A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots

A simple and complete solution to determine the distributions of queue lengths at different observation epochs for the model GI^X/Geo/c/N is presented. In the past, various discrete-time queueing models, particularly the multi-server bulk-arrival queues with finite-buffer have been solved using complicated methods that lead to results in a non-explicit form. The purpose of this paper is to present a simple derivation for the model GI^X/Geo/c/N that leads to a complete solution in an explicit form. The same method can also be used to solve the GI^X/Geo/c/N queues with heavy-tailed inter-batch-arrival time distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue lengths at different time epochs. All queue-length distributions are in the form of sums of geometric terms.

     

Diffusion Approximations for Queues with Markovian Bases

Consider a base family of state-dependent queues whose queue-length process can be formulated by a continuous-time Markov process. In this paper, we develop a piecewise-constant diffusion model for an enlarged family of queues, each of whose members has arrival and service distributions generalized from those of the associated queue in the base. The enlarged family covers many standard queueing systems with finite waiting spaces, finite sources and so on. We provide a unifying explicit expression for the steady-state distribution, which is consistent with the exact result when the arrival and service distributions are those of the base. The model is an extension as well as a refinement of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a birth-and-death process. As a typical base, we still focus on birth-and-death processes, but we also consider a class of continuous-time Markov processes with lower-triangular infinitesimal generators.     

Finite-Buffer Queues with Workload-Dependent Service and Arrival Rates

We consider M/G/1 queues with workload-dependent arrival rate, service speed, and restricted accessibility. The admittance of customers typically depends on the amount of work found upon arrival in addition to its own service requirement. Typical examples are the finite dam, systems with customer impatience and queues regulated by the complete rejection discipline. Our study is motivated by queueing scenarios where the arrival rate and/or speed of the server depends on the amount of work present, like production systems and the Internet.First, we compare the steady-state distribution of the workload in two finite-buffer models, in which the ratio of arrival and service speed is equal. Second, we find an explicit expression for the cycle maximum in an M/G/1 queue with workload-dependent arrival and service rate. And third, we derive a formal solution for the steady-state workload density in case of restricted accessibility. The proportionality relation between some finite and infinite-buffer queues is extended. Level crossings and Volterra integral equations play a key role in our approach.AMS subject classification: 60K25, 90B22     

Queues with waiting time dependent service

Motivated by service levels in terms of the waiting-time distribution seen, for instance, in call centers, we consider two models for systems with a service discipline that depends on the waiting time. The first model deals with a single server that continuously adapts its service rate based on the waiting time of the first customer in line. In the second model, one queue is served by a primary server which is supplemented by a secondary server when the waiting of the first customer in line exceeds a threshold. Using level crossings for the waiting-time process of the first customer in line, we derive steady-state waiting-time distributions for both models. The results are illustrated with numerical examples.     

Moderate deviations for queues in critical loading

We establish logarithmic asymptotics of moderate deviations for queue-length and waiting-time processes in single server queues and open queueing networks in critical loading. Our results complement earlier diffusion approximation results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Factorization and Stochastic Decomposition Properties in Bulk Queues with Generalized Vacations

This paper considers a class of stationary batch-arrival, bulk-service queues with generalized vacations. The system consists of a single server and a waiting room of infinite capacity. Arrivals of customers follow a batch Markovian arrival process. The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in groups of fixed size B. For this class of queues, we show that the vector probability generating function of the stationary queue length distribution is factored into two terms, one of which is the vector probability generating function of the conditional queue length distribution given that the server is on vacation. The special case of batch Poisson arrivals is carefully examined, and a new stochastic decomposition formula is derived for the stationary queue length distribution.AMS subject classification: 60K25, 90B22, 60K37     

A two-queue model with Bernoulli service schedule and switching times

In this paper, we present a detailed analysis of a cyclic-service queueing system consisting of two parallel queues, and a single server. The server serves the two queues with a Bernoulli service schedule described as follows. At the beginning of each visit to a queue, the server always serves a customer. At each epoch of service completion in the ith queue at which the queue is not empty, the server makes a random decision: with probability p_i, it serves the next customer; with probability 1-p_i, it switches to the other queue. The server takes switching times in its transition from one queue to the other. We derive the generating functions of the joint stationary queue-length distribution at service completion instants, by using the approach of the boundary value problem for complex variables. We also determine the Laplace-Stieltjes transforms of waiting time distributions for both queues, and obtain their mean waiting times. This revised version was published online in June 2006 with corrections to the Cover Date.     

Transient flows in queueing systems via closure approximations

We present a method for obtaining approximations to the distribution of flow times of customers in arbitrary queueing systems. We first propose approximations for uni-variate and multi-variate distributions of non-negative random variables. Then using a closure approximation, we show that the distribution of flow time can be calculated recursively. Computational results for the single server, multi-server and tandem queues are encouraging, with less than 5%average error in the mean flow time in most cases. The average error in the variance of flow times is found to be less than 10% for the more regular distributions.     

Analyzing a multiserver bulk-service finite-buffer queue

This paper analyzes a finite-buffer multiserver bulk-service queueing system in which the interarrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the queue-length distributions at prearrival and arbitrary epochs. We also present Laplace–Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.     

Reliability Analysis of the Retrial Queue with Server Breakdowns and Repairs

Retrial queues have been widely used to model many problems arising in telephone switching systems, telecommunication networks, computer networks and computer systems, etc. It is of basic importance to study reliability of retrial queues with server breakdowns and repairs because of limited ability of repairs and heavy influence of the breakdowns on the performance measure of the system. However, so far the repairable retrial queues are analyzed only by queueing theory. In this paper we give a detailed analysis for reliability of retrial queues. By using the supplementary variables method, we obtain the explicit expressions of some main reliability indexes such as the availability, failure frequency and reliability function of the server. In addition, some special queues, for instance, the repairable M/G/1 queue and repairable retrial queue can be derived from our results. These results may be generalized to the repairable multi-server retrial models.     

