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The Maximum Likelihood Estimation (MLE) method is an established statistical method to estimate unknown parameters of a distribution. A disadvantage of the MLE method is that it requires an analytically tractable density, which is not available in many cases. This is the case, for example, with applications in service systems, since waiting models from queueing theory typically have no closed-form solution for the underlying density. This problem is addressed in this paper. MLE is used in combination with Stochastic Approximation (SA) to calibrate the arrival parameter θ of a G/G/1 queue via waiting time data. Three different numerical examples illustrate the application of the proposed estimator. Data sets of an M/G/1 queue, G/M/1 queue and model mismatch are considered. In a model mismatch, a mismatch is present between the used data and the postulated queuing model. The results indicate that the estimator is versatile and can be applied in many different scenarios.

The analysis of Weibull and Pareto distribution functions in the approximation of the density of a sum of damping functions, PMRQ approximation and approximation of service distribution in the peak mode of the flow. There was found one of the characteristics of the network, the average waiting time of packets in the queue after application of spectral method of the Lindley’s integral equation. Finally, PMRQ approximation of the average waiting time in TES+/G/1, MMPR, was compared with the analytic values of the same value in the spectral solution of the Lindley’s integral equation obtained by simulation with real traffic. Traffic is captured using the Wireshark protocol analyzer program. The time distributions were obtained in the EasyFit data analysis program, designed for quick statistical data analysis and decision making. Methods of approximation of network traffic allows us to estimate the average packet latency using statistical data analysis, which will improve the quality of service and predict the behavior of traffic.

Perel and Yechiali 2014 [16] considered a multi-queue single-server retrial polling system with batch service of an unlimited size, i.e., the so called “Israeli queue” with retrial, where the system consists of a main queue and an orbit queue, and only the customer at the head of the orbit queue is allowed to try to access the main queue. In this present paper, we aim to give a further study on this queueing model, in which each of the retrial customers in the orbit queue independently repeatedly tries for receiving service, and customers may abandon at the arrival instant from outside or at the departure epoch from the orbit queue. By analysing this model, we find that it is difficult to obtain an explicit closed form solution for the joint stationary probability distribution of the number of retrial customers in the orbit queue and the number of groups in the main queue. Therefore, by making use of the matrix analytic approach and the censoring technique, we derive the tail asymptotics result for the stationary joint probabilities.

This paper presents a novel and computationally efficient methodology for approximating the queue length (the number of customers in the system) distributions of time-varying non-Markovian many-server queues (e.g., Gt/Gt/nt queues), where the number of servers (nt) is large. Our methodology consists of two steps. The first step uses phase-type distributions to approximate the general interarrival and service times, thus generating an approximating Pht/Pht/nt queue. The second step develops strong approximation theory to approximate the Pht/Pht/nt queue with fluid and diffusion limits whose mean and variance can be computed using ordinary differential equations. However, by naively representing the Pht/Pht/nt queue as a Markov process by expanding the state space, we encounter the lingering phenomenoneven when the queue is overloaded. Lingering typically occurs when the mean queue length is equal or near the number of servers, however, in this case it also happens when the queue is overloaded and this time is not of zero measure. As a result, we develop an alternative representation for the queue length process that avoids the lingering problem in the overloaded case, thus allowing for the derivation of a Gaussian diffusion limit. Finally, we compare the effectiveness of our proposed method with discrete event simulation in a variety parameter settings and show that our approximations are very accurate. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0760 .

We consider an infinite-server queue, where the arrival and service rates are both governed by a continuous-time Markov chain that is independent of all other aspects of the queue. Through constructing a pretty Markov process, several performance measures are derived by using the supplementary variable technique. In addition, special cases and numerical examples are studied in detail.

The B M A P ∕ S M ∕ 1 queue is the most general single-server queueing model which can be analysed analytically. Problem of computation of stationary distributions of queue length is solved in the literature. However, the problem of computation of the moments of these distributions is not enough addressed. This problem is more complicated than its particular case when the service times are independent identically distributed random variables due to reducibility of some involved matrices. In this communication, we solve this problem.

