Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions

This paper considers the queue length distribution in a class of FIFO single-server queues with (possibly correlated) multiple arrival streams, where the service time distribution of customers may be different for different streams. It is widely recognized that the queue length distribution in a FIFO queue with multiple non-Poissonian arrival streams having different service time distributions is very hard to analyze, since we have to keep track of the complete order of customers in the queue to describe the queue length dynamics. In this paper, we provide an alternative way to solve the problem for a class of such queues, where arrival streams are governed by a finite-state Markov chain. We characterize the joint probability generating function of the stationary queue length distribution, by considering the joint distribution of the number of customers arriving from each stream during the stationary attained waiting time. Further we provide recursion formulas to compute the stationary joint queue length distribution and the stationary distribution representing from which stream each customer in the queue arrived.     

Single-Server Queues with Markov-Modulated Arrivals and Service Speed

This paper considers single-server queues with several customer classes. Arrivals of customers are governed by the underlying continuous-time Markov chain with finite states. The distribution of the amount of work brought into the system on arrival is assumed to be general, which may differ with different classes. Further, the service speed depends on the state of the underlying Markov chain. We first show that given such a queue, we can construct the corresponding new queue with constant service speed by means of a change of time scale, and the time-average quantities of interest in the original queue are given in terms of those in the new queue. Next we characterize the joint distribution of the length of a busy period and the number of customers served during the busy period in the original queue. Finally, assuming the FIFO service discipline, we derive the Laplace–Stieltjes transform of the actual waiting time distribution in the original queue.     

Matrix Product-Form Solution for an LCFS-PR Single-Server Queue with Multiple Arrival Streams Governed by a Markov Chain

This paper considers a stationary single-server queue with multiple arrival streams governed by a Markov chain, where customers are served on an LCFS preemptive-resume basis. Service times of customers from each arrival stream are generally distributed and service time distributions for different arrival streams may be different. Under these assumptions, it is shown that the stationary joint distribution of queue strings representing from which arrival stream each customer in the system arrived and remaining service times of respective customers in the system has a matrix product-form solution, where matrices constituting the solution are given in terms of the infinitesimal generator of a certain Markov chain. Compared with the previous works, the result in this paper is more general in the sense that general service time distributions are allowed, and it has the advantage of computational efficiency. Note also that the result is a natural extension of the classical result for the LCFS-PR M/G/1 queue. Further, utilizing the matrix product-form solution, we derive a new expression of the vector Laplace–Stieltjes transform of the stationary distribution of unfinished work in the work-conserving single-server queue with multiple arrival streams governed by a Markov chain, which is given by the sum of matrix-geometric series.     

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     

Analysis of an Infinite-Server Queue with Batch Markovian Arrival Streams

This paper considers an infinite-server queue with multiple batch Markovian arrival streams. The service time distribution of customers may be different for different arrival streams, and simultaneous batch arrivals from more than one stream are allowed. For this queue, we first derive a system of ordinary differential equations for the time-dependent matrix joint generating function of the number of customers in the system. Next assuming phase-type service times, we derive explicit and numerically feasible formulas for the time-dependent and limiting joint binomial moments. Further, some numerical examples are provided to discuss the impact of system parameters on the performance.     

Invariance relations in single server queues with LCFS service discipline

This paper is concerned with single server queues having LCFS service discipline. We give a condition to hold an invariance relation between time and customer average queue length distributions in the queues. The relation is a generalization of that in an ordinary GI/M/1 queue. We compare the queue length distributions for different single server queues with finite waiting space under the same arrival process and service requirement distribution of customer and derive invariance relations among them.This research was supported in part by a grant from the Tokyo Metropolitan Government. The latter part of this paper was written while the author resided at the University of California, Berkeley.     

A Communication Multiplexer Problem: Two Alternating Queues with Dependent Randomly-Timed Gated Regime

Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling instants and at an arbitrary moment, mean busy period duration and mean cycle time length, expected number of messages transmitted during a busy period and mean waiting times. Finally, we present graphs of numerical results comparing the mean waiting times in the two queues as functions of the relative loads, showing the effect of the RTG regime.     

Queues with Service Rate Controlled by a Delayed Feedback

We consider a single queue with a Markov modulated Poisson arrival process. Its service rate is controlled by a scheduler. The scheduler receives the workload information from the queue after a delay. This queue models the buffer in an earth station in a satellite network where the scheduler resides in the satellite. We obtain the conditions for stability, rates of convergence to the stationary distribution and the finiteness of the stationary moments. Next we extend these results to the system where the scheduler schedules the service rate among several competing queues based on delayed information about the workloads in the different queues.     

