Showing papers in "Queueing Systems in 1999"
TL;DR: A queueing system with r non‐identical servers working in parallel, exogenous arrivals into m different job classes, and linear holding costs for each class is considered, and the Brownian solution suggests the following: virtually all backlogged work should be held in one particular job class.
Abstract: We consider a queueing system with r non-identical servers working in parallel, exogenous arrivals into m different job classes, and linear holding costs for each class Each arrival requires a single service, which may be provided by any of several different servers in our general formulations the service time distribution depends on both the job class being processed and the server selected The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs onto available servers A linear program involving only first-moment data (average arrival rates and mean service times) is used to define heavy traffic for a system of this form, and also to articulate a condition of overlapping server capabilities which leads to resource pooling in the heavy traffic limit Assuming that the latter condition holds, we rescale time and state space in standard fashion, then identify a Brownian control problem that is the formal heavy traffic limit of our rescaled scheduling problem Because of the assumed overlap in server capabilities, the limiting Brownian control problem is effectively one-dimensional, and it admits a pathwise optimal solution That is, in the limiting Brownian control problem the multiple servers of our original model merge to form a single pool of service capacity, and there exists a dynamic control policy which minimizes cumulative cost incurred up to any time t with probability one Interpreted in our original problem context, the Brownian solution suggests the following: virtually all backlogged work should be held in one particular job class, and all servers can and should be productively employed except when the total backlog is small It is conjectured that such ideal system behavior can be approached using a family of relatively simple scheduling policies related to the c\mu rule
240 citations
TL;DR: The well-known Pollaczec–Khinchine formula and some other known results including M/D/1, M/Ek/1 and M/M/1 have been derived as particular cases.
Abstract: We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate l (>0) all demand the first ‘essential’ service, whereas only some of them demand the second ‘optional’ service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B(v) and that of the second optional service are exponential with mean service time 1/m_2 (m_2>0). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly. The well-known Pollaczec–Khinchine formula and some other known results including M/D/1, M/E_{k}/1} and M/M/1 have been derived as particular cases.
170 citations
143 citations
TL;DR: A queueing model with reneging that provides a stylistic representation of the transplant waiting list and closed-form asymptotic expressions for the stationary waiting time, stationary waited time until transplantation, and fraction of patients who receive transplantation for each patient class are developed.
Abstract: Motivated by the problem of organ allocation, we develop a queueing model with reneging that provides a stylistic representation of the transplant waiting list. The model assumes that there are several classes of patients, several classes of organs, and patient reneging due to death. We focus on randomized organ allocation policies and develop closed-form asymptotic expressions for the stationary waiting time, stationary waiting time until transplantation, and fraction of patients who receive transplantation for each patient class. Analysis of these expressions identifies the main factors that underlie the performance of the transplant waiting list and demonstrates that queueing models can prove useful when evaluating the organ allocation system.
106 citations
TL;DR: In this article, the authors considered the scheduling problem for multiclass queueing networks and optimization of Markov decision processes, and showed that the value iteration algorithm may perform poorly when the algorithm is not initialized properly.
Abstract: This paper considers in parallel the scheduling problem for multiclass queueing networks, and optimization of Markov decision processes. It is shown that the value iteration algorithm may perform poorly when the algorithm is not initialized properly. The most typical case where the initial value function is taken to be zero may be a particularly bad choice. In contrast, if the value iteration algorithm is initialized with a stochastic Lyapunov function, then the following hold: (i) a stochastic Lyapunov function exists for each intermediate policy, and hence each policy is regular (a strong stability condition), (ii) intermediate costs converge to the optimal cost, and (iii) any limiting policy is average cost optimal. It is argued that a natural choice for the initial value function is the value function for the associated deterministic control problem based upon a fluid model, or the approximate solution to Poisson’s equation obtained from the LP of Kumar and Meyn. Numerical studies show that either choice may lead to fast convergence to an optimal policy.
70 citations
TL;DR: It is shown that in the large deviations limiting regime in which the number of inputs increases and the service rate and buffer size are increased in proportion, the statistical characteristics of a flow are essentially unchanged by passage through the switch.