We consider a model consisting of two fluid queues driven by the same background continuous-time Markov chain, such that the rates of change of the fluid in the second queue depend on whether the first queue is empty or not: when the first queue is nonempty, the content of the second queue increases, and when the first queue is empty, the content of the second queue decreases. We analyze the stationary distribution of this tandem model using operator-analytic methods. The various densities (or Laplace–Stieltjes transforms thereof) and probability masses involved in this stationary distribution are expressed in terms of the stationary distribution of some embedded process. To find the latter from the (known) transition kernel, we propose a numerical procedure based on discretization and truncation. For some examples we show the method works well, although its performance is clearly affected by the quality of these approximations, both in terms of accuracy and runtime.

This paper deals with an M / G / 1 queue with vacations and multiple phases of operation. If there are no customers in the system at the instant of a service completion, a vacation commences, that is, the system moves to vacation phase 0. If none is found waiting at the end of a vacation, the server goes for another vacation. Otherwise, the system jumps from phase 0 to some operative phase i with probability $$q_i$$ , $$i = 1,2, \ldots ,n.$$ In operative phase i, $$i = 1,2, \ldots ,n$$ , the server serves customers according to the discipline of FCFS (First-come, first-served). Using the method of supplementary variables, we obtain the stationary system size distribution at arbitrary epoch. The stationary sojourn time distribution of an arbitrary customer is also derived. In addition, the stochastic decomposition property is investigated. Finally, we present some numerical results.

We propose a vacation queue model for a signalized traffic intersection to elucidate intra-cycle queue size variations. Each incoming street to the intersection is modeled to experience Poisson arrivals, and to have finite queue capacity; vehicles arriving to full queue are dropped. Under a fixed-time policy, the queue size dynamics on each street are decoupled. Motivated by vacation queues, an imbedded Markov chain corresponding to queue sizes at the end of cycles is considered, whose transition probabilities are computed from analytical transient solutions of M/D/1/N queues. Sufficient conditions for convergence to steady-state distributions are provided for fixed-time policy. Simulations suggest consistency between the queue sizes computed by the proposed model, and the Webster and time-dependent traffic queue models.

We study a system of two non-identical and separate M/M/1/• queues with capacities (buffers) C1 < ∞ and C2 = ∞, respectively, served by a single server that alternates between the queues. The server’s switching policy is threshold-based, and, in contrast to other threshold models, is determined by the state of the queue that is not being served. That is, when neither queue is empty while the server attends Qi (i = 1, 2), the server switches to the other queue as soon as the latter reaches its threshold. When a served queue becomes empty we consider two switching scenarios: (i) Work-Conserving, and (ii) Non-Work-Conserving. We analyze the two scenarios using Matrix Geometric methods and obtain explicitly the rate matrix R, where its entries are given in terms of the roots of the determinants of two underlying matrices. Numerical examples are presented and extreme cases are investigated.

This paper is concerned with a discrete-time Geo/G/1 repairable queueing system with Bernoulli feedback and randomized p,N-policy. The service station may be subject to failures randomly during serving customers and therefore is sent for repair immediately. The p,N-policy means that when the number of customers in the system reaches a given threshold value N, the deactivated server is turned on with probability p or is still left off with probability 1−p. Applying the law of total probability decomposition, the renewal theory and the probability generating function technique, we investigate the queueing performance measures and reliability indices simultaneously in our work. Both the transient queue length distribution and the recursive expressions of the steady-state queue length distribution at various epochs are explicitly derived. Meanwhile, the stochastic decomposition property is presented for the proposed model. Various reliability indices, including the transient and the steady-state unavailability of the service station, the expected number of the service station breakdowns during the time interval 0+,n+ and the equilibrium failure frequency of the service station are also discussed. Finally, an operating cost function is formulated, and the direct search method is employed to numerically find the optimum value of N for minimizing the system cost. Copyright © 2017 John Wiley & Sons, Ltd.

We consider a $d$-node tandem queue with arrival process and light-tailed service processes at all queues i.i.d. and independent of each other. We consider three variations of the probability that the number of customers in the system reaches some high level $N$, namely during a busy cycle, in steady state, and upon arrival of a new customer. We show that their decay rates for large $N$ have the same value and give an expression for this value.

