Search

In this paper, we develop a diffusion approximation for the transient distribution of the workload process in a standard single-server queue with a non-stationary Polya arrival process, which is a path-dependent Markov point process. The path-dependent arrival process model is useful because it has the arrival rate depending on the history of the arrival process, thus capturing a self-reinforcing property that one might expect in some applications. The workload approximation is based on heavy-traffic limits for (i) a sequence of Polya processes, in which the limit is a Gaussian-Markov process, and (ii) a sequence of

This paper studies tight upper bounds for the mean and higher moments of the steady-state waiting time in the GI/GI/1 queue given the first two moments of the interarrival-time and service-time distributions. We apply the theory of Tchebycheff systems to obtain sufficient conditions for classical two-point distributions to yield the extreme values. These distributions are determined by having one mass at 0 or at the upper limit of support.

It has long been conjectured that the tight upper bound for the mean steady-state waiting time in the GI/GI/1 queue given the first two moments of the interarrival-time and service-time distributions is attained asymptotically by two-point distributions. The two-point distribution for the interarrival time has one mass point at 0, but the service-time distribution involves a limit; there is one mass point at a high value, but that upper mass point must increase to infinity while the probability on that point must decrease to 0 appropriately. In this paper, we develop effective numerical and simulation algorithms to compute the value of this conjectured tight bound. The algorithms are aided by reductions of the special queues with extremal interarrival-time and extremal service-time distributions to D/GI/1 and GI/D/1 models. Combining these reductions yields an overall representation in terms of a D/RS(D)/1 discrete-time model involving a geometric random sum of deterministic random variables (the RS(D)), where the two deterministic random variables in the model may have different values, so that the extremal steady-state waiting time need not have a lattice distribution. Efficient computational methods are developed. The computational results show that the conjectured tight upper bound offers a significant improvement over established bounds.

This paper studies stationary customer flows in an open queueing network. The flows are the processes counting customers flowing from one queue to another or out of the network. We establish the existence of unique stationary flows in generalized Jackson networks and convergence to the stationary flows as time increases. We establish heavy-traffic limits for the stationary flows, allowing an arbitrary subset of the queues to be critically loaded. The heavy-traffic limit with a single bottleneck queue is especially tractable because it yields limit processes involving one-dimensional reflected Brownian motion. That limit plays an important role in our new nonparametric decomposition approximation of the steady-state performance using indices of dispersion and robust optimization.

We develop a general framework for stationary marked point processes in discrete time. We start with a careful analysis of the sample paths. Our initial representation is a sequence $$\{(t_j,k_j): j\in {\mathbb {Z}}\}$$ of times $$t_j\in {\mathbb {Z}}$$ and marks $$k_j\in {\mathbb {K}}$$ , with batch arrivals (i.e., $$t_j=t_{j+1}$$ ) allowed. We also define alternative interarrival time and sequence representations and show that the three different representations are topologically equivalent. Then, we develop discrete analogs of the familiar stationary stochastic constructs in continuous time: time-stationary and point-stationary random marked point processes, Palm distributions, inversion formulas and Campbell’s theorem with an application to the derivation of a periodic-stationary Little’s law. Along the way, we provide examples to illustrate interesting features of the discrete-time theory.

We establish a central-limit-theorem (CLT) version of the periodic Little’s law (PLL) in discrete time, which complements the sample-path and stationary versions of the PLL we recently established, motivated by data analysis of a hospital emergency department. Our new CLT version of the PLL extends previous CLT versions of LL. As with the LL, the CLT version of the PLL is useful for statistical applications.

To better understand what stochastic model might be appropriate in applications with system data, we study the consequences of fitting a stationary birth-and-death (BD) process to the sample path of a periodic $$M_t/GI/\infty $$Mt/GI/ź model. The fitted BD process will necessarily have the correct steady-state distribution (appropriately defined), but will not have the correct transient behavior. Nevertheless, the fitted birth-rate and death-rate functions have structure determined by the $$M_t/GI/\infty $$Mt/GI/ź model that should be seen with data if the $$M_t/GI/\infty $$Mt/GI/ź model is appropriate. In this paper, we establish heavy-traffic fluid limits that yield explicit approximation formulas for the fitted birth-rate and death-rate functions that can help evaluate whether a periodic $$M_t/GI/\infty $$Mt/GI/ź model is appropriate. We also establish many-server heavy-traffic fluid limits for the steady-state distribution in the periodic $$M_t/GI/\infty $$Mt/GI/ź model. For the special case of sinusoidal arrival rates, the limiting steady-state distribution has an arcsine law.