Abstract: Consider a switch which queues traffic from many independent input flows. We show that in the large deviations limiting regime in which the number of inputs increases and the service rate and buffer size are increased in proportion, the statistical characteristics of a flow are essentially unchanged by passage through the switch. This significantly simplifies the analysis of networks of switches. It means that each traffic flow in a network can be assigned an effective bandwidth, independent of the other flows, and the behaviour of any switch in the network depends only on the effective bandwidths of the flows using it.
68 citations
TL;DR: A large deviation approach provides a powerful method of studying the tail behavior of the increase in the buffer content during a busy period of the M/G/∞ queue feeding the buffer.
Abstract: A fluid queue with ON periods arriving according to a Poisson process and having a long-tailed distribution has long range dependence. As a result, its performance deteriorates. The extent of this performance deterioration depends on a quantity determined by the average values of the system parameters. In the case when the the performance deterioration is the most extreme, we quantify it by studying the time until the amount of work in the system causes an overflow of a large buffer. This turns out to be strongly related to the tail behavior of the increase in the buffer content during a busy period of the M/G/\infty queue feeding the buffer. A large deviation approach provides a powerful method of studying such tail behavior.
68 citations
TL;DR: It is shown that, unlike the two‐station networks in Dai and Vande Vate [18], the global stability region of the three‐station network is not the intersection of its stability regions under static buffer priority disciplines, and the “worst” or extremal disciplines are not static bufferpriority disciplines.
Abstract: This paper studies the stability of a three-station fluid network. We show that, unlike the two-station networks in Dai and Vande Vate l18r, the global stability region of our three-station network is not the intersection of its stability regions under static buffer priority disciplines. Thus, the “worst” or extremal disciplines are not static buffer priority disciplines. We also prove that the global stability region of our three-station network is not monotone in the service times and so, we may move a service time vector out of the global stability region by reducing the service time for a class. We introduce the monotone global stability region and show that a linear program (LP) related to a piecewise linear Lyapunov function characterizes this largest monotone subset of the global stability region for our three-station network. We also show that the LP proposed by Bertsimas et al. l1r does not characterize either the global stability region or even the monotone global stability region of our three-station network. Further, we demonstrate that the LP related to the linear Lyapunov function proposed by Chen and Zhang l11r does not characterize the stability region of our three-station network under a static buffer priority discipline.
67 citations
TL;DR: A general framework for obtaining asymptotic distributional bounds on the stationary backlog in a buffer fed by a combined fluid process A1 + A2 and drained at a constant rate c is proposed.
Abstract: We propose a general framework for obtaining asymptotic distributional bounds on the stationary backlog W^{A_1+A_2,c} in a buffer fed by a combined fluid process A_1+A_2 and drained at a constant rate c The fluid process A_1 is an (independent) on–off source with average and peak rates \rho_1 and r_1, respectively, and with distribution G for the activity periods The fluid process A_2 of average rate \rho_2 is arbitrary but independent of A_1 These bounds are used to identify subexponential distributions G and fairly general fluid processes A_2 such that the asymptotic equivalence \mathbf{P}l[W^{A_1+A_2,c}>x \sim \mathbf{P}l[W^{A_1,c-\rho_2}>x]\quad (x\to\infty) holds under the stability condition \rho_1+\rho_2 and the non-triviality condition c-\rho_2 In these asymptotics the stationary backlog W^{A_1,c-\rho_2} results from feeding source A_1 into a buffer drained at reduced rate c-\rho_2 This reduced load asymptotic equivalence extends to a larger class of distributions G a result obtained by Jelenkovic and Lazar l19r in the case when G belongs to the class of regular intermediate varying distributions
67 citations
TL;DR: A finite‐capacity single‐server vacation model with close‐down/setup times and Markovian arrival processes and Laplace–Stieltjes transforms of both the virtual and actual waiting time distributions is considered.
