Asymptotics of waiting time distributions in the accumulating priority queue

Queueing Systems - We analyze the asymptotics of waiting time distributions in the two-class accumulating priority queue with general service times. The accumulating priority queue was suggested by...     

Waiting time analysis of the multiple priority dual queue with a preemptive priority service discipline

The dual queue consists of two queues, called the primary queue and the secondary queue. There is a single server in the primary queue but the secondary queue has no service facility and only serves as a holding queue for the overloaded primary queue. The dual queue has the additional feature of a priority scheme to help reduce congestion. Two classes of customers, class 1 and 2, arrive to the dual queue as two independent Poisson processes and the single server in the primary queue dispenses an exponentially distributed service time at the rate which is dependent on the customer’s class. The service discipline is preemptive priority with priority given to class 1 over class 2 customers. In this paper, we use matrix-analytic method to construct the infinitesimal generator of the system and also to provide a detailed analysis of the expected waiting time of each class of customers in both queues.     

Waiting time and queue length analysis of Markov-modulated fluid priority queues

Queueing Systems - This paper considers a multi-type fluid queue with priority service. The input fluid rates are modulated by a Markov chain, which is common for all fluid types. The service rate...     

Strategic bidding in a discrete accumulating priority queue

We consider an unobservable M/G/1 accumulating priority queue where homogeneous customers choose one of a finite number of priority classes. We show that there are either one or two pure Nash equilibrium strategies. In the latter case they are two consecutive classes and there exists an equilibrium strategy mixing between these two classes. We find the best-response function and show that it is unimodal, with follow-the-crowd and avoid-the-crowd instances.     

Strategic bidding in an accumulating priority queue: equilibrium analysis

This paper considers the pricing decisions of substitutable products which are produced by duopolistic manufacturers respectively and then sold by one common retailer to the consumers. Both the manufacturing cost and the customer demand for each product are characterized as fuzzy variables. Six expected value models are developed to explore the effects of the duopolistic manufacturers’ three pricing strategies (i.e. Bertrand competition, Stackelberg competition, cooperation) and vertical competition between the manufacturers and the retailer on the optimal pricing decisions and channel profit-split of a two echelon supply chain, and the corresponding analytical solutions are derived. Finally, a numerical example is given to illustrate the effectiveness of the proposed models, and to gain additional managerial insights.     

Flow time distributions in aK classM/G/1 priority feedback queue

In this paper, aK classM/G/1 queueing system with feedback is examined. Each arrival requires at least one, and possibly up toK service phases. A customer is said to be in classk if it is waiting for or receiving itskth phase of service. When a customer finishes its phasek ≤K service, it either leaves the system with probabilityp _k, or it instantaneously reenters the system as a classk + 1 customer with probability (1 −p _k). It is assumed thatp _k = 1. Service is non-preemptive and FCFS within a specified priority ordering of the customer classes. Level crossing analysis of queues and delay cycle results are used to derive the Laplace-Stieltjes Transform (LST) for the PDF of the sojourn time in classes 1,…,k;k ≤K.     

Waiting time analysis for a tandem queue with correlated service

The effect on overall waiting time including service is investigated when there is positive correlation between the first and second stage service times in a two stage tandem system. The importance of understage waiting (or “buffer”) space is underlined. The analysis is an application of the authors' work reported in [1].     

Waiting time distributions of runs in higher order Markov chains

We consider a {0,1}-valuedm-th order stationary Markov chain. We study the occurrences of runs where two 1’s are separated byat most/exactly/at least k 0’s under the overlapping enumeration scheme wherek≥0 and occurrences of scans (at leastk ₁ successes in a window of length at mostk, 1≤k ₁≤k) under both non-overlapping and overlapping enumeration schemes. We derive the generating function of first two types of runs. Under the conditions, (1) strong tendency towards success and (2) strong tendency towards reversing the state, we establish the convergence of waiting times of ther-th occurrence of runs and scans to Poisson type distributions. We establish the central limit theorem and law of the iterated logarithm for the number of runs and scans up to timen.     