We consider a class of general $$G_t/G_t/1$$Gt/Gt/1 single-server queues, including the $$M_t/M_t/1$$Mt/Mt/1 queue, with unlimited waiting space, service in order of arrival, and a time-varying arrival rate, where the service rate at each time is subject to control. We study the rate-matching control, where the service rate is made proportional to the arrival rate. We show that the model with the rate-matching control can be regarded as a deterministic time transformation of a stationary G / G / 1 model, so that the queue length distribution is stabilized as time evolves. However, the time-varying virtual waiting time is not stabilized. We show that the time-varying expected virtual waiting time with the rate-matching service-rate control becomes inversely proportional to the arrival rate in a heavy-traffic limit. We also show that no control that stabilizes the queue length asymptotically in heavy traffic can also stabilize the virtual waiting time. Then we consider two square-root service-rate controls and show that one of these stabilizes the waiting time when the arrival rate changes slowly relative to the average service time, so that a pointwise stationary approximation is appropriate.

This paper explores grey-box modeling of queueing systems. A stationary birth-and-death (BD) process model is fitted to a segment of the sample path of the number in the system in the usual way. The birth (death) rates in each state are estimated by the observed number of arrivals (departures) in that state divided by the total time spent in that state. Under minor regularity conditions, if the queue length (number in the system) has a proper limiting steady-state distribution, then the fitted BD process has that same steady-state distribution asymptotically as the sample size increases, even if the actual queue-length process is not nearly a BD process. However, the transient behavior may be very different. We investigate what we can learn about the actual queueing system from the fitted BD process. Here we consider the standard $$GI/GI/s$$GI/GI/s queueing model with $$s$$s servers, unlimited waiting room and general independent, non-exponential, interarrival-time and service-time distributions. For heavily loaded $$s$$s-server models, we find that the long-term transient behavior of the original process, as partially characterized by mean first passage times, can be approximated by a deterministic time transformation of the fitted BD process, exploiting the heavy-traffic characterization of the variability.

In previous papers we developed a deterministic fluid approximation for an overloaded Markovian queueing system having two customer classes and two service pools, known in the call-center literature as the X model. The system uses the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload. Under FQR-T, customers are served by their own service pool until a threshold is exceeded. Then, one-way sharing is activated with customers from one class allowed to be served in both pools. The control aims to keep the two queues at a pre-specified fixed ratio. We supported the fluid approximation by establishing a many-server heavy-traffic functional weak law of large numbers (FWLLN) involving an averaging principle. In this paper we develop a refined diffusion approximation for the same model based on a many-server heavy-traffic functional central limit theorem (FCLT).

This paper is a sequel to our 2010 paper in this journal in which we established heavy-traffic limits for two-parameter processes in infinite-server queues with an arrival process that satisfies an FCLT and i.i.d. service times with a general distribution. The arrival process can have a time-varying arrival rate. In particular, an FWLLN and an FCLT were established for the two-parameter process describing the number of customers in the system at time t that have been so for a duration y. The present paper extends the previous results to cover the case in which the successive service times are weakly dependent. The deterministic fluid limit obtained from the new FWLLN is unaffected by the dependence, whereas the Gaussian process limit (random field) obtained from the FCLT has a term resulting from the dependence. Explicit expressions are derived for the time-dependent means, variances, and covariances for the common case in which the limit process for the arrival process is a (possibly time scaled) Brownian motion.

This paper introduces a deterministic fluid model that approximates the many-server G t /GI/s t +GI queueing model, and determines the time-dependent performance functions. The fluid model has time-varying arrival rate and service capacity, abandonment from queue, and non-exponential service and patience distributions. Two key assumptions are that: (i) the system alternates between overloaded and underloaded intervals, and (ii) the functions specifying the fluid model are suitably smooth. An algorithm is developed to calculate all performance functions. It involves the iterative solution of a fixed-point equation for the time-varying rate that fluid enters service and the solution of an ordinary differential equation for the time-varying head-of-line waiting time, during each overloaded interval. Simulations are conducted to confirm that the algorithm and the approximation are effective.

We establish heavy-traffic limits for stationary waiting times and other performance measures in G n /G n /1 queues, where G n indicates that an original point process is modified by cyclic thinning of order n, i.e., the thinned process contains every nth point from the original point process. The classical example is the Erlang E n /E n /1 queue, where cyclic thinning of order n is applied to both the interarrival times and the service times, starting from a "base" M/M/1 model. The models G n /D/1 and D/G n /1 are special cases of G n /G n /1. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated G n /D/1 model, where the interarrival times are the sum of n consecutive interarrival times in the original model, the G/D/n model is a special case as well. As n??, the G n /G n /1 models approach the deterministic D/D/1 model. We obtain revealing limits by letting ? n ?1 as n??, where ? n is the traffic intensity in model n.

We previously introduced and analyzed the G t /M t /s t +GI t many-server fluid queue with time-varying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper, we establish an asymptotic loss of memory (ALOM) property for that fluid model, i.e., we show that there is asymptotic independence from the initial conditions as time t evolves, under regularity conditions. We show that the difference in the performance functions dissipates over time exponentially fast, again under the regularity conditions. We apply ALOM to show that the stationary G/M/s+GI fluid queue converges to steady state and the periodic G t /M t /s t +GI t fluid queue converges to a periodic steady state as time evolves, for all finite initial conditions.

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.

We establish many-server heavy-traffic limits for G/M/n+M queueing models, allowing customer abandonment (the +M), subject to exogenous regenerative service interruptions. With unscaled service interruption times, we obtain a FWLLN for the queue-length process, where the limit is an ordinary differential equation in a two-state random environment. With asymptotically negligible service interruptions, we obtain a FCLT for the queue-length process, where the limit is characterized as the pathwise unique solution to a stochastic integral equation with jumps. When the arrivals are renewal and the interruption cycle time is exponential, the limit is a Markov process, being a jump-diffusion process in the QED regime and an O---U process driven by a Levy process in the ED regime (and for infinite-server queues). A stochastic-decomposition property of the steady-state distribution of the limit process in the ED regime (and for infinite-server queues) is obtained.

This paper studies the last departure time from a queue with a terminating arrival process. This problem is motivated by a model of two-stage inspection in which finitely many items come to a first stage for screening. Items failing first-stage inspection go to a second stage to be examined further. Assuming that arrivals at the second stage can be regarded as an independent thinning of the departures from the first stage, the arrival process at the second stage is approximately a terminating Poisson process. If the failure probabilities are not constant, then this Poisson process will be nonhomogeneous. The last departure time from an M t /G/? queue with a terminating arrival process serves as a remarkably tractable approximation, which is appropriate when there are ample inspection resources at the second stage. For this model, the last departure time is a Poisson random maximum, so that it is possible to give exact expressions and develop useful approximations based on extreme-value theory.

We establish heavy-traffic stochastic-process limits for the queue-length and overflow stochastic processes in the standard single-server queue with finite waiting room (G/G/1/K). We show that, under regularity conditions, the content and overflow processes in related single-server models with finite waiting room, such as the finite dam, satisfy the same heavy-traffic stochastic-process limits. As a consequence, we obtain heavy-traffic limits for the proportion of customers or input lost over an initial interval. Except for an interchange of the order of two limits, we thus obtain heavy-traffic limits for the steady-state loss proportions. We justify the interchange of limits in M/GI/1/K and GI/M/1/K special cases of the standard GI/GI/1/K model by directly establishing local heavy-traffic limits for the steady-state blocking probabilities.

We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Levy process. When the Levy process representing the net input has no negative jumps, the steady-state distribution of the reflected Levy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently approximated.