Abstract: We consider a finite-capacity single-server vacation model with close-down/setup times and Markovian arrival processes (MAP). The queueing model has potential applications in classical IP over ATM or IP switching systems, where the close-down time corresponds to an inactive timer and the setup time to the time delay to set up a switched virtual connection (SVC) by the signaling protocol. The vacation time may be considered as the time period required to release an SVC or as the time during which the server goes to set up other SVCs. By using the supplementary variable technique, we obtain the queue length distribution at an arbitrary instant, the loss probability, the setup rate, as well as the Laplace–Stieltjes transforms of both the virtual and actual waiting time distributions.
63 citations
TL;DR: The deterministic, worst-case analysis of [25] is extended to the case where a detailed statistical model of the input traffic is available and can be used as a basis for an admission control mechanism.
Abstract: In this paper we consider a stochastic server (modeling a multiclass communication switch) fed by a set of parallel buffers. The dynamics of the system evolve in discrete-time and the generalized processor sharing (GPS) scheduling policy of l25r is implemented. The arrival process in each buffer is an arbitrary, and possibly autocorrelated, stochastic process. We obtain a large deviations asymptotic for the buffer overflow probability at each buffer. In the standard large deviations methodology, we provide a lower and a matching (up to first degree in the exponent) upper bound on the buffer overflow probabilities. We view the problem of finding a most likely sample path that leads to an overflow as an optimal control problem. Using ideas from convex optimization we analytically solve the control problem to obtain both the asymptotic exponent of the overflow probability and a characterization of most likely modes of overflow. These results have important implications for traffic management of high-speed networks. They extend the deterministic, worst-case analysis of l25r to the case where a detailed statistical model of the input traffic is available and can be used as a basis for an admission control mechanism.
TL;DR: This work shows that a properly normalized d-dimensional queue length process converges in distribution to a fd-dimensional semimartingale reflecting Brownian motion (RBM) in a d- dimensional box under a heavy traffic condition.
Abstract: We consider a queueing network of d single server stations. Each station has a finite capacity waiting buffer, and all customers served at a station are homogeneous in terms of service requirements and routing. The routing is assumed to be deterministic and hence feedforward. A server stops working when the downstream buffer is full. We show that a properly normalized d-dimensional queue length process converges in distribution to a d-dimensional semimartingale reflecting Brownian motion (RBM) in a d-dimensional box under a heavy traffic condition. The conventional continuous mapping approach does not apply here because the solution to our Skorohod problem may not be unique. Our proof relies heavily on a uniform oscillation result for solutions to a family of Skorohod problems. The oscillation result is proved in a general form that may be of independent interest. It has the potential to be used as an important ingredient in establishing heavy traffic limit theorems for general finite buffer networks.
TL;DR: The Laplace–Stieltjes transform of the sojourn-time distribution in steady state is derived, and it is shown that the expectedSojourn time is not proportional to the service requirement and scaled by the traffic load.
Abstract: We study the sojourn times of customers in an M/M/1 queue with the processor sharing service discipline and a server that is subject to breakdowns. The lengths of the breakdowns have a general distribution, whereas the “on-periods” are exponentially distributed. A branching process approach leads to a decomposition of the sojourn time, in which the components are independent of each other and can be investigated separately. We derive the Laplace–Stieltjes transform of the sojourn-time distribution in steady state, and show that the expected sojourn time is not proportional to the service requirement. In the heavy-traffic limit, the sojourn time conditioned on the service requirement and scaled by the traffic load is shown to be exponentially distributed. The results can be used for the performance analysis of elastic traffic in communication networks, in particular, the ABR service class in ATM networks, and best-effort services in IP networks.
TL;DR: A retrial queueing system with the batch Markovian arrival process and semi-Markovian service is investigated and asymptotically quasi-Toeplitz 2-dimensional Markov chains are introduced into consideration and applied for solving the problem.
Abstract: A retrial queueing system with the batch Markovian arrival process and semi-Markovian service is investigated. We suppose that the intensity of retrials linearly depends on the number of repeated calls. The distribution of the number of calls in the system is the subject of research. Asymptotically quasi-Toeplitz 2-dimensional Markov chains are introduced into consideration and applied for solving the problem.
TL;DR: The results indicate that scheduling policies can have a significant influence on the extent to which long-range dependence in the arrivals influences the performance of communication networks.