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.     

Provable bounds for the mean queue lengths in a heterogeneous priority queue

We analyze a two-class two-server system with nonpreemptive heterogeneous priority structures. We use matrix–geometric techniques to determine the stationary queue length distributions. Numerical solution of the matrix–geometric model requires that the number of phases be truncated and it is shown how this affects the accuracy of the results. We then establish and prove upper and lower bounds for the mean queue lengths under the assumption that the classes have equal mean service times. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximating queue size and waiting time distributions in general polling systems

Polling system models are extensively used to model a large variety of computer and communication networks as well as production and service systems in which multiple customer classes or a number of distinct items compete for the capacity of a common server or production facility. In this paper we describe an efficient approximation method for the steady state distributions of the queue sizes and waiting times. This method is highly accurate as demonstrated by an extensive numerical study. In addition, it is highly adaptable to a variety of arrival patterns and switching protocols, including exhaustive and gated regimes, simple cyclical systems as well as general polling tables. For a system withN stations, one finds the firstK probability density function values of the steady state queue size in any given station inO(max(N, K ²) time only. When executed on an IBM system RS/6000, we have observed an average CPU time of less than 1 second for systems with as many as 50 stations over a large variety of parameter settings.     

Waiting time for consecutive repetitions of a pattern and related distributions

Annals of the Institute of Statistical Mathematics - Let k be a positive integer. Some exact distributions of the waiting time random variables for k consecutive repetitions of a pattern are...     

Sojourn time distributions for the M/M/1 queue in a Markovian environment

In this paper, we study a single server queue in which both the arrival rate and service rate depend on the state of an external Markov process (called the environment) with a finite state space. Given that the environment is in state j, the mean arrival and service rates are λ_j and μ_j respectively. For such a queue, the queue length distribution is known to be matrix geometric. In this paper, we characterize the Laplace-Stieltjes transform of the sojourn time distribution under four disciplines - last come first served preemptive resume, last come first served, processor sharing and round robin. We also discuss a potential application of this queue in the are of data communication.     

Repeated random insertion into a priority queue

The average cost of inserting n elements into an initially empty heap is analyzed, under the assumption that the n! orders in which the n elements can be inserted are equally likely. It is proved that this average, expressed in number of exchanges per insertion, is bounded by a constant about 1.7645.     

The permutational power of a priority queue

A priority queue transforms an input permutation of some set of sizen into an output permutation. It is shown that the number of such pairs (, ) is (n + 1) ^n–1. Some related enumerative and algorithmic questions are also considered.Supported by the National Science and Engineering Research Council of Canada under Grant A4219.     

The discrete-time preemptive repeat identical priority queue

Priority queueing systems come natural when customers with diversified delay requirements have to wait to get service. The customers that cannot tolerate but small delays get service priority over customers which are less delay-sensitive. In this contribution, we analyze a discrete-time two-class preemptive repeat identical priority queue with infinite buffer space and generally distributed service times. Newly arriving high-priority customers interrupt the on-going service of a low-priority customer. After all high-priority customers have left the system, the interrupted service of the low-priority customer has to be repeated completely. By means of a probability generating functions approach, we analyze the system content and the delay of both types of customers. Performance measures (such as means and variances) are calculated and the impact of the priority scheduling is discussed by means of some numerical examples.     

On the single server retrial queue with priority customers

We consider anM ₂/G ₂/1 type queueing system which serves two types of calls. In the case of blocking the first type customers can be queued whereas the second type customers must leave the service area but return after some random period of time to try their luck again. This model is a natural generalization of the classicM ₂/G ₂/1 priority queue with the head-of-theline priority discipline and the classicM/G/1 retrial queue. We carry out an extensive analysis of the system, including existence of the stationary regime, embedded Markov chain, stochastic decomposition, limit theorems under high and low rates of retrials and heavy traffic analysis.Visiting from: Department of Probability, Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia.