Transient Queue Length Distribution Research Articles

Performance analysis of an M/G/1 queue with bi-level randomized (p, N1, N2)-policy

This paper proposes an M/G/1 queueing model with bi-level randomized (p, N1, N2)-policy. That is, after all of the customers in the system are served, the server is closed down immediately. If N1(≥ 1) customers are accumulated in the queue, the server is activated for service with probability p(0 ≤ p ≤ 1) or still left off with probability 1 − p. When the number of customers in the system becomes N2(≥ N1), the server begins serving the waiting customers until the system becomes empty again. Using the total probability decomposition technique and the Laplace transform, we study the transient queue length distribution and obtain the expressions of the Laplace transform of the transient queue-length distribution with respect to time t. Then, employing L’Hospital’s rule and some algebraic operations, the explicit recursive formulas of the steady-state queue-length distribution, which can be used to accurately evaluate the probabilities of queue length, are presented. Moreover, some other important queuing performance indices, such as the explicit expressions of its probability generating function of the steady-state queue-length distribution, the expected queue size and so on, are derived. Meanwhile, we investigate the system capacity optimization design by the steady-state queue-length distribution. Finally, an operating cost function is constructed, and by numerical calculation, we find the minimum of the long-run average cost rate and the optimal bi-level threshold policy (N*1,N*2) that satisfies the average waiting time constraints.

Queues with path-dependent arrival processes

AbstractWe study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \rightarrow \infty$. We establish a heavy-traffic diffusion limit for the $\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.

An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

Performance and reliability analysis of a repairable discrete‐time Geo/G/1 queue with Bernoulli feedback and randomized policy

This paper is concerned with a discrete‐time Geo/G/1 repairable queueing system with Bernoulli feedback and randomized ‐policy. The service station may be subject to failures randomly during serving customers and therefore is sent for repair immediately. The ‐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 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.

Analysis ofD-policy discrete-timeGeo/G/1queue with secondJ-optional service and unreliable server

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.

Queue size distribution of Geo/G/1 queue under the Min(N,D)-policy

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.

Queue size distribution and capacity optimum design for [formula omitted]-policy [formula omitted] queue with setup time and variable input rate

In this paper we consider a discrete-time Geo/G/1 queue with N-policy and setup time, where the customers’ input rate varies according to the server’s status: idle, setup and busy states. By using the total probability decomposition technique, we study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursion expressions of the z-transformation of the transient queue length distribution and the recursion expressions of the steady state queue length distribution at arbitrary time epoch n+. The results obtained in this paper indicate that the queue length distribution in equilibrium no longer follows the stochastic decomposition discipline. The important relations between the steady state queue length distributions at different time epochs (n−,n,n+) are discovered. Finally, by numerical examples, we discuss the sensitivity of the steady state queue length distribution towards system parameters, and illustrate the application of the expressions for the steady state queue length distribution in the system capacity design.

Poster

Large scale systems such as customer contact centers, like telephone call centers, as well as healthcare centers, like hospitals, have customer inflow-outflow dynamics with many common features. The customer arrival patterns may have time of day or seasonal effects. Moreover, customer population sizes tend to be large where the individual actions are intrinsic and independent of other customer actions and there are multiple service agents, so many customers have access to services in parallel. Finally, arriving customers engaging in service may be delayed if all the available agents are busy. Moreover, these waiting customers may decide to leave the systems if they feel that their delay in receiving service is excessively long. A fundamental Markov process, with dynamic rates, queueing model for large service systems is a multiserver queue with non-homogeneous Poisson arrivals as well as service and customer abandonment times that are exponentially distributed. An asymptotic scaling for these queues leads to both functional strong law of large numbers and central limit theorems. The first yields a dynamical system that we call a fluid model. The second scaled limit yields a diffusion process model that is Gaussian under conditions involving the fluid model not lingering too closely to the number of servers. Finally, the fluid model coupled with the mean and covariance of the Gaussian model is also a dynamical system. These results are a special case of a general asymptotic theory for Markovian service networks as derived in Mandelbaum, Massey, and Reiman [1]. In practice, these results yield useful Gaussian approximations to the transient queue length distribution. However, these estimates tend not to work as well when the lingering effect is significant. We can improve these methods by introducing a new technique that is called the Gaussian-skewness approximation (GSA). It is the special case of a general Hermite polynomial expansion for a Gaussian random variable. We then obtain dynamical systems that approximate the mean, variance and higher cumulant moments of the queueing process for a more accurate, non-Gaussian estimation. The numerical example that we consider in this paper, to illustrate our approximation methods, has a time interval (0, 20], an arrival rate function λ(t) = 20 + 10 sin(t), a service rate ? = 1.0, abandonment rate = 0.5, and the number of servers is c = 20. We highlight the specific time t = 4.25 in Figure 1 since the simulated histogram for the queue length distribution here is when the mean number in the queueing system has gone from being greater than the number of servers, or overloaded, to equaling the number exactly at this time. This is when our queue length is the least Gaussian in appearance. Note that this is also a time where the skewness attains a locally maximal value. Figure 1 also illustrates that GSA (solid curve) skews the its density distribution more to the left compared to the other two Gaussian approximations (both dotted lines where the smaller one at the peak, which also is the largest one left of the peak, uses the fluid model for the mean at this time and the diffusion model for the variance). Moreover, GSA yields an asymmetric density so it does a better job of fitting both queueing distributional tails here than any Gaussian method.

Analysis of [formula omitted] queue with [formula omitted]-entering discipline during adaptive multistage vacations

This paper studies the M X / G / 1 queue with adaptive multistage server vacations in which the customers who arrive during server vacations enter the system with probability p ( 0 ≤ p ≤ 1 ) . Using a simple method, the recursive formulae of the transient queue-length distribution and the equilibrium queue-length distribution are obtained. Furthermore, the distributions of the server idle period and the server circular busy period are also discussed.