Abstract: Consider a single server queue with unit service rate fed by an arrival process of the following form: sessions arrive at the times of a Poisson process of rate \lambda, with each session lasting for an independent integer time \tau \geq 1, where P(\tau = k ) = p_k with p_k \sim \alpha k^{-(1 +\alpha)}L(k), where 1<\alpha<2 and L(\cdot) is a slowly varying function. Each session brings in work at unit rate while it is active. Thus the work brought in by each arrival is regularly varying, and, because 1 < \alpha < 2, the arrival process of work is long-range dependent. Assume that the stability condition \lambda E[\tau] < 1 holds. By simple arguments we show that for any stationary nonpreemptive service policy at the queue, the stationary sojourn time of a typical session must stochastically dominate a regularly varying random variable having infinite means this is true even if the duration of a session is known at the time it arrives. On the other hand, we show that there exist causal stationary preemptive policies, which do not need knowledge of the session durations at the time of arrival, for which the stationary sojourn time of a typical session is stochastically dominated by a regularly varying random variable having finite mean. These results indicate that scheduling policies can have a significant influence on the extent to which long-range dependence in the arrivals influences the performance of communication networks.
TL;DR: In this article, the authors consider a GI/G/1 queue with a heavy tail and prove a heavy-traffic limit theorem for the distribution of the stationary actual waiting time.
Abstract: We consider a GI/G/1 queue in which the service time distribution and/or the interarrival time distribution has a heavy tail, i.e., a tail behaviour like t^{-
u} with 1<
u \leq 2, so that the mean is finite but the variance is infinite. We prove a heavy-traffic limit theorem for the distribution of the stationary actual waiting time \mathbf{W}. If the tail of the service time distribution is heavier than that of the interarrival time distribution, and the traffic load a \rightarrow 1, then \mathbf{W}, multiplied by an appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the Kovalenko distribution. If the tail of the interarrival time distribution is heavier than that of the service time distribution, and the traffic load a \rightarrow 1, then \mathbf{W}, multiplied by another appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the negative exponential distribution.
TL;DR: In this paper, the stationary expected loss rate for a single server queue with i.i.d. arrival and service processes has been shown to be asymptotically constant.
Abstract: Consider a single server queue with i.i.d. arrival and service processes, \{A,\ A_n,n\geq 0\} and \{C,\ C_n,n\geq 0\}, respectively, and a finite buffer B. The queue content process \{Q^B_n,\ n\geq 0\} is recursively defined as Q^B_{n+1}=\min((Q^B_n+A_{n+1}-C_{n+1})^+,B), q^+=\max(0,q). When \mathbb{E}(A-C)<0, and A has a subexponential distribution, we show that the stationary expected loss rate for this queue \mathbb{E}(Q^B_n+A_{n+1}-C_{n+1}-B)^+ has the following explicit asymptotic characterization: \mathbb{E}(Q^B_n+A_{n+1}-C_{n+1}-B)^+\sim \mathbb{E}(A-B)^+ \quad \hbox{as} \ B\rightarrow \infty, independently of the server process C_n. For a fluid queue with capacity c, M/G/\infty arrival process A_t, characterized by intermediately regularly varying on periods \tau^{\mathrm{on}}, which arrive with Poisson rate \Lambda, the average loss rate \lambda_{\mathrm{loss}}^B satisfies {\lambda_{\mathrm{loss}}^B}\sim \Lambda \mathbb{E}(\tau^{\mathrm{on}}\eta-B)^+ \quad \hbox{as}\ B\rightarrow \infty, where \eta=r+\rho-c, \rho=\mathbb{E}A_ts r (c\leq r) is the rate at which the fluid is arriving during an on period. Accuracy of the above asymptotic relations is verified with extensive numerical and simulation experiments. These explicit formulas have potential application in designing communication networks that will carry traffic with long-tailed characteristics, e.g., Internet data services.
TL;DR: A generalized criterion for the stability of Markovian queueing systems in terms of stochastic fluid limits is introduced and an example in which this criterion may be applied: a polling system with two stations and two heterogeneous servers is considered.
