X(t)), where ?(x) is some bounded function and μ>0 We first establish the differential system for the conditional probability density functions of the couple (L(t),X(t)) in the stationary regime, where L(t) is the number of customers in the system at time t By assuming that ?(x) is defined by ?(x)=1??((x ? a/?)?(?b/?)) for some positive real numbers a, b and ?, we show that the above differential system has a unique solution under some condition on a and b We then show that this solution is close, in some appropriate sense, to the solution to the differential system obtained when ? is replaced with ?(x)=1?? x for sufficiently small ? We finally perform a perturbation analysis of this latter solution for small ? This allows us to check at the first order the validity of the so-called reduced service rate approximation, stating that everything happens as if the server rate were constant and equal to $\mu(1-\varepsilon {\mathbb{E}}(X(t)))$

Abstract: A two-station network with controllable inputs and sequencing control, proposed by Wein (Oper. Res. 38:1065---1078, 1990), is analyzed. A control is sought to minimize holding cost subject to a throughput constraint. In a Lagrangian formulation, input vanishes in the fluid limit. Several alternative fluid models, including workload formulations, are analyzed to develop a heuristic policy for the stochastic network. Both the fluid heuristic and Wein's diffusion solution are compared with the optimal policy by solving the dynamic program. Examples with up to six customer classes, using Poisson arrival and service processes, are presented. The fluid heuristic does well at sequencing control but the diffusion gives additional, and better, information on input control. The fluid analysis, in particular whether the fluid priorities are greedy, aids in determining whether the fluid heuristic contains useful information.

Abstract: The approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several improvements and one correction are made, which enables the constructed bound to be significantly tighter. Several examples are given, showing how to implement the theoretical result.

Abstract: We consider a discrete-time Geo/G/1 retrial queue where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically geometric. Remarkably, the result is inconsistent with the corresponding result in the continuous-time counterpart, the M/G/1 retrial queue, where the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function.

Abstract: We consider the maximum queue length and the maximum number of idle servers in the classical Erlang delay model and the generalization allowing customer abandonment--the M/M/n+M queue. We use strong approximations to show, under regularity conditions, that properly scaled versions of the maximum queue length and maximum number of idle servers over subintervals [0,t] in the delay models converge jointly to independent random variables with the Gumbel extreme value distribution in the quality-and-efficiency-driven (QED) and ED many-server heavy-traffic limiting regimes as n and t increase to infinity together appropriately; we require that t n ?? and t n =o(n 1/2?? ) as n?? for some ?>0.

Abstract: This paper analyzes a generic class of two-node queueing systems. A first queue is fed by an on---off Markov fluid source; the input of a second queue is a function of the state of the Markov fluid source as well, but now also of the first queue being empty or not. This model covers the classical two-node tandem queue and the two-class priority queue as special cases. Relying predominantly on probabilistic argumentation, the steady-state buffer content of both queues is determined (in terms of its Laplace transform). Interpreting the buffer content of the second queue in terms of busy periods of the first queue, the (exact) tail asymptotics of the distribution of the second queue are found. Two regimes can be distinguished: a first in which the state of the first queue (that is, being empty or not) hardly plays a role, and a second in which it explicitly does. This dichotomy can be understood by using large-deviations heuristics.

Abstract: We consider a single-class queueing network in which the functional network primitives describe the cumulative exogenous arrivals, service times and routing decisions of the queues. The behavior of the network consisting of the cumulative total arrival, cumulative idle time, and queue length developments for each node is specified by conditions which relate the network primitives to the network behavior. For a broad class of network primitives, including discrete customer and fluid models, a network behavior exists, but need not be unique. Nevertheless, the mapping from network primitives to the set of associated network behavior is upper semicontinuous at network primitives with continuous routing. As an application we consider a sequence of random network primitives satisfying a sample path large deviation principle. We take advantage of the partial functional set-valued upper semicontinuity in order to derive a large deviation principle for the sequence of associated random queue length processes and to identify the rate function. This extends the results of Puhalskii (Markov Process. Relat. Fields 13(1), 99---136, 2007) about large deviations for the tail probabilities of generalized Jackson networks. Since the analysis is carried out on the doubly-infinite time axis ?, we can directly treat stationary situations.

Abstract: Transient solutions for M/M/c queues are important for staffing call centers, police stations, hospitals and similar institutions. In this paper we show how to find transient solutions for M/M/c queues with finite buffers by using eigenvalues and eigenvectors. To find the eigenvalues, we create a system of difference equations where the coefficients depend on a parameter x. These difference equations allow us to search for all eigenvalues by changing x. To facilitate the search, we use Sturm sequences for locating the eigenvalues. We also show that the resulting method is numerically stable.

Abstract: The purpose of this paper is to survey techniques for constructing effective policies for controlling complex networks, and to extend these techniques to capture special features of wireless communication networks under different networking scenarios. Among the key questions addressed are: The relationship between static network equilibria, and dynamic network control. The effect of coding on control and delay through rate regions. Routing, scheduling, and admission control. Through several examples, ranging from multiple-access systems to network coded multicast, we demonstrate that the rate region for a coded communication network may be approximated by a simple polyhedral subset of a Euclidean space. The polyhedral structure of the rate region, determined by the coding, enables a powerful workload relaxation method that is used for addressing complexity--the relaxation technique provides approximations of a highly complex network by a far simpler one. These approximations are the basis of a specific formulation of an h-MaxWeight policy for network routing. Simulations show a 50% improvement in average delay performance as compared to methods used in current practice.