Predicting Time-Dependent Distributions of Queues and Delays for Road Traffic at Roundabouts and Priority Junctions

Vehicle queues and delays at busy road junctions have to be treated time-dependently when the traffic demand and the available capacity are approximately equal. Existing methods allow the queue length at a given time to be directly estimated as an average over all possible evolutions of the queueing system consistent with the given initial conditions and the time-dependent arrival and service rates. The paper describes the development of methods to predict the underlying distributions. Estimates of the variance and the overall frequency distribution for queue length and delay are obtained by simulating an M/M/1 queueing model with parameters varying with time. Predictive models are developed to represent the simulation results. They require as input values of parameters describing the duration of the peak and the time-average traffic intensities and capacities.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Queues with Markov Modulated Service Rates

In this paper, we consider a queue whose service speed changes according to an external environment that is governed by a Markov process. It is possible that the server changes its service speed many times while serving a customer. We derive first and second moments of the service time of customers in system using first step analysis to obtain an insight on the service process. In fact, we obtain an intriguing result in that the moments of service time actually depend on the arrival process! We also show that the mean service rate is not the reciprocal of the mean service time. Further, since it is not possible to obtain a closed form expression for the queue length distribution, we use matrix geometric methods to compute performance measures such as average queue length and waiting time. We apply the method of large deviations to obtain tail distributions of the workload in the queue using the concept of effective bandwidth. We present two applications in computer systems: (1) Web server with multi-class requests and (2) CPU with multiple processes. We illustrate the analysis and various methods discussed with the help of numerical examples for the above two applications. AMS subject classification: 90B22, 68M20     

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     

Analysis of a finite MAP/G/1 queue with group services

The finite capacity queues, GI/PH/1/N and PH/G/1/N, in which customers are served in groups of varying sizes were recently introduced and studied in detail by the author. In this paper we consider a finite capacity queue in which arrivals are governed by a particular Markov renewal process, called a Markovian arrival process (MAP). With general service times and with the same type of service rule, we study this finite capacity queueing model in detail by obtaining explicit expressions for (a) the steady-state queue length densities at arrivals, at departures and at arbitrary time points, (b) the probability distributions of the busy period and the idle period of the server and (c) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures when services are of phase type are discussed. An illustrative numerical example is presented.     

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.     

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     

Mixed gated/exhaustive service in a polling model with priorities

In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service discipline, i.e. only customers standing in front of the gate are served during this visit. In contrast, high priority customers arriving during the visit period of the queue are allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type. Through numerical examples we illustrate that the mixed gated/exhaustive service discipline can significantly decrease waiting times of high priority jobs. In many cases there is a minimal negative impact on the waiting times of low priority customers but, remarkably, it turns out that in polling systems with larger switch-over times there can be even a positive impact on the waiting times of low priority customers.     

A two-queue,one-server model with priority for the longer queue

The queueing model studied consists of one server and two queues. Each queue has its own Poisson arrival stream and service time distribution. After a service completion, the server proceeds with a customer from the longer queue, if the queues are unequal; if the queues are equal, the server chooses with some probability a customer from one of the queues. The model is of practical interest in performance analysis, but also of theoretical interest because the functional equation to be solved has not yet been studied in the queueing literature. A basic analysis of this functional equation is presented. Some numerical results are given to assess the influence of the present service discipline. Some new properties of L.S. transforms of service time distributions are discussed in the appendix.Dr. T. Katayama has formulated the present problem and brought it to the author's attention during his visit in October/November 1984 to the NTT-Electr. Comm. Lab.'s Musashino, Tokyo 180.     

A unified algorithm for computing the stationary queue length distributions inM(k)/G/1/N and GI/M(k)/1/N queues

An iterative algorithm is developed for computing numerically the stationary queue length distributions in M/G/1/N queues with arbitrary state-dependent arrivals, or simply M(k)/G/1/N queues. The only input requirement is the Laplace-Stieltjes transform of the service time distribution.In addition, the algorithm can also be used to obtain the stationary queue length distributions in GI/M/1/N queues with state-dependent services, orGI/M(k)/1/N, after establishing a relationship between the stationary queue length distributions inGI/M(k)/1/N and M(k)/G/1/N+1 queues.Finally, we elaborate on some of the well studied special cases, such asM/G/1/N queues,M/G/1/N queues with distinct arrival rates (which includes the machine interference problems), andGI/M/C/N queues. The discussions lead to a simplified algorithm for each of the three cases.     

Sojourn time distributions in the queue defined by a general QBD process

We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.