Transient solution for queue-length distribution of Geometry/G/1 queueing model

In this paper, the Geometry/G/1 queueing model with inter-arrival times generated by a geometric(parameter p) distribution according to a late arrival system with delayed access and service times independently distributed with distribution {gj}, j ≥ 1 is studied. By a simple method (techniques of probability decomposition, renewal process theory) that is different from the techniques used by Hunter (1983), the transient property of the queue with initial state i(i ≥ 0) is discussed. The recursion expression for u-transform of transient queue-length distribution at any time point n+ is obtained, and the recursion expression of the limiting queue length distribution is also obtained.

ALGORITHMIC COMPUTATION OF THE TRANSIENT QUEUE LENGTH DISTRIBUTION IN THE BMAP/D/c QUEUE

This paper proposes a numerically feasible algorithm for the transient queue length distribution in the BMAP/D/c queue. The proposed algorithm ensures the accuracy of the computational result and it is applicable not only to the stable case but also to the unstable case. This paper also discusses a numerical procedure to compute moments of the transient queue length distribution. Finally, some numerical examples are presented to demonstrate the applicability of the proposed algorithm.

Lattice paths combinatorics applied to transient queue length distribution of C2/M/1 queues and busy period analysis of bulk queues C2b/M/1

Abstract This paper aims at deriving explicit transient queue length distribution for GI/M/1 system and busy period analysis of bulk queue GIb/M/1 through lattice paths (LPs) combinatorics. The general interarrival time distribution is approximated by two-phase Cox distribution, C2, that has Markovian property, enabling us to represent the processes by two-dimensional LPs. As distributions C2 cover a wide class of distributions that have rational Laplace–Stieltjes transforms (LSTs) with square coefficient of variation lying in [ 1 2 ,∞) , the results obtained are applicable to a large class of real life situations. Some numerical results for the C2b/M/1 model are also given.

Transient analysis applied to traffic modeling

Traffic modeling has been an extensive area of research in the last few years, and a lot of modeling effort has been devoted to better understand the issues involved in multiplexing traffic over high speed links. The goals of the performance analyst include the development of accurate traffic models to predict, with sufficient accuracy, the impact of the traffic generated by applications over the network resources, and the evaluation of the quality of service (QoS) being achieved. Performance studies include determining buffer behavior, evaluate cell loss probability, admission control algorithms, and many others.One performance study issue is the calculation of descriptors from different traffic models. In the literature, one can find a large number of models that have been proposed, including Markov and non-Markovian models [1]. Although not possessing the long-range dependence property, Markov models are still attractive not only due to their mathematical tractability but also because it has been shown that long-range correlations can be approximately obtained from certain kinds of Markovian models (e.g. [11]). Furthermore, works such as [8] show that Markov models can be used to accurately predict performance metrics.Once a set of traffic models is chosen, the modeler should obtain the desired performance measures. Hopefully the measures should be calculated analytically using efficient algorithms.The modeling steps briefly outlined above may require the transient analysis of general Markovian models, including Markov reward models. One of the goals of this work is to present new algorithms we developed to obtain efficiently measures such as the transient queue length distribution (and from that, the packet loss ratio as a function of time) directly from the model of the source feeding the queue. We also obtain second order descriptors such as the index of dispersion and the autocovariance from the source models. Using these algorithms the modeler can evaluate the efficacy of different Markovian models to predict performance metrics.

The MX/G/1 queue with queue length dependent service times

We deal with the MX/G/1 queue where service times depend on the queue length at the service initiation. By using Markov renewal theory, we derive the queue length distribution at departure epochs. We also obtain the transient queue length distribution at time t and its limiting distribution and the virtual waiting time distribution. The numerical results for transient mean queue length and queue length distributions are given.

THE QUEUE-LENGTH DISTRIBUTION FOR Mx/G/1 QUEUE WITH SINGLE SERVER VACATION

This paper studies the bulk-arrival Mx/G/1 queue with single server vacation. By introducing the server busy period and using the Laplace transform, the recursion expression of the Laplace transform of the transient queue-length distribution is derived. Furthermore, the distribution and stochastic decomposition result of the queue length at a random point in equilibrium are obtained. Especially some results for the single-arrival M/G/1 queue with single server vacation and bulk-arrival Mx/G/1 queue but with no server vacation can be derived directly by the results obtained in this paper.

The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

Transient solutions for denumerable-state Markov processes

The algorithm for the transient solution for the denumerable state Markov process with an arbitrary initial distribution is given in this paper. The transient queue length distribution for a general Markovian queueing system can be obtained by this algorithm. As examples, some numerical results are presented.

Transient solutions for denumerable-state Markov processes

The algorithm for the transient solution for the denumerable state Markov process with an arbitrary initial distribution is given in this paper. The transient queue length distribution for a general Markovian queueing system can be obtained by this algorithm. As examples, some numerical results are presented.

A family of bounds for the transient behavior of a Jackson network

Using operator methods, we derive a family of stochastic bounds for the Jackson network. For its transient joint queue-length distribution, we can stochastically bound it above by various networks that decouple into smaller independent Jackson networks. Each bound is determined by a distinct partitioning of the index set for the nodes. Except for the trivial cases, none of these bounds can be extended to a sample path ordering between it and the original network. Finally, we can partially order the bounds themselves whenever one partition of the index set is the refinement of another. These results suggest new types of partial orders for stochastic processes that are not equivalent to sample-path orderings.

New results for the Jackson network

Using operator methods, we prove a general decomposition theorem for Jackson networks. For its transient joint queue-length distribution, we can stochastically bound it above by a network that decouples into smaller independent Jackson networks.