In this paper, analysis of single queue, multi-server with limited system capacity under first come first served discipline was carried out using iterative method. The arrivals of customers and service times of customers are assumed poisson and exponential distributions respectively. This queuing model is an extension of single queue, single server with limited system capacity. Performance measures of the model, such as the expected number of customers in the queue and in the system, the expected waiting times of customers in the queue and in the system respectively were derived. The performance measures so derived were compared with that of single queue, single server with limited capacity {M/M/1:(N/FCFS)} model. The numerical illustration indicates that single queue, multi-server with limited capacity {M/M/c:(N/FCFS)} model is more effective and efficient in handling congestions.Keywords: Iterative method, Single queue, Multi-Server queue model, Limited capacity, Performance measures

This paper considers an infinite-buffer batch-service queue with Markovian arrival process, generally distributed and batch-size-dependent service time. We obtain a bivariate vector generating function of queue length and server content distribution at departure epoch of a batch. The complete joint distribution of queue length, server content and phase of the arrival process at departure epoch is extracted in terms of roots of the associated characteristic equation. By employing these probability vectors we also perceive the joint distribution at arbitrary and pre-arrival epochs. Our analytic procedure and results are demonstrated using some numerical examples for phase-type as well as deterministic service time distributions with high and low values of model parameters. The occurrence of multiple roots are also investigated in case of phase-type service time distribution. Finally, we also investigate the influence of correlation of the arrival process on the behavior of the system performance.

In this paper, we propose a numerical method, based on Taylor series expansion, for computing the uncertainty in performance measures of vacation queueing model due to epistemic uncertainties in its parameters. Specifically, we consider the M/G/1/N queue with vacations, where we assume that vacation parameter is derived from finite number of observations, they themselves will have epistemic uncertainty associated with them. Using this method, we compute the expected value and the variance of the stationary distribution associated with the M/G/1/N queue with vacations. In order to illustrate the applicability of the proposed method, a series of numerical examples are discussed and the obtained results are compared to the corresponding Monte Carlo simulations ones.

We consider a finite population processor-sharing (PS) queue, with Markovian arrivals and an exponential server. Such a queue can model an interactive computer system consisting of a bank of terminals in series with a central processing unit (CPU). For systems with a large population $N$ and a commensurately rapid service rate, or infrequent arrivals, we obtain various asymptotic results. We analyze the conditional sojourn time distribution of a tagged customer, conditioned on the number $n$ of others in the system at the tagged customer's arrival instant, and also the unconditional distribution. The asymptotics are obtained by a combination of singular perturbation methods and spectral methods. We consider several space/time scales and parameter ranges, which lead to different asymptotic behaviors. We also identify precisely when the finite population model can be approximated by the standard infinite population $M/M/1$-PS queue., 60 pages and 3 figures

We consider a finite-buffer single-server queue with batch Markovian arrival process. In the case of finite-buffer batch arrival queue, there are different customer rejection/acceptance strategies such as partial batch rejection, total batch rejection, and total batch acceptance policy. We consider partial batch rejection strategy throughout our paper. We obtain queue length distributions at various epochs such as pre-arrival, arbitrary, and post-departure as well as some important performance measures, like probability of loss for the first, an arbitrary, and the last customer of a batch, mean queue lengths, and mean waiting times. The corresponding queueing model under single and multiple vacation policy has also been investigated. Some numerical results have been presented in the form of tables by considering phase-type and Pareto service time distributions. The proposed analysis is based on the successive substitutions in the Markov chain equations of the queue-length distribution at an embedded post-departure epoch of a customer. We also establish relationships among the queue-length distributions at post-departure, arbitrary, and pre-arrival epochs using the classical argument based on Markov renewal theory and semi-Markov processes. Such queueing systems find applications in the performance evaluation of teletraffic part in 4G A11-IP networks.