Abstract: We introduce a generalized criterion for the stability of Markovian queueing systems in terms of stochastic fluid limits. We consider an example in which this criterion may be applied: a polling system with two stations and two heterogeneous servers.
TL;DR: A single-server finite-capacity queue with general bulk service rule where customers arrive according to a Poisson process and service times of the batches are arbitrarily distributed is considered.
Abstract: In this paper, we consider a single-server finite-capacity queue with general bulk service rule where customers arrive according to a Poisson process and service times of the batches are arbitrarily distributed. The queue is analyzed using both the supplementary variable and imbedded Markov chain techniques. The relations between state probabilities at departure and arbitrary epochs have been presented in explicit forms.
TL;DR: A new approach is derived for the computation of the stationary buffer content of an infinite buffer fluid queue that leads to a numerically stable algorithm for which the precision of the result can be given in advance.
Abstract: We consider an infinite buffer fluid queue receiving its input from the output of a Markovian queue with finite or infinite waiting room. The input is characterized by a Markov modulated rate process. We derive a new approach for the computation of the stationary buffer content. This approach leads to a numerically stable algorithm for which the precision of the result can be given in advance.
TL;DR: Stability is proved by combining elementary large deviations theory with an analysis of an associated fluid control problem to show that policies in the family considered are shown to be stable.
Abstract: This paper describes a family of discrete-review policies for scheduling open multiclass queueing networks. Each of the policies in the family is derived from what we call a dynamic reward function: such a function associates with each queue length vector q and each job class k a positive value r_k(q), which is treated as a reward rate for time devoted to processing class k jobs. Assuming that each station has a traffic intensity parameter less than one, all policies in the family considered are shown to be stable. In such a policy, system status is reviewed at discrete points in time, and at each such point the controller formulates a processing plan for the next review period, based on the queue length vector observed. Stability is proved by combining elementary large deviations theory with an analysis of an associated fluid control problem. These results are extended to systems with class dependent setup times as well as systems with alternate routing and admission control capabilities.
TL;DR: The set of feasible values for the autocorrelation decay parameter and the first two or three moments of the interarrival time distribution are determined and a method is derived for matching these parameters to a MAP of order two.
Abstract: We show that the autocorrelation sequence of interarrival times for a Markovian arrival process (MAP) of order two is geometric. We determine the set of feasible values for the autocorrelation decay parameter and the first two or three moments of the interarrival time distribution. A method is derived for matching these parameters to a MAP of order two and some numerical examples are included to illustrate approximating higher dimensional MAPs by two dimensional ones. The numerical examples have helped us pose important questions regarding the significance of correlation in a MAP of order two when it is used as input to a queueing model.
TL;DR: The deviant behavior of a queue fed by a large number of traffic streams is analyzed and the most likely trajectory (or ‘optimal path’) to buffer overflow is given by applying large deviations techniques.
Abstract: We analyse the deviant behavior of a queue fed by a large number of traffic streams. In particular, we explicitly give the most likely trajectory (or ‘optimal path’) to buffer overflow, by applying large deviations techniques. This is done for a broad class of sources, consisting of Markov fluid sources and periodic sources. Apart from a number of ramifications of this result, we present guidelines for the numerical evaluation of the optimal path.
TL;DR: It is established that logarithmic asymptotics of moderate deviations for queue-length and waiting-time processes in single server queues and open queueing networks in critical loading complement earlier diffusion approximation results.
Abstract: We establish logarithmic asymptotics of moderate deviations for queue-length and waiting-time processes in single server queues and open queueing networks in critical loading. Our results complement earlier diffusion approximation results.
TL;DR: In the case of light-tailed summands, a simple proof of the famous Cramér asymptotic formula is given via the change of probability measure and bounds of the tails of geometric convolution are given.
Abstract: Tails of distributions having the form of the geometric convolution are considered. In the case of light-tailed summands, a simple proof of the famous Cramer asymptotic formula is given via the change of probability measure. Some related results are obtained, namely, bounds of the tails of geometric convolutions, expressions for the distribution of the 1st failure time and failure rate in regenerative systems, and others. In the case of heavy-tailed summands, two-sided bounds of the tail of the geometric convolution are given in the cases where the summands have either Pareto or Weibull distributions. The results obtained have the property that the corresponding lower and upper bounds are tailed-equivalent.
