Tail Probability of Low-Priority Queue Length in a Discrete-Time Priority BMAP/PH/1 Queue

In this paper, a versatile Markovian queueing system is considered. Given a fixed threshold level c, the server serves customers one a time when the queue length is less than c, and in batches of fixed size c when the queue length is greater than or equal to c. The server is subject to failure when serving either a single or a batch of customers. Service rates, failure rates, and repair rates, depend on whether the server is serving a single customer or a batch of customers. While the analytical method provides the initial probability vector, we use the entropy principle to obtain both the initial probability vector (for comparison) and the tail probability vector. The comparison shows the results obtained analytically and approximately are in good agreement, especially when the first two moments are used in the entropy approach.

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Asymptotic Throughput in Discrete‐Time Cyclic Networks with Queue‐Length‐Dependent Service Rates

Stochastic Models ◽

10.1081/stm-120025401 ◽

2003 ◽

Vol 19 (4) ◽

pp. 483-506 ◽

Cited By ~ 1

Author(s):

Hans Daduna ◽

Victor Pestien ◽

S. Ramakrishnan

Keyword(s):

Discrete Time ◽

Queue Length ◽

Service Rates ◽

Cyclic Networks

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The stationary tail asymptotics in the GI/G/1-type queue with countably many background states

Advances in Applied Probability ◽

10.1239/aap/1103662965 ◽

2004 ◽

Vol 36 (4) ◽

pp. 1231-1251 ◽

Cited By ~ 43

Author(s):

Masakiyo Miyazawa ◽

Yiqiang Q. Zhao

Keyword(s):

State Space ◽

Discrete Time ◽

Priority Queue ◽

Decay Rates ◽

Single Server ◽

Renewal Theorem ◽

Tail Probabilities ◽

Markov Renewal Equation ◽

Geometric Decay ◽

Background State

We consider the asymptotic behaviour of the stationary tail probabilities in the discrete-time GI/G/1-type queue with countable background state space. These probabilities are presented in matrix form with respect to the background state space, and shown to be the solution of a Markov renewal equation. Using this fact, we consider their decay rates. Applying the Markov renewal theorem, it is shown that certain reasonable conditions lead to the geometric decay of the tail probabilities as the level goes to infinity. We exemplify this result using a discrete-time priority queue with a single server and two types of customer.

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Traffic light queues as a generalization to queueing theory

Journal of Applied Probability ◽

10.2307/3212172 ◽

1971 ◽

Vol 8 (3) ◽

pp. 480-493 ◽

Cited By ~ 1

Author(s):

Hisashi Mine ◽

Katsuhisa Ohno

Keyword(s):

Generating Function ◽

Discrete Time ◽

Queue Length ◽

Length Distribution ◽

Probability Generating Function ◽

Traffic Light ◽

Queue Length Distribution ◽

Mean Delay ◽

Stationary Queue Length ◽

Stationary Queue Length Distribution

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.

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Computational Procedures for a Class of GI/D/kSystems in Discrete Time

Journal of Probability and Statistics ◽

10.1155/2009/716364 ◽

2009 ◽

Vol 2009 ◽

pp. 1-18

Author(s):

Md. Mostafizur Rahman ◽

Attahiru Sule Alfa

Keyword(s):

Discrete Time ◽

Queue Length ◽

Computation Time ◽

Single Server ◽

The Matrix ◽

Interarrival Times ◽

Computational Procedures ◽

Set Up ◽

First In First Out ◽

Service Duration

A class of discrete time GI/D/ksystems is considered for which the interarrival times have finite support and customers are served in first-in first-out (FIFO) order. The system is formulated as a single server queue with new general independent interarrival times and constant service duration by assuming cyclic assignment of customers to the identical servers. Then the queue length is set up as a quasi-birth-death (QBD) type Markov chain. It is shown that this transformed GI/D/1 system has special structures which make the computation of the matrixRsimple and efficient, thereby reducing the number of multiplications in each iteration significantly. As a result we were able to keep the computation time very low. Moreover, use of the resulting structural properties makes the computation of the distribution of queue length of the transformed system efficient. The computation of the distribution of waiting time is also shown to be simple by exploiting the special structures.

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Algorithmic analysis of the BMAP/D/k system in discrete time

Advances in Applied Probability ◽

10.1239/aap/1067436338 ◽

2003 ◽

Vol 35 (4) ◽

pp. 1131-1152 ◽

Cited By ~ 6

Author(s):

Attahiru Sule Alfa

Keyword(s):

Structural Properties ◽

Customer Service ◽

Discrete Time ◽

Queue Length ◽

Busy Period ◽

Marginal Distribution ◽

Waiting Times ◽

Single Server ◽

Waiting Time Distribution ◽

Algorithmic Analysis

We exploit the structural properties of the BMAP/D/k system to carry out its algorithmic analysis. Specifically, we use these properties to develop algorithms for studying the distributions of waiting times in discrete time and the busy period. One of the structural properties used results from considering the system as having customers assigned in a cyclic order—which does not change the waiting-time distribution—and then studying only one arbitrary server. The busy period is defined as the busy period of an arbitrary single server based on this cyclic assignment of customers to servers. Finally, we study the marginal distribution of the joint queue length and phase of customer arrival. The structural property used for studying the queue length is based on the observation of the system every interval that is the length of one customer service time.

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