This paper is concerned with a discrete-time Geo /G / 1 queueing system with D -policy and J -optional services in which the service station may be subject to failures at random during serving the customers. All the arriving customers require the first essential service, whereas some of them may opt for a second service from the J additional services with some probability. As soon as the system becomes empty, the server will not restart the service until the sum of the service times of the waiting customers in the system reaches or exceeds some given positive integer D . Applying the total probability decomposition law, renewal theory, and probability generating function technique, the queueing indices and reliability measures are investigated simultaneously in our work. Both the probability generating function of the transient queue length distribution and the explicit formulas of the steady-state queue length distribution at time epoch n + are derived. Meanwhile, the stochastic decomposition property is presented for the proposed model. Various reliability indices, including the transient and steady-state unavailability, the expected number of breakdowns during (0+ ,n + ], and the equilibrium failure frequency, are discussed. Finally, the optimum value of D for minimizing the system cost is numerically discussed under a given cost structure.

A queueing system of the M/G/n-type, n ≥ 1, with a bounded total volume is considered. It is assumed that the volumes of the arriving packets are generally distributed random variables. Moreover, the AQM-type mechanism is used to control the actual buffer state: each of the arriving packets is dropped with a probability depending on its volume and the occupied volume of the system at the pre-arrival epoch. The explicit formulae for the stationary queue-size distribution and the loss probability are found. Numerical examples illustrating theoretical formulae are given as well.

Following recent developments in the application of Stein’s method in queueing theory, this paper is intended to be a short treatment showing how Stein’s method can be developed and applied to the single server queue in heavy traffic. Here we provide two approaches to this approximation: one based on equilibrium couplings and another involving comparison of generators.

This paper studies a control problem for the multiclass G/G/1 queue for a risk-sensitive cost of the form $$n^{-1}\log E\exp \sum _ic_iX^n_i(T)$$n-1logEexpźiciXin(T), where $$c_i>0$$ci>0 and $$T>0$$T>0 are constants, $$X^n_i$$Xin denotes the class-i queue length process, and the numbers of arrivals and service completions per unit time are of order n. The main result is the asymptotic optimality, as $$n\rightarrow \infty $$nźź, of a priority policy, provided that $$c_i$$ci are sufficiently large. Such a result has been known only in the Markovian (M/M/1) case. The index which determines the priority is explicitly computed in the case of Gamma-distributed interarrival and service times.

This work suggests a new analysis approach for a discrete-time GI/G/1 queue with multiple vacations. The method used is called a modified supplementary variable technique and our result is an exact transform-free expression for the steady state queue length distribution. Utilizing this result, we propose a simple two-moment approximation for the queue length distribution. From this, approximations for the mean queue length and the probabilities of the number of customers in the system are also obtained. To evaluate the approximations, we conduct numerical experiments which show that our approximations are remarkably simple yet provide fairly good performance, especially for a Bernoulli arrival process.

This article develops a computational algorithm for the loss probability in the stationary M/G/1 queue with impatient customers whose impatience times follow a phase-type distribution (M/G/1+PH). T...

Time-dependent solutions to queuing models are very useful for evaluating the performance of real-world systems. However, because of their mathematical complexity, few available results exist. In this paper, we derive the time-dependent performance measures for an M / D / 1 queue starting with a positive number of initial customers. Using the limiting property of an Erlang distribution, we obtain closed-form time-dependent formulas for the queue length and the waiting time. Furthermore, the time-dependent queue length probability in a busy period is derived.

This note discusses the inter-loss time ofan M/G/1/1 queuing system. The inter-loss time is defined as the time duration between two consecutive losses of arriving customers. In this study, we present the explicit Laplace transform of the inter-loss time distribution of an M/G/1/1 queuing system.

We consider an infinite-buffer single-server queue with renewal input and Markovian service process where customers are served in batches according to a general bulk service rule. Queue-length distributions at epochs of pre-arrival, arbitrary and post-departure have been obtained along with some important performance measures such as mean queue lengths and mean waiting times in both the system as well as the queue. We also obtain the steady-state service batch size distributions as well as system-length distributions. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of queue-length distribution at a pre-arrival epoch. Also, we provide analytical and numerical comparison between the roots method used in this paper and the matrix geometric method in terms of computational complexities and required computation time to evaluate pre-arrival epoch probabilities for both the methods. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.

We propose a method to analyze M/G/1/m queuing systems with the function of random dropping of customers and service time distribution dependent on the queue length. We obtain formulas to determine the Laplace transforms of the distribution of the number of customers in the system during busy period and of the distribution function of the busy period and to calculate the stationary characteristics. The relations for the stationary characteristics are tested using simulation models constructed with the use of the GPSS World tools. The results of the use of various control tools of system parameters are compared in the example.