TL;DR: This work considers the slotted ALOHA protocol on a channel with a capture effect, which contains the CDMA protocols as special cases and obtains sufficient rate conditions, which are close to necessary for stability of the system, when the arrival streams are stationary ergodic.
Abstract: We consider the slotted ALOHA protocol on a channel with a capture effect. There are M<∞ users each with an infinite buffer. If in a slot, i packets are transmitted, then the probability of a successful reception of a packet is q_i. This model contains the CDMA protocols as special cases. We obtain sufficient rate conditions, which are close to necessary for stability of the system, when the arrival streams are stationary ergodic. Under the same rate conditions, for general regenerative arrival streams, we obtain the rates of convergence to stationarity, finiteness of stationary moments and various functional limit theorems. Our arrival streams contain all the traffic models suggested in the recent literature, including the ones which display long range dependence. We also obtain bounds on the stationary moments of waiting times which can be tight under realistic conditions. Finally, we obtain several results on the transient performance of the system, e.g., first time to overflow and the limits of the overflow process. We also extend the above results to the case of a capture channel exhibiting Markov modulated fading. Most of our results and proofs will be shown to hold also for the slotted ALOHA protocol without capture.
TL;DR: Various models within the queueing framework which have been suggested for teletraffic data aim to capture certain stylised features of the data, such as variability of arrival rates, heavy-tailedness of on- and off-periods and long-range dependence in teletrail transmission.
Abstract: This article reviews various models within the queueing framework which have been suggested for teletraffic data. Such models aim to capture certain stylised features of the data, such as variability of arrival rates, heavy-tailedness of on- and off-periods and long-range dependence in teletraffic transmission. Subexponential distributions constitute a large class of heavy-tailed distributions, and we investigate their (sometimes disastrous) influence within teletraffic models. We demonstrate some of the above effects in an explorative data analysis of Munich Universities’ intranet data.
TL;DR: The asymptotic tail of the cycle maximum during a regenerative cycle, and the asymPTotic Tail of the stationary random variable L∞ of the level process are given and shown to be subexponential with L ∞ having the heavier tail.
Abstract: Bivariate regenerative Markov modulated queueing processes \{I_n,L_n\} are described. \{I_n\} is the phase process, and \{L_n\} is the level process. Increments in the level process have subexponential distributions. A general boundary behavior at the level 0 is allowed. The asymptotic tail of the cycle maximum, M_{C^{\mathrm{reg}}}, during a regenerative cycle, C^{\mathrm{reg}}, and the asymptotic tail of the stationary random variable L_\infty, respectively, of the level process are given and shown to be subexponential with L_{\infty} having the heavier tail.
TL;DR: This paper deals with the statistical analysis of bulk arrival queues from a Bayesian point of view, focusing on prediction of the usual measures of performance of the system in equilibrium.
Abstract: This paper deals with the statistical analysis of bulk arrival queues from a Bayesian point of view. The focus is on prediction of the usual measures of performance of the system in equilibrium. Posterior predictive distribution of the number of customers in the system is obtained through its probability generating function. Posterior distribution of the waiting time, in the queue and in the system, of the first customer of an arriving group is expressed in terms of their Laplace and Laplace–Stieltjes transform. Discussion of numerical inversion of these transforms is addressed.
TL;DR: The aim of this paper is to prove the convergence of the stationary buffer content process {XN*(t)} in the fNth model to the buffercontent process { XN(t)} in the limiting Gaussian model.
Abstract: We consider fluid models with infinite buffer size. Let \{Z_N(t)\} be the net input rate to the buffer, where \{Z_N(t)\} is a superposition of N homogeneous alternating on–off flows. Under heavy traffic environment \{Z_N (t)\} converges in distribution to a centred Gaussian process with covariance function of a single flow. The aim of this paper is to prove the convergence of the stationary buffer content process \{X^*_N(t)\} in the Nth model to the buffer content process \{X^*(t)\} in the limiting Gaussian model.