By exploiting an infinite-server-model lower bound, we show that the tails of the steady-state and transient waiting-time distributions in the M/GI/s queue with unlimited waiting room and the first-come first-served discipline are bounded below by tails of Poisson distributions. As a consequence, the tail of the steady-state waiting-time distribution is bounded below by a constant times the sth power of the tail of the service-time stationary-excess distribution. We apply that bound to show that the steady-state waiting-time distribution has a heavy tail (with appropriate definition) whenever the service-time distribution does. We also establish additional results that enable us to nearly capture the full asymptotics in both light and heavy traffic. The difference between the asymptotic behavior in these two regions shows that the actual asymptotic form must be quite complicated.

Multiprocessor load balancing aims to improve performance by moving jobs from highly loaded processors to more lightly loaded processors. Some schemes allow only migration of new jobs upon arrival, while other schemes allow migration of jobs in progress. A difficulty with all these schemes, however, is that they require continuously maintaining detailed state information. In this paper we consider the alternative of periodic load balancing, in which the loads are balanced only at each T time units for some appropriate T. With periodic load balancing, state information is only needed at the balancing times. Moreover, it is often possible to use slightly stale information collected during the interval between balancing times. In this paper we study the performance of periodic load balancing. We consider multiple queues in parallel with unlimited waiting space to which jobs come either in separate independent streams or by assignment (either random or cyclic) from a single stream. Resource sharing is achieved by periodically redistributing the jobs or the work in the system among the queues. The performance of these systems of queues coupled by periodic load balancing depends on the transient behavior of a single queue. We focus on useful approximations obtained by considering a large number of homogeneous queues and a heavy load. When the number of queues is sufficiently large, the number of jobs or quantity of work at each queue immediately after redistribution tends to evolve deterministically, by the law of large numbers. The steady-state (limiting) value of this deterministic sequence is obtained as the solution of a fixed point equation, where the initial value is equal to the expected transient value over the interval between successive redistributions conditional on the initial value. A refined approximation based on the central limit theorem is a normal distribution, where the mean and variance are obtained by solving a pair of fixed-point equations. With higher loads, which is natural to consider when load balancing is performed, a heavy-traffic limit theorem shows that one-dimensional reflected Brownian motion can be used to approximately describe system performance, even with general arrival and service processes. With these approximations, we show how performance depends on the assumed arrival pattern of jobs and the model parameters. We do numerical calculations and conduct simulation experiments to show the accuracy of the approximations.

We develop deterministic fluid approximations to describe the recovery from rare congestion events in a large multi-server system in which customer holding times have a general distribution. There are two cases, depending on whether or not we exploit the age distribution (the distribution of elapsed holding times of customers in service). If we do not exploit the age distribution, then the rare congestion event is a large number of customers present. If we do exploit the age distribution, then the rare event is an unusual age distribution, possibly accompanied by a large number of customers present. As an approximation, we represent the large multi-server system as an M/G/\infty model. We prove that, under regularity conditions, the fluid approximations are asymptotically correct as the arrival rate increases. The fluid approximations show the impact upon the recovery time of the holding-time distribution beyond its mean. The recovery time may or not be affected by the holding-time distribution having a long tail, depending on the precise definition of recovery. The fluid approximations can be used to analyze various overload control schemes, such as reducing the arrival rate or interrupting services in progress. We also establish large deviations principles to show that the two kinds of rare events have the same exponentially small order. We give numerical examples showing the effect of the holding-time distribution and the age distribution, focusing especially on the consequences of long-tail distributions.

We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptive-resume disciplines. We show that the low-priority steady-state waiting-time can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waiting-time distribution. We exploit this structures to determine the asymptotic behavior of the tail probabilities. Unlike the FIFO case, there is routinely a region of the parameters such that the tail probabilities have non-exponential asymptotics. This phenomenon even occurs when both service-time distributions are exponential. When non-exponential asymptotics holds, the asymptotic form tends to be determined by the non-exponential asymptotics for the high-priority busy-period distribution. We obtain asymptotic expansions for the low-priority waiting-time distribution by obtaining an asymptotic expansion for the busy-period transform from Kendall’s functional equation. We identify the boundary between the exponential and non-exponential asymptotic regions. For the special cases of an exponential high-priority service-time distribution and of common general service-time distributions, we obtain convenient explicit forms for the low-priority waiting-time transform. We also establish asymptotic results for cases with long-tail service-time distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the non-exponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waiting-time transform.