This paper studies a system consisting of two parallel queues with transfers of customers. In the system, one queue is called main queue and the other one is called auxiliary queue. The main queue is monitored at exponential time instances. At a monitoring instant, if the number of customers in main queue reaches L\((>K)\), a batch of \(L-K\) customers is transferred from the main queue to the auxiliary queue, and if the number of customers in main queue is less than or equal to K, the transfers will not happen. For this system, by using a Foster-Lyapunov type condition, we establish a sufficient stability condition. Then, we provide a sufficient condition under which, for any fixed number of customers in the auxiliary queue, the stationary probability of the number of customers in the main queue has an exact geometric tail asymptotic as the number of customers in main queue increases to infinity. Finally, we give some numerical results to illustrate the impact of some critical model parameters on the decay rate.

We consider a single-server polling system consisting of two queues of fluid with arrival process generated by a big number of heavy-tailed On/Off sources, and application in road traffic and communication systems. Class-j fluid is assigned to queue j, $$j=1,2$$j=1,2. Server 2 visits both queues to process or let pass the corresponding fluid class. If there is class-2 fluid in the system, it is processed by server 2 until the queue is empty, and only then server 2 visits queue 1, revisiting queue 2 and restarting the cycle as soon as new class-2 fluid arrives, with zero switchover times. Server 1 is an "extra" server which continuously processes class-1 fluid (if there is any). During the visits of server 2 to queue 1, class-1 fluid is simultaneously processed by both servers (possibly at different speeds). We prove a heavy-traffic limit theorem for a suitable workload process associated with this model. Our limit process is a two-dimensional reflected fractional Brownian motion living in a convex polyhedron. A key ingredient in the proof is a version of the Invariance Principle of Semimartingale reflecting Brownian motions which, in turn, is also proved.

This paper considers a discrete-time Geo/G/1 queue under the Min(N,D)-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D, whichever occurs first (Min(N,D)-policy). By using renewal process theory and total probability decomposition technique, the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursive expression of the z-transformation of the transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n +. Meanwhile, the authors obtain the explicit expressions of the additional queue length distribution. Furthermore, the important relations between the steady state queue length distributions at different time epochs n -, n and n + are also reported. Finally, the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution, and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.

This paper studies a finite buffer size Geo X / G / 1 / N queue with single working vacation and different input rates. By combining two classic methods of supplementary variable and embedded Markov chain, the matrix-form solutions of queue-length distributions at departure, arbitrary, pre-arrival and outside observer's observation epochs are obtained. Based on these queue-length distributions, some performance measures are also derived and some numerical experiments are carried out. Then, an optimal control on state-dependent operating cost of a production to order is presented. HighlightsAnalyze a new queue model raised from practical production system.Obtain the solutions in form of formulae for queue length distributions at different epochs.Optimal analysis for the state-dependent cost of system.

This paper explores a discrete-time Geo / G / 1 queueing system with N-policy and D-policy wherein customers arrive at the system in variable input rates according to the states of the server. When the system becomes empty, the server stays idle until the number of waiting customers reaches N or the sum of the service times of the waiting customers in the system reaches or exceeds a predetermined positive integer D, whichever happens first (referred to as Min(N, D)-policy). Employing the renewal theory, the total probability decomposition law and the probability generating function technique we obtain the probability generating function for the transient queue size distribution at time epoch $$n^+$$ . Also, the explicit recursive formulas of the steady-state queue length distribution at time epochs $$n^+$$ , n, $$n^-$$ and outside observer’s time epoch are derived, respectively. Finally, some numerical experiments are conducted to investigate the sensitivity of system performance measures and the optimization of system capacity.

A discrete-timeGeo/G/1queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type-icycle, and mean time that the system resides in phase0. In addition, we show that theM/G/1queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.

A generalised Pollaczek–Khinchin formula for the Polya/G/1 queue, with a Polya peaked arrival process, general distributed service times, and infinite number of waiting positions, is obtained. It is shown that the peakedness of the number of arrivals and the variance of the service time lead to a significant increase in the service delay and queue length.