Service Time Distribution Function Research Articles

Shannonian Maximum Entropy Balking Threshold Mechanism (BTM) for a Stable M/G/1 Queue with Significant Applications of M/G/1 Queue Theory to Augmented Reality (AR)

An exposition is undertaken to analytically derive validate the Shannonian Maximum Entropy BTM for the underlying stable queue. Most importantly, the analytic derivation of the upper and lower bounds 0f the absolute difference between Shannonian Cumulative service time distribution functions (CDFs) with and without balking. Typical numerical experiments are provided. Additionally, some applications of M/G/1 queue theory to AR are given. Some challenging open problems are addressed combined with closing remarks and future research directions.

Adaptive minimax estimation of service time distribution in the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from departure data

This article deals with the problem of estimating the service time distribution of the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from observation of the departure epochs. We develop minimax optimal estimators of G and study behavior of the minimax pointwise risk over a suitable family of service time distribution functions. In addition, we address the problem of adaptive estimation and propose a data–driven estimation procedure that adapts to unknown smoothness of the service time distribution function G. Lastly, a numerical study is presented to illustrate practical performance of the developed adaptive procedure.

Experiences with implementing parallel discrete-event simulation on GPU

Modern graphics processing units (GPUs) offer much more computational power than recent CPUs by providing a vast number of simple, data-parallel, multithreaded cores. In this study, we focus on the use of a GPU to perform parallel discrete-event simulation. Our approach is to use a modified service time distribution function to allow more independent events to be processed in parallel. The implementation issues and alternative strategies will be discussed in detail. We describe and compare our experience and results in using Thrust and CUB, two open-source parallel algorithms libraries which resemble the C $${++}$$ Standard Template Library, to build our tool. The experimental results show that our implementation can be two orders of magnitude faster than the sequential simulation for large-scale simulation models.

Two-parameter process limits for infinite-server queues with dependent service times via chaining bounds

We prove two-parameter process limits for infinite-server queues with weakly dependent service times satisfying the $$\rho $$ -mixing condition. The two-parameter processes keep track of the elapsed or residual service times of customers in the system. We use the new methodology developed in Pang and Zhou (Stoch Process Appl 127(5):1375–1416, 2017) to prove weak convergence of two-parameter stochastic processes. Specifically, we employ the maximal inequalities for two-parameter queueing processes resulting from the method of chaining. This new methodology requires a weaker mixing condition on the service times than the $$\phi $$ -mixing condition in Pang and Whitt (Queueing Syst 73(2):119–146, 2013), as well as fewer regularity conditions on the service time distribution function.

The potentials method for a closed queueing system with hysteretic strategy of the service time change

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Multiserver systems with rejections and the equiprobable distribution of customers

The queueing systems with rejections, arbitrary service time distributions, and two ways of the distributions of customers over heterogeneous servers (the equiprobable customer distribution over all (idle and busy) servers and the equiprobable customer distribution over idle servers) are discussed and used to find the stationary distributions of the number of customers. Their invariance with respect to service time distribution functions is proved. For an MX/M/n/0 system with homogeneous servers and the equiprobable customer distribution over all servers, the algorithm is proposed for determining the stationary distributions of the number of customers when their number in the batch is less than six. MX/M/2/0 and MX/M/3/0 systems with the equiprobable customer distribution over all servers and the arbitrary distribution of the number of arriving customers are studied. The results are checked using simulation models based on GPSS World software.

Nonparametric inference from the M/G/1 workload

Consider an M/G/1 queue with unknown service-time distribution and unknown traffic intensity ρ. Given systematically sampled observations of the workload, we construct estimators of ρ and of the service-time distribution function, and we study asymptotic properties of these estimators.

Maximum entropy analysis to the N policy M/G/1 queueing system with a removable server

We study a single removable server in an M/G/1 queueing system operating under the N policy in steady-state. The server may be turned on at arrival epochs or off at departure epochs. Using the maximum entropy principle with several well-known constraints, we develop the approximate formulae for the probability distributions of the number of customers and the expected waiting time in the queue. We perform a comparative analysis between the approximate results with exact analytic results for three different service time distributions, exponential, 2-stage Erlang, and 2-stage hyper-exponential. The maximum entropy approximation approach is accurate enough for practical purposes. We demonstrate, through the maximum entropy principle results, that the N policy M/G/1 queueing system is sufficiently robust to the variations of service time distribution functions.