In this paper we consider the M_t /G/\infty queueing model with infinitely many servers and a nonhomogeneous Poisson arrival process. Our goal is to obtain useful insights and formulas for nonstationary finite-server systems that commonly arise in practice. Here we are primarily concerned with the peak congestion. For the infinite-server model, we focus on the maximum value of the mean number of busy servers and the time lag between when this maximum occurs and the time that the maximum arrival rate occurs. We describe the asymptotic behavior of these quantities as the arrival changes more slowly, obtaining refinements of previous simple approximations. In addition to providing improved approximations, these refinements indicate when the simple approximations should perform well. We obtain an approximate time-dependent distribution for the number of customers in service in associated finite-server models by using the modified-offered-load (MOL) approximation, which is the finite-server steady-state distribution with the infinite-server mean serving as the offered load. We compare the value and lag in peak congestion predicted by the MOL approximation with exact values for M_t/M/s delay models with sinusoidal arrival-rate functions obtained by numerically solving the Chapman–Kolmogorov forward equations. The MOL approximation is remarkably accurate when the delay probability is suitably small. To treat systems with slowly varying arrival rates, we suggest focusing on the form of the arrival-rate function near its peak, in particular, on its second and third derivatives at the peak. We suggest estimating these derivatives from data by fitting a quadratic or cubic polynomial in a suitable interval about the peak.

In this paper we compare the sliding-window (SW) and leaky-bucket (LB) input regulators. These regulators reject, or treat as lower priority, certain arrivals to a queueing system, so as to reduce congestion in the queueing system. Such regulators are currently of interest for access control in emerging high-speed communication networks. The SW admits no more than a specified numberW of arrivals in any interval of specified lengthL. The LB is a counter that increases by one up to a maximum capacityC for each arrival and decreases continuously at a given drain rate to as low as zero; an arrival is admitted if the counter is less than or equal toC−1. To indirectly represent the impact of the regulator on the performance of the queueing system, we focus on the maximum bursts admissible at the peak rate. We show that the SW admits larger bursts than the LB at any given peak rate and admissible average rate. To make the comparison, we use a special construction: We start with a sample path of an arrival process with a given peak rate. We choose a window lengthL for the SW and find the minimum window contentW that is just conforming (so there are no rejections). We then set the LB drain rate equal toW/L, so that the two admissible average rates are identical. Finally, we choose the LB capacityC so that the given arrival process is also just conforming for the LB. With this construction, we show that the SW will admit larger bursts at the peak rate than the LB. We also develop approximations for these maximum burst sizes and their ratio over long time intervals based on extreme-value asymptotics. We use simulations to confirm that these approximations do indeed enable us to predict the burst ratios for typical stochastic arrival processes.

We consider the standardGI/G/1 queue with unlimited waiting room and the first-in first-out service discipline. We investigate the steady-state waiting-time tail probabilitiesP(W>x) when the service-time distribution has a long-tail distribution, i.e., when the service-time distribution fails to have a finite moment generating function. We have developed algorithms for computing the waiting-time distribution by Laplace transform inversion when the Laplace transforms of the interarrival-time and service-time distributions are known. One algorithm, exploiting Pollaczek's classical contourintegral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient two-parameter family of long-tail distributions on the positive half line with explicit Laplace transforms. This family is a Pareto mixture of exponential (PME) distributions. These PME distributions have monotone densities and Pareto-like tails, i.e., are of orderx −r forr>1. We use this family of long-tail distributions to investigate the quality of approximations based on asymptotics forP(W>x) asx→∞. We show that the asymptotic approximations with these long-tail service-time distributions can be remarkably inaccurate for typicalx values of interest. We also derive multi-term asymptotic expansions for the waiting-time tail probabilities in theM/G/1 queue. Even three terms of this expansion can be remarkably inaccurate for typicalx values of interest. Thus, we evidently must rely on numerical algorithms for determining the waiting-time tail probabilities in this case. When working with service-time data, we suggest using empirical Laplace transforms.

