Two-parameter heavy-traffic limits for infinite-server queues with dependent service times

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.     

Tandem queues with deterministic service times

In this paper we consider a tandem queueing model for a sequence of multiplexers at the edge of an ATM network. All queues of the tandem queueing model have unit service times. Each successive queue receives the output of the previous queue plus some external arrivals. For the case of two queues in series, we study the end-to-end delay of a cell (customer) arriving at the first queue, and the covariance of its delays at both queues. The joint queue length process at all queues is studied in detail for the 2-queue and 3-queue cases, and we outline an approach to the case of an arbitrary number of queues in series.Part of the research of this author has been supported by the European Grant BRA-QMIPS of CEC DG XIII.The research of this author was done during the time that he was affiliated with CWI, in a joint project with PTT Research.     

Tail asymptotics of a Markov-modulated infinite-server queue

This paper analyzes large deviation probabilities related to the number of customers in a Markov-modulated infinite-server queue, with state-dependent arrival and service rates. Two specific scalings are studied: in the first, just the arrival rates are linearly scaled by \(N\) (for large \(N\) ), whereas in the second in addition the Markovian background process is sped up by a factor \(N^{1+\varepsilon }\) , for some \(\varepsilon >0\) . In both regimes (transient and stationary) tail probabilities decay essentially exponentially, where the associated decay rate corresponds to that of the probability that the sample mean of i.i.d. Poisson random variables attains an atypical value.     

Two-parameter heavy-traffic limits for infinite-server queues

In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables Q ^e (t,y) and Q ^r (t,y) representing the number of customers in the system at time t that have elapsed service times less than or equal to time y, or residual service times strictly greater than y. We also consider W ^r (t,y) representing the total amount of work in service time remaining to be done at time t+y for customers in the system at time t. The two-parameter stochastic-process limits in the space D([0,∞),D) of D-valued functions in D draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (Adv. Appl. Probab. 23, 188–209, 1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (Queueing Syst. 25, 235–280, 1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c.d.f.     

Networks of infinite-server queues with nonstationary Poisson input

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.     

On queues with service and interarrival times depending on waiting times

We consider an extension of the standard G/G/1 queue, described by the equation $W\stackrel{ \mathcal {D}}{=}\max\mathrm{max}\,\{0,B-A+YW\}$ , where ?[Y=1]=p and ?[Y=?1]=1?p. For p=1 this model reduces to the classical Lindley equation for the waiting time in the G/G/1 queue, whereas for p=0 it describes the waiting time of the server in an alternating service model. For all other values of p, this model describes a FCFS queue in which the service times and interarrival times depend linearly and randomly on the waiting times. We derive the distribution of W when A is generally distributed and B follows a phase-type distribution, and when A is exponentially distributed and B deterministic.     

A simple policy for multiple queues with size-independent service times

We consider a service system with two Poisson arrival queues. A server chooses which queue to serve at each moment. Once a queue is served, all the customers will be served within a fixed amount of time. This model is useful in studying airport shuttling or certain online computing systems. We propose a simple yet optimal state-independent policy for this problem which is not only easy to implement, but also performs very well.     

Calculation of delay characteristics for multiserver queues with constant service times

We consider a discrete-time infinite-capacity queueing system with a general uncorrelated arrival process, constant-length service times of multiple slots, multiple servers and a first-come-first-served queueing discipline. Under the assumption that the queueing system can reach a steady state, we first establish a relationship between the steady-state probability distributions of the system content and the customer delay. Next, by means of this relationship, an explicit expression for the probability generating function of the customer delay is obtained from the known generating function of the system content, derived in previous work. In addition, several characteristics of the customer delay, namely the mean value, the variance and the tail distribution of the delay, are derived through some mathematical manipulations. The analysis is illustrated by means of some numerical examples.     

Scheduling in a multi-class series of queues with deterministic service times

We consider a problem of scheduling in a multi-class network of single-server queues in series, in which service times at the nodes are constant and equal. Such a model has potential application to automated manufacturing systems or packet-switched communication networks, where a message is divided into packets (or cells) of fixed lengths. The network is a series-type assembly or transfer line, with the exception that there is an additional class of jobs that requires processing only at the first node (class 0). There is a holding cost per unit time that is proportional to the total number of customers in the system. The objective is to minimize the (expected) total discounted holding cost over a finite or an infinite horizon. We show that an optimal policy gives priority to class-0 jobs at node 1 when at least one of a set ofm–1 inequalities on partial sums of the components of the state vector is satisfied. We solve the problem by two methods. The first involves formulating the problem as a (discrete-time) Markov decision process and using induction on the horizon length. The second is a sample-path approach using an interchange argument to establish optimality.The research of this author was supported by the National Science Foundation under Grant No. DDM-8719825. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.     