Subexponential asymptotics of a Markov-modulated random walk with queueing applications

Let {(Xn,Jn)} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {Fij}, Fij(B) = ℙ[X1 ∈ B, J1 = j | J0 = i], B ∈ B(ℝ), i, j ∈ E. If Fij([x,∞))/(1-H(x)) → Wij ∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G+(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼Xn < 0, at least one Wij > 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[supn≥0Sn > x] → (−𝔼Xn)−1 ∫x∞ ℙ[Xn > u]du as x → ∞, where Sn = ∑1nXk, S0 = 0. Two general queueing applications of this result are given.First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.

Subexponential asymptotics of a Markov-modulated random walk with queueing applications

Let {(X n ,J n )} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {F ij }, F ij (B) = ℙ[X 1 ∈ B, J 1 = j | J 0 = i], B ∈ B (ℝ), i, j ∈ E. If F ij ([x,∞))/(1-H(x)) → W ij ∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G +(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼X n < 0, at least one W ij > 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[sup n≥0 S n > x] → (−𝔼X n )−1 ∫ x ∞ ℙ[X n > u]du as x → ∞, where S n = ∑1 n X k , S 0 = 0. Two general queueing applications of this result are given. First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.

Heterogeneous-server loss systems with ordered entry: An anomaly

We consider a loss system with Poisson arrivals and ordered entry, and we suppose that server i has service-time distribution function F i . Consistent with the results of others, we verify that, in contrast with the classical Erlang loss system ( F i = F for all i), when F i ≠ F j then the loss probability is not insensitive to the form of F i or Fj (even when F i and F j have the same first moment). Our main point is to note a remarkable counterintuitive fact that is an easy consequence of this result: the policy of assigning arrivals to the fastest idle server does not necessarily produce the lowest loss probability.

On the stability condition of a precedence-based queueing discipline

The queueing model known as precedence-based queueing discipline, proposed in [8] by Tsitsiklis et al. is addressed. In this queueing model, there are infinitely many servers and for any pair of customers i and j such that i arrived later than j, there is a fixed probability p that i will have to wait for the end of the execution of j before starting its execution. In the case where the customer service times are deterministic and the arrival process is Poisson, these authors have derived the stability condition which determines the maximum arrival rate that keeps the system stable. For more general statistics, they conjectured that the stability condition would possibly depend on the complete service time distribution functions but only on the first moment of the inter-arrival times.In this paper, we consider this queueing model and relax the restrictive statistical assumptions mentioned above by only assuming that the service times and the inter-arrival times are stationary and ergodic sequences, so that these variables can receive general distribution functions and be correlated.We derive a general expression for the stability condition which in turn proves the above conjectures. The results are obtained by establishing pathwise evolution equations for these queueing systems, and then a schema which, in certain sense, generalizes the schema of Loynes for the G/G/1 queues.

On the stability condition of a precedence-based queueing discipline

The queueing model known as precedence-based queueing discipline, proposed in [8] by Tsitsiklis et al. is addressed. In this queueing model, there are infinitely many servers and for any pair of customers i and j such that i arrived later than j, there is a fixed probability p that i will have to wait for the end of the execution of j before starting its execution. In the case where the customer service times are deterministic and the arrival process is Poisson, these authors have derived the stability condition which determines the maximum arrival rate that keeps the system stable. For more general statistics, they conjectured that the stability condition would possibly depend on the complete service time distribution functions but only on the first moment of the inter-arrival times. In this paper, we consider this queueing model and relax the restrictive statistical assumptions mentioned above by only assuming that the service times and the inter-arrival times are stationary and ergodic sequences, so that these variables can receive general distribution functions and be correlated. We derive a general expression for the stability condition which in turn proves the above conjectures. The results are obtained by establishing pathwise evolution equations for these queueing systems, and then a schema which, in certain sense, generalizes the schema of Loynes for the G/G/1 queues.

Mixtures of exponential distributions in the gi/g/1 queue (with and without warming up)

We consider the GI/G/1 queue and its generalization with warming up in the stationary state. There are shown so called conservation theorems for mixtures of ex-ponential distributions where also negative weights Fk are allowed. By means of studying the corresponding characteristic functions at its poles and zeros we obtain the following results: If the service time distribution function (d. f.) is such a mixture and its characteristic function satisfies a certain condition then both the waiting time d.f. and the sojourn time d.f. are such mixtures in the GI/G/1

Bounds for the variance of the busy period of the M/G/∞ queue

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

Bounds for the variance of the busy period of the M/G/∞ queue

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

The Infinite Server Queue with Arrivals Generated by a Non-Homogeneous Compound Poisson Process

This paper considers the infinite server queue with arrivals generated by a non-homogeneous compound Poisson process. In such a system, customers arrive in groups of variable size, the arrival epochs of groups being points of a non-homogeneous Poisson process, and they are served without delay. The service times of customers who belong to the same group need not be independent nor identically distributed. Assuming that the system is initially empty, the transient distribution of the queue size and of the counting departure process is obtained. Also the limiting queue size distribution (when it exists) is determined and it is found to be insensitive to the form the service time distribution functions.

A Numerical Method to Obtain the Equilibrium Results for the Multiple Finite Source Priority Queueing Model

In this paper a numerical method is presented to obtain the equilibrium results of the multiple finite source queueing model having a single server using a nonpreemptive fixed priority service discipline and where each class of customers has a different exponential interarrival density function and arbitrary service time distribution function. This solution method uses the method of imbedded Markov Chain and the renewal reward theorem to find the proportion of time the server is busy with the customers of different classes. The equilibrium results are then related to these proportions using an extension of Little's formula.

Approximating the stationary waiting time distribution function of GI/GI/1-queues with arithmetic interarrival time and service time distribution function

This paper deals with the stationary waiting time distributionW of GI/GI/1 queues. It is restricted to the case that interarival time and service time have lattice distributions. First some estimations forW are presented. These estimations allow to construct bounds forW if an approximation ofW is given, where good approximations yield good bounds. Next we describe in detail some methods for calculation of approximations ofW. Finally we compare these methods, using a number of test problems. Diese Arbeit beschaftigt sich mit der stationaren WartezeitverteilungW von GI/GI/1 Warteschlangen. Die Darstellung beschrankt sich auf den Fall, daβ Zwischenankunftszeit und Bedienungszeit gitterformig verteilt sind. Zunachst werden Abschatzungen fur die stationare Wartezeitverteilung hergeleitet. Diese Abschatzungen gestatten es, aus einer gegebenen Naherung beidseitige Schranken furW zu berechnen. Dabei ist gesichert, daβ gute Naherungen gute Schranken liefern. Anschlieβend werden einige Moglichkeiten zur Berechnung vonW angegeben und mit Hilfe von Testbeispielen, unter Verwendung der beschriebenen Abschatzungen, verglichen.

A Note on Product-Form Solution for Queuing Networks with Poisson Arrivals and General Service-Time Distributions with Finite Means

Product form was first introduced by Jackson [Operations Research, 1957; Management Science, 1963] who considered only negative exponential service distributions with queue length dependent service rates. Posner and Bernholtz [Operations Research, 1968] showed that one has product form in the case where there are many classes of customers, service times are exponential with class queue length dependent service rates, and lag times (the time for a customer to travel between two servers is a random variable with continuously differentiable distribution function). Baskett, Chandy, Muntz and Palacios [JACM, 1975] extended these results to networks with multiple classes of customers and service time distributions which have rational Laplace transforms vanishing at ∞. Chandy, Howard and Towsley [JACM, 1977] extended this further to the case where the service time distribution functions are continuously differentiable. They also introduced the concept of station balance and established some important relationships between station balance, local balance and product form. In particular, for the closed queueing network they showed that if each queue in the network satisfies local balance in isolation with Poisson arrivals then one has product form