Non-stationary Queueing Systems Research Articles

An approximate approach to fuzzy stochastic differential equations under sub-fractional Brownian motion

In this paper, we introduce fuzzy stochastic differential equations (FSDEs) driven by sub-fractional Brownian motion (SFBM) which are applied to describe phenomena subjected to randomness and fuzziness simultaneously. The SFBM is an extension of the Brownian motion that retains many properties of fractional Brownian motion (FBM), but not the stationary increments. This property makes SFBM a possible candidate for models that include long-range dependence, self-similarity, and non-stationary increments which is suitable for the construction of stochastic models in finance and non-stationary queueing systems. We apply an approximation method to stochastic integrals, and a decomposition of the SFBM to find the existence and uniqueness of the solutions.

Strong Embeddings for Transitory Queueing Models

In this paper, we establish strong embedding theorems, in the sense of the Komlós-Major-Tusnády framework, for the performance metrics of a general class of transitory queueing models of nonstationary queueing systems. The nonstationary and non-Markovian nature of these models makes the computation of performance metrics hard. The strong embeddings yield error bounds on sample path approximations by diffusion processes in the form of functional strong approximation theorems.

Modeling single-channel non-stationary queueing systems presented in the form of a cyclic graph of states

Modeling single-channel non-stationary queueing systems presented in the form of a cyclic graph of states

A model for waiting times for non-stationary queueing systems

Queueing models are stochastic models that represent the probability that a queueing system will be found in a particular configuration or state. Several interesting stationary queueing systems have been solved analytically; on the other hand, non-stationary queueing systems are relatively unexplored. The present study analyses the waiting times of a non-stationary M/M/1 queueing system using simulation methods.

Risk measures and their application to staffing nonstationary service systems

In this paper, we explore the use of static risk measures from the mathematical finance literature to assess the performance of some standard nonstationary queueing systems. To do this we study two important queueing models, namely the infinite server queue and the multi-server queue with abandonment. We derive exact expressions for the value of many standard risk measures for the Mt/M/∞, Mt/G/∞, and Mt/Mt/∞ queueing models. We also derive Gaussian based approximations for the value of risk measures for the Erlang-A queueing model. Unlike more traditional approaches of performance analysis, risk measures offer the ability to satisfy the unique and specific risk preferences or tolerances of service operations managers. We also show how risk measures can be used for staffing nonstationary systems with different risk preferences and assess the impact of these staffing policies via simulation.

Program Implementation of the Numerical-Analytical Method for Computation of Non-Stationary Service System Models

A numerical-analytical method for non-stationary queueing systems models computation is presented. The solution of Chapman—Kolmogorov equations is found in the analytical form. The algorithm and its practical implementation with Java language are discussed. Computation time and results precision for the presented method and the Runge—Kutta type method used in Matlab are compared.

A Poisson–Charlier approximation for nonstationary queues

In this paper, we develop a new approximation for nonstationary multiserver queues with abandonment. Our method uses the Poisson–Charlier polynomials, which are a discrete orthogonal polynomial sequence that is orthogonal with respect to the Poisson distribution. We show that by appealing to the Poisson–Charlier polynomials that we can estimate the mean, variance, and probability of delay of our nonstationary queueing system with good accuracy. Lastly, we provide a numerical example that illustrates that our approximations are effective.

Effects of state-dependent balking on multi-server non-stationary queueing systems

A combination of discrete-time modelling and theoretical analysis is used to develop an easy-to-use Normal approximation for the time-dependent behaviour of multi-server queueing systems subject to state-dependent balking. Key findings that underpin the approximation are that time lags between peaks in arrival rates and congestion levels can be ignored; queue behaviour is insensitive to distribution of service time beyond its mean; and distribution of number in the system is near Normal. This model is then used to derive valuable management implications for service-orientated systems where delays are frequent and abandonments are an important feature. Key insights are that such systems can often adopt very ‘sub-optimal’ behaviour, and that customer impatience can be very beneficial where management wishes to improve system performance. ‘Optimal’ performance is defined, and practical measures for queue managers to move system performance towards ‘optimality’ are identified.

Modelling Waiting Times for Non-Stationary Queueing Systems

Queueing models are generally constructed to represent the steady state of a queueing system, that is, the typical, long run or average state of the system. As a consequence, these are stochastic models that represent the probability that a queueing system will be found in a particular configuration or state. Several interesting stationary queueing systems have been solved analytically; on the other hand, non-stationary queueing systems are relatively unexplored. The present study analyses the waiting times of a non-stationary M/M/1 queueing system.

Approximation of the non-stationary M(t)/M(t)/c(t)-queue using stationary queueing models: The stationary backlog-carryover approach

This paper proposes a new approach for the time-dependent analysis of stochastic and non-stationary queueing systems. The analysis of a series of stationary queueing models leads to a new approximation of time-dependent performance measures. Based on a stationary backlog-carryover (SBC) approximation of the time-dependent expected utilization, different approximations of the time-dependent expected queue length and the number of customers in the system are discussed. Limiting results are given for the case of constant rates. The accuracy of the SBC approach is shown for non-stationary M(t)/M(t)/c(t) queueing systems with time-dependent and piecewise constant arrival rates. In numerical experiments we demonstrate the reliability of this approach and compare it with the (lagged) stationary independent period by period (SIPP) approach. In addition, the approximation is applied to temporarily overloaded systems that cannot be analyzed by the variants of the SIPP approach.

Optimal Pricing and Admission Control in a Queueing System with Periodically Varying Parameters

We consider congestion control in a nonstationary queueing system. Assuming that the arrival and service rates are bounded, periodic functions of time, a Markov decision process (MDP) formulation is developed. We show under the infinite horizon discounted and average reward optimality criteria, for each fixed time, optimal pricing and admission control strategies are nondecreasing in the number of customers in the system. This extends stationary results to the nonstationary setting. Despite this result, the problem still seems intractable. We propose an easily implementable pointwise stationary approximation (PSA) to approximate the optimal policies, suggest a heuristic to improve the implementation of the PSA and verify its usefulness via a numerical study.

Ergodicity and stability of nonstationary queueing systems

Ergodicity and stability of nonstationary queueing systems

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.

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.