Analytic approximations of queues with lightly- and heavily-correlated autoregressive service times

We consider a single-server queueing system. The arrival process is modelled as a Poisson process while the service times of the consecutive customers constitute a sequence of autoregressive random variables. Our interest into autoregressive service times comes from the need to capture temporal correlation of the channel conditions on wireless network links. If these fluctuations are slow in comparison with the transmission times of the packets, transmission times of consecutive packets are correlated. Such correlation needs to be taken into account for an accurate performance assessment. By means of a transform approach, we obtain a functional equation for the joint transform of the queue content and the current service time at departure epochs in steady state. To the best of our knowledge, this functional equation cannot be solved by exact mathematical techniques, despite its simplicity. However, by means of a Taylor series expansion in the parameter of the autoregressive process, a “light-correlation” approximation is obtained for performance measures such as moments of the queue content and packet delay. We illustrate our approach by some numerical examples, thereby assessing the accuracy of our approximations by simulation. For the heavy correlation case, we give differential equation approximations based on the time-scale separation technique, and present numerical examples in support of this approximation.     

Geo/G/1 queues with disasters and general repair times

This paper discusses discrete-time single server Geo/G/1 queues that are subject to failure due to a disaster arrival. Upon a disaster arrival, all present customers leave the system. At a failure epoch, the server is turned off and the repair period immediately begins. The repair times are commonly distributed random variables. We derive the probability generating functions of the queue length distribution and the FCFS sojourn time distribution. Finally, some numerical examples are given.     

Approximate analysis of non-stationary loss queues and networks of loss queues with general service time distributions

A Fixed Point Approximation (FPA) method has recently been suggested for non-stationary analysis of loss queues and networks of loss queues with Exponential service times. Deriving exact equations relating time-dependent mean numbers of busy servers to blocking probabilities, we generalize the FPA method to loss systems with general service time distributions. These equations are combined with associated formulae for stationary analysis of loss systems in steady state through a carried load to offered load transformation. The accuracy and speed of the generalized methods are illustrated through a wide set of examples.     

Departures from queues with changeover times

We consider the M/M ^ij /1 queue as a model of queues with changeover times, i.e., the service is exponential with parameter _ij depending on the previous job type (i) and the current job type (j). It is shown that the departure process is renewal and Poisson iff _ij = (constant). In this case, types of departures are dependent renewal processes. Crosscovariance and crosscorrelations are given.     

Priority queues with semi-exhaustive service

This paper studies a new type of multi-class priority queues with semi-exhaustive service and server vacations, which operates as follows: A single server continues serving messages in queuen until the number of messages decreases toone less than that found upon the server's last arrival at queuen, where 1nN. In succession, messages of the highest class present in the system, if any, will be served according to this semi-exhaustive service. Applying the delay cycle analysis and introducing a super-message composed of messages served in a busy period, we derive explicitly the Laplace-Stieltjes transforms (LSTs) and the first two moments of the message waiting time distributions for each class in the M/G/1-type priority queues with multiple and single vacations. We also derive a conversion relationship between the LSTs for waiting times in the multiple and single vacation models.     

Asymptotic behaviour of the tandem queueing system with identical service times at both queues

Consider a tandem queue consisting of two single-server queues in series, with a Poisson arrival process at the first queue and arbitrarily distributed service times, which for any customer are identical in both queues. For this tandem queue, we relate the tail behaviour of the sojourn time distribution and the workload distribution at the second queue to that of the (residual) service time distribution. As a by-result, we prove that both the sojourn time distribution and the workload distribution at the second queue are regularly varying at infinity of index 1−ν, if the service time distribution is regularly varying at infinity of index −ν (ν>1). Furthermore, in the latter case we derive a heavy-traffic limit theorem for the sojourn time S ⁽²⁾ at the second queue when the traffic load ρ↑ 1. It states that, for a particular contraction factor Δ (ρ), the contracted sojourn time Δ (ρ) S ⁽²⁾ converges in distribution to the limit distribution H(·) as ρ↑ 1 where .     

Waiting times in queues with relative priorities

This paper determines the mean waiting times for a single server multi-class queueing model with Poisson arrivals and relative priorities. If the server becomes idle, the probability that the next job is from class-i is proportional to the product between the number of class-i jobs present and their priority parameter.