We show, under regularity conditions, that a counting process satisfies a large deviations principle in ℝ or the Gartner-Ellis condition (convergence of the normalized logarithmic moment generating functions) if and only if its inverse process does. We show, again under regularity conditions, that embedded regenerative structure is sufficient for the counting process or its inverse process to have exponential asymptotics, and thus satisfy the Gartner-Ellis condition. These results help characterize the small-tail asymptotic behavior of steady-state distributions in queueing models, e.g., the waiting time, workload and queue length.

This paper establishes functional central limit theorems describing the heavy-traffic behavior of open single-class queueing networks with service interruptions. In particular, each station has a single server which is alternatively up and down. There are two treatments of the up and down times. The first treatment corresponds to fixed up and down times and leads to a reflected Brownian motion, just as when there are no service interruptions, but with different parameters. To represent long rare interruptions, the second treatment has growing up and down times with the up and down times being of ordern andn 1/2, respectively, when the traffic intensities are of order 1-n−1/2. In this case we establish convergence in the SkorohodM 1 topology to a multidimensional reflection of multidimensional Brownian motion plus a multidimensional jump process.

In this paper we focus on networks of infinite-server queues with nonhomogeneous Poisson arrival processes. We start by introducing a more general Poisson-arrival-location model (PALM) in which arrivals move independently through a general state space according to a location stochastic process after arriving according to a nonhomogeneous Poisson process. The usual open network of infinite-server queues, which is also known as a linear population process or a linear stochastic compartmental model, arises in the special case of a finite state space. The mathematical foundation is a Poisson-random-measure representation, which can be obtained by stochastic integration. It implies a time-dependent product-form result: For appropriate initial conditions, the queue lengths (numbers of customers in disjoint subsets of the state space) at any time are independent Poisson random variables. Even though there is no dependence among the queue lengths at each time, there is important dependence among the queue lengths at different times. We show that the joint distribution is multivariate Poisson, and calculate the covariances. A unified framework for constructing stochastic processes of interest is provided by stochastically integrating various functionals of the location process with respect to the Poisson arrival process. We use this approach to study the flows in the queueing network; e.g., we show that the aggregate arrival and departure processes at a given queue (to and from other queues as well as outside the network) are generalized Poisson processes (without necessarily having a rate or unit jumps) if and only if no customer can visit that queue more than once. We also characterize the aggregate arrival and departure processes when customers can visit the queues more frequently. In addition to obtaining structural results, we use the stochastic integrals to obtain explicit expressions for time-dependent means and covariances. We do this in two ways. First, we decompose the entire network into a superposition of independent networks with fixed deterministic routes. Second, we make Markov assumptions, initially for the evolution of the routes and finally for the entire location process. For Markov routing among the queues, the aggregate arrival rates are obtained as the solution to a system of input equations, which have a unique solution under appropriate qualifications, but not in general. Linear ordinary differential equations characterize the time-dependent means and covariances in the totally Markovian case.

We constructG/G/1/k queueing models that fail to have anticipated monotonicity properties with respect to the capacityk. In one model the long-run average number of customers in the system is arbitrarily close to the capacityk, but it decreases to an arbitrarily small value when the capacity is increased. In another model the throughput is arbitrarily close to the arrival rate when the capacity isk, but the throughput decreases to an arbitrarily small value when the capacity is increased. These examples involving non i.i.d. service times, which are associated with external arrivals instead of being assigned when service begins, show that stochastic assumptions and arguments involving more than direct sample-path comparisons are essential for obtaining useful bounds and positive comparison results.

A fundamental principle of queueing theory isL=λW (Little's law), which states that the time-average or expected time-stationary number of customers in a system is equal to the product of the arrival rate and the customer-average or expected customer-stationary time each customer spends in the system. This principle is now well known and frequently applied. However, in recent years there have been extensions, such as H=λG and the continuous, distributional, ordinal and central-limit-theorem versions, which show that theL=λW relation, when viewed properly, has much more power than was previously realized. Moreover, connections have been established between H=λG and other fundamental relations, such as the rate conservation law and PASTA (Poisson arrivals see time averages), which show that there is a much greater unity in the overall theory than was previously realized. This paper provides a review.

Two decompositions are established for the probability transition function of the queue length process in the M/M/1 queue by a simple probabilistic argument. The transition function is expressed in terms of a zero-avoiding probability and a transition probability to zero in two different ways. As a consequence, the M/M/1 transition function can be represented as a positive linear combination of convolutions of the busy-period density. These relations provide insight into the transient behavior and facilitate establishing related results, such as inequalities and asymptotic behavior.

We consider a modification of the standardG/G/1 queue with unlimited waiting space and the first-in first-out discipline in which the service times and interarrival times depend linearly and randomly on the waiting times. In this model the waiting times satisfy a modified version of the classical Lindley recursion. We determine when the waiting-time distributions converge to a proper limit and we develop approximations for this steady-state limit, primarily by applying previous results of Vervaat [21] and Brandt [4] for the unrestricted recursionYn+1=CnYn+Xn. Particularly appealing for applications is a normal approximation for the stationary waiting time distribution in the case when the queue only rarely becomes empty. We also consider the problem of scheduling successive interarrival times at arrival epochs, with the objective of achieving nearly maximal throughput with nearly bounded waiting times, while making the interarrival time sequence relatively smooth. We identify policies depending linearly and deterministically upon the work in the system which meet these objectives reasonably well; with these policies the waiting times are approximately contained in a specified interval a specified fraction of time.

Underlying the fundamental queueing formulaL=λW is a relation between cumulative processes in continuous time (the integral of the queue length process) and in discrete time (the sum of the waiting times of successive customers). Except for remainder terms which usually are asymptotically negligible, each cumulative process is a random time-transformation of the other. As a consequence, in addition to the familiar relation between the with-prob ability-one limits of the averages, roughly speaking, the customer-average wait obeys a central limit theorem if and only if the time-average queue length obeys a central limit theorem, in which case both averages, properly normalized, converge in distribution jointly, and the individual limiting distributions are simply related. This relation between the central limit theorems is conveniently expressed in terms of functional central limit theorems, using the continuous mapping theorem and related arguments. The central limit theorems can be applied to compare the asymptotic efficiency of different estimators of queueing parameters. For example, when the arrival rateλ is known and the interarrivai times and waiting times are negatively correlated, it is more asymptotically efficient to estimate the long-run time-average queue lengthL indirectly by the sample-average of the waiting times, invokingL=λW, than it is to estimate it by the sample-average of the queue length. This variance-reduction principle extends a corresponding result for the standard GI/G/s model established by Carson and Law [2].

This paper shows that certain basic descriptions of the time-dependent behavior of the M/M/1 queue have very simple representations as mixtures of exponentials. In particular, this is true for the busy-period density, the probability that the server is busy starting at zero, the expected queue length starting at zero and the autocorrelation function of the stationary queue-length process. In each case the mixing density is a minor modification of a beta density. The last two representations also apply to regulated or reflected Brownian motion (RBM) by virtue of the heavy-traffic limit. Connections are also established to the classical spectral representations of Ledermann and Reuter (1954) and Karlin and McGregor (1958) and the associated trigonometric integral representations of Ledermann and Reuter, Vaulot (1954), Morse (1955), Riordan (1961) and Takacs (1962). Overall, this paper aims to provide a more unified view of the M/M/1 transient behavior and show how several different approaches are related.

We study the convergence of finite-capacity open queueing systems to their infinite-capacity counterparts as the capacity increases. Convergence of the transient behavior is easily established in great generality provided that the finite-capacity system can be identified with the infinite-capacity system up to the first time that the capacity is exceeded. Convergence of steady-state distributions is more difficult; it is established here for the GI/GI/c/n model withc servers,n-c extra waiting spaces and the first-come first-served discipline, in which all arrivals finding the waiting room full are lost without affecting future arrivals, via stochastic dominance and regenerative structure.

The familiar queueing principle expressed by the formulaL=λW can be interpreted as a relation among strong laws of large numbers. In a previous paper, we showed that this principle can be extended to include relations among other classical limit theorems such as central limit theorems and laws of the iterated logarithm. Here we provide sufficient conditions for these limit theorems using regenerative structure.