Nonstationary Queueing Research Articles

The Critical Few Minutes in Scheduling Time‐Varying Queuing Systems

ABSTRACTFor nonstationary queuing systems where demand varies over time, an important practical issue is scheduling the number of servers to be available at various times of the day. Widely used scheduling procedures typically involve adding servers at natural time points (e.g., on the hour or at half past the hour) during peak demand periods. Scheduling is often complicated by restrictions on the minimum amount of time (human) servers must work, the earliest (or latest) time a server is available, and limits on the maximum number of servers that can be used at any one time. This paper was motivated by experience with actual queuing systems that embodied such complications. For these systems common scheduling methods that used “natural” starting times for servers resulted in needlessly long customer waits. This research demonstrates that changing the starting times of servers by only a few minutes can have dramatic impacts on customer waiting times for extended periods. In addition, the results highlight the importance of server punctuality.

Bias optimal admission control policies for a multiclass nonstationary queueing system

We consider a finite-capacity queueing system where arriving customers offer rewards which are paid upon acceptance into the system. The gatekeeper, whose objective is to ‘maximize’ rewards, decides if the reward offered is sufficient to accept or reject the arriving customer. Suppose the arrival rates, service rates, and system capacity are changing over time in a known manner. We show that all bias optimal (a refinement of long-run average reward optimal) policies are of threshold form. Furthermore, we give sufficient conditions for the bias optimal policy to be monotonic in time. We show, via a counterexample, that if these conditions are violated, the optimal policy may not be monotonic in time or of threshold form.

Bias optimal admission control policies for a multiclass nonstationary queueing system

We consider a finite-capacity queueing system where arriving customers offer rewards which are paid upon acceptance into the system. The gatekeeper, whose objective is to ‘maximize’ rewards, decides if the reward offered is sufficient to accept or reject the arriving customer. Suppose the arrival rates, service rates, and system capacity are changing over time in a known manner. We show that all bias optimal (a refinement of long-run average reward optimal) policies are of threshold form. Furthermore, we give sufficient conditions for the bias optimal policy to be monotonic in time. We show, via a counterexample, that if these conditions are violated, the optimal policy may not be monotonic in time or of threshold form.

Transient laws of non-stationary queueing systems and their applications

In this paper we consider the general class of non-stationary queueing models and identify structural relationships between the number of customers in the system and the delay at time t, denoted by L(t) and S(t), respectively. In particular, we first establish a transient Little’s law at the same level of generality as the classical stationary version of Little’s law. We then obtain transient distributional laws for overtake free non-stationary systems. These laws relate the distributions of L(t) and S(t) and constitute a complete set of equations that describes the dynamics of overtake free non-stationary queueing systems. We further extend these laws to multiclass systems as well. Finally, to demonstrate the power of the transient laws we apply them to a variety queueing systems: Infinite and single server systems with non-stationary Poisson arrivals and general non-stationary services, multiclass single server systems with general non-stationary arrivals and services, and multiserver systems with renewal arrivals and deterministic services, operating in the transient domain. For all specific systems we relate the performance measures using the established set of laws and obtain a complete description of the system in the sense that we have a sufficient number of integral equations and unknowns. We then solve the set of integral equations using asymptotic expansions and exact numerical techniques. We also report computational results from our methods.

An analysis of the congestion effects of link failures in wide area networks

The authors present the results of a study to determine the effects of link failures on the performance of a network in terms of the occurrence of congestion due to traffic restoration after a failure. The network studied is a virtual circuit based packet switched wide area network. A generic queueing framework is developed to study the effect of failures and the subsequent traffic restoration on network performance. In general, the congestion resulting after a failure is a transient phenomenon. Hence, a numerical methods based nonstationary queueing analysis is conducted in order to quantify the effects of failures in terms of the transient behavior of queue lengths and packet loss probabilities. A bounding relationship is developed whereby a network node can determine whether or not congestion will occur as the result of traffic restoration after a failure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonstationary queues with Interrrupted Poisson arrivals and unreliable/repairable servers

A queueing model having a nonstationary Interrupted Poisson arrival process (IPP(t)),s time-dependent exponential unreliable/repairable servers and finite capacityc is introduced, and an approximation method for analysis of it is developed and tested. Approximations are developed for the time-dependent queue length moments and the system viewpoint waiting time distributions and moments. The approximation involves state-space partitioning and numerically integrating partial-moment differential equations (PMDEs). Surrogate distribution approximations (SDA's) are used to close the system of PMDEs. The approximations allow for analysis using only (s + 1)(s + 6) differential equations for the queue length moments rather than the 2(c + 1)(s +1) equations required by the classic method of numerically integrating the full set of Kolmogorov-forward equations. Effectively hours of cpu time are reduced to minutes for even modest capacity systems. Approximations for waiting time distributions and moments are developed.

Approximating nonstationaryPh(t)/M(t)/s/c queueing systems

Nonstationary phase processes are defined and a surrogate distribution approximation (SDA) method for analyzing transient and nonstationary queueing systems with nonstationary phase arrival processes is presented. Regardless of system capacityc, the SDA method requires the numerical solution of only 6K differential equations, whereK is the number of phases in the arrival process, compared to theK(c+1) Kolmogorov forward equations required for the classical method of solution. Time-dependent approximations of mean and variance of the number of entities in the system and the number of busy servers are obtained. Empirical test results over a wide range of systems indicate the SDA is quite accurate.

Stationary Poisson departure processes from non-stationary queues

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

Stationary Poisson departure processes from non-stationary queues

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt /∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

Control of entry to a nonstationary queuing system

AbstractIn this article the control of entry of customers to a queuing system with s servers is considered. It is assumed that the arrivals form a nonstationary Poisson process with a periodic rate. The service times are assumed to be exponentially distributed with a parameter independent of time. The cost structure considered is the same as that of Naor. It is demonstrated numerically that, like the stationary cases, the average expected benefit of customers per unit of time is a unimodal function of the critical point. And, also, the social critical point is smaller than the individual critical point. These suggest the use of a search technique for finding the social critical point. The results show successful application of the discrete version of the Fibonacci search.

Nonstationary Queuing Probabilities for Landing Congestion of Aircraft

Illustration is made, with reference to a case study of airport landing congestion in the New York area, of how numerical solutions may be easily obtained for the transient behavior of nonstationary waiting lines characterized by Poisson arrivals, constant holding time, and service in order of arrival. In the case study only the mean arrival rate varies as a given function of time, but the methods used as easily provide for variation also of servicing rate and of number of servers. Exact numerical values are sought and obtained only at intervals of the servicing time. While these times were short in the case study, so that little information is sacrificed, the method may be adapted to yield more frequent values in the case of long servicing times. Subject to this restriction, the entire probability distribution of the queue is calculated as the main numerical routine, from which the distribution of waiting time is directly obtained. Use of a stored-program computer is simple, cheap, and especially effective under saturating arrival rates.