供应链中的MQ(t)/Gj/∞排队群

本文研究由一个非时齐Poisson到达过程驱动的多个MQ(t)／Gj／∞型排队系统．当一个K型到达发生时(K(?){1,…,m}),对每一个j∈K,顾客按批量QjK进入排队系统j,(Q1K,…,QmK)是相依的随机向量．我们导出了联合队长过程和联合输出过程的多变量母函数,并给出各排队系统之间的相关性分析．这些结果为非时齐供应链分析提供了理论依据     

Computation of the quasi-stationary distributions inM(n)/GI/1/K andGI/M(n)/1/K queues

In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.     

Nonpreemptive Multi-Server Priority Queues

Hokstad recently published an approximate method for calculating the behaviour of an M/G/m queue. This note applies his results to the nonpreemptive priority situation with two priority classes having the same service-time distribution. Laplace transforms and the first two moments of the waiting-time distributions are given.     

The Asymptotic Workload Behavior of Two Coupled Queues

We consider a system of two coupled queues Q ₁ and Q ₂. When both queues are backlogged, they are each served at unit rate. However, when one queue empties, the service rate at the other queue increases. Thus, the two queues are coupled through the mechanism for dynamically sharing surplus service capacity. We derive the asymptotic workload behavior at Q ₁ for various scenarios where at least one of the two queues has a heavy-tailed service time distribution. First of all, we consider a situation where the traffic load at Q ₁ is below the nominal unit service rate. We show that if the service time distribution at Q ₁ is heavy-tailed, then the workload behaves exactly as if Q ₁ is served in isolation at a constant rate, which only depends on the service time distribution at Q ₂ through its mean. In addition, we establish that if the service time distribution at Q ₁ is exponential, then the workload distribution is either exponential or semi-exponential, depending on whether the traffic load at Q ₂ exceeds the nominal service rate or not. Next, we focus on a regime where the traffic load at Q ₁ exceeds the nominal service rate, so that Q ₁ relies on the surplus capacity from Q ₂ to maintain stability. In that case, the workload distribution at Q ₁ is determined by the heaviest of the two service time distributions, so that Q ₁ may inherit potentially heavier-tailed characteristics from Q ₂.     

Large Deviations of Queues Sharing a Randomly Time-Varying Server

We consider a discrete-time model where multiple queues, each with its own exogenous arrival process, are served by a server whose capacity varies randomly and asynchronously with respect to different queues. This model is primarily motivated by the problem of efficient scheduling of transmissions of multiple data flows sharing a wireless channel. We address the following problem of controlling large deviations of the queues: find a scheduling rule, which is optimal in the sense of maximizing

Analysis of multiserver retrial queueing system: A martingale approach and an algorithm of solution

The paper studies a multiserver retrial queueing system withm servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter μ₁. A time between retrials is exponentially distributed with parameter μ₂ for each customer. Using a martingale approach the paper provides an analysis of this system. The paper establishes the stability condition and studies a behavior of the limiting queue-length distributions as μ₂ increases to infinity. As μ₂→∞, the paper also proves the convergence of appropriate queue-length distributions to those of the associated “usual” multiserver queueing system without retrials. An algorithm for numerical solution of the equations, associated with the limiting queue-length distribution of retrial systems, is provided. AMS 2000 Subject classifications: 60K25 60H30.     

Heavy-traffic limits for many-server queues with service interruptions

We establish many-server heavy-traffic limits for G/M/n+M queueing models, allowing customer abandonment (the +M), subject to exogenous regenerative service interruptions. With unscaled service interruption times, we obtain a FWLLN for the queue-length process, where the limit is an ordinary differential equation in a two-state random environment. With asymptotically negligible service interruptions, we obtain a FCLT for the queue-length process, where the limit is characterized as the pathwise unique solution to a stochastic integral equation with jumps. When the arrivals are renewal and the interruption cycle time is exponential, the limit is a Markov process, being a jump-diffusion process in the QED regime and an O–U process driven by a Levy process in the ED regime (and for infinite-server queues). A stochastic-decomposition property of the steady-state distribution of the limit process in the ED regime (and for infinite-server queues) is obtained.     

Batch service systems with heterogeneous servers

Bulk-service multi-server queues with heterogeneous server capacity and thresholds are commonly seen in several situations such as passenger transport or package delivery services. In this paper, we develop a novel decomposition-based solution approach for such queues using arguments from renewal theory. We then obtain the distribution of the waiting time measure for multi-type server systems. We also obtain other useful performance measures such as utilization, expected throughput time, and expected queue lengths.

     

Two Queues with Weighted One-Way Overflow

We consider an open queueing network consisting of two queues with Poisson arrivals and exponential service times and having some overflow capability from the first to the second queue. Each queue is equipped with a finite number of servers and a waiting room with finite or infinite capacity. Arriving customers may be blocked at one of the queues depending on whether all servers and/or waiting positions are occupied. Blocked customers from the first queue can overflow to the second queue according to specific overflow routines. Using a separation method for the balance equations of the two-dimensional server and waiting room demand process, we reduce the dimension of the problem of solving these balance equations substantially. We extend the existing results in the literature in three directions. Firstly, we allow different service rates at the two queues. Secondly, the overflow stream is weighted with a parameter p ∈ [0,1], i.e., an arriving customer who is blocked and overflows, joins the overflow queue with probability p and leaves the system with probability 1 − p. Thirdly, we consider several new blocking and overflow routines. An erratum to this article can be found at