Monotonicity results for MR/GI/1 queues

This paper considers queues with a Markov renewal arrival process and a particular transition matrix for the underlying Markov chain. We study the effect that the transition matrix has on the waiting time of the nth customer as well as on the stationary waiting time. The main theorem generalizes results of Szekli et al. (1994a) and partly confirms their conjecture. In this context we show the importance of a new stochastic ordering concept.

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M/M/1 Model with Unreliable Service

International Journal of Statistics and Probability ◽

10.5539/ijsp.v7n1p125 ◽

2017 ◽

Vol 7 (1) ◽

pp. 125

Author(s):

Joshua Patterson, Andrzej Korzeniowski

Keyword(s):

Markov Chain ◽

Waiting Time ◽

Queue Length ◽

Sufficient Conditions ◽

Probability Generating Function ◽

Embedded Markov Chain ◽

Queueing Models ◽

Stieltjes Transform ◽

Stationary Queue Length ◽

Stationary Waiting Time

We define a new term ”unreliable service” and construct the corresponding embedded Markov Chain to an M/M/1 queue with so defined protocol. Sufficient conditions for positive recurrence and closed form of stationary distribution are provided. Furthermore, we compute the probability generating function of the stationary queue length and Laplace-Stieltjes transform of the stationary waiting time. In the course of the analysis an interesting decomposition of both the queue length and waiting time has emerged. A number of queueing models can be recovered from our work by taking limits of certain parameters.

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A relation between stationary queue and waiting time distributions

Journal of Applied Probability ◽

10.1017/s0021900200035713 ◽

1971 ◽

Vol 8 (03) ◽

pp. 617-620 ◽

Cited By ~ 14

Author(s):

Rasoul Haji ◽

Gordon F. Newell

Keyword(s):

Waiting Time ◽

Queueing System ◽

Arrival Process ◽

Time Interval ◽

Queue Size ◽

Random Time Interval ◽

Queue Discipline ◽

Stationary Waiting Time ◽

Number Of Customers ◽

First In First Out

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

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COMPARISONS OF DEPENDENCE FOR STATIONARY MARKOV PROCESSES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800143025 ◽

2000 ◽

Vol 14 (3) ◽

pp. 299-315 ◽

Cited By ~ 17

Author(s):

Taizhong Hu ◽

Xiaoming Pan

Keyword(s):

Markov Process ◽

Markov Processes ◽

Waiting Time ◽

Continuous Time ◽

Queueing Theory ◽

Stationary Markov Process ◽

Stationary Waiting Time ◽

Markov Modulated ◽

Stationary Workload ◽

Monotonicity Results

Results and conditions which quantify the decrease in dependence with lag for a stationary Markov process and enable one to compare the dependence for two stationary Markov processes are obtained. The notions of dependence used in this article are the supermodular ordering and the concordance ordering. Both discrete-time and continuous-time Markov processes are considered. Some applications of the main results are given. In queueing theory, the monotonicity results of the waiting time of the nth customer as well as the stationary waiting time in an MR/GI/1 queue and the stationary workload in a Markov-modulated queue are established, thus strengthening previous results while simplifying their derivation. This article is a continuation of those by Fang et al. [7] and Hu and Joe [10].

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A relation between stationary queue and waiting time distributions

Journal of Applied Probability ◽

10.2307/3212186 ◽

1971 ◽

Vol 8 (3) ◽

pp. 617-620 ◽

Cited By ~ 49

Author(s):

Rasoul Haji ◽

Gordon F. Newell

Keyword(s):

Waiting Time ◽

Queueing System ◽

Arrival Process ◽

Time Interval ◽

Queue Size ◽

Random Time Interval ◽

Queue Discipline ◽

Stationary Waiting Time ◽

Number Of Customers ◽

First In First Out

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

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A finite capacity queue with Markovian arrivals and two servers with group services

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953394000171 ◽

1994 ◽

Vol 7 (2) ◽

pp. 161-178 ◽

Cited By ~ 8

Author(s):

S. Chakravarthy ◽

Attahiru Sule Alfa

Keyword(s):

Steady State ◽

Waiting Time ◽

Queue Length ◽

Time Distribution ◽

Arrival Process ◽

Finite Capacity ◽

Waiting Time Distribution ◽

Phase Type ◽

Stationary Waiting Time ◽

Number Of Customers

In this paper we consider a finite capacity queuing system in which arrivals are governed by a Markovian arrival process. The system is attended by two exponential servers, who offer services in groups of varying sizes. The service rates may depend on the number of customers in service. Using Markov theory, we study this finite capacity queuing model in detail by obtaining numerically stable expressions for (a) the steady-state queue length densities at arrivals and at arbitrary time points; (b) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. The stationary waiting time distribution is shown to be of phase type when the interarrival times are of phase type. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures are discussed. A conjecture on the nature of the mean waiting time is proposed. Some illustrative numerical examples are presented.

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Hitting times of Markov chains, with application to state-dependent queues

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700025508 ◽

1977 ◽

Vol 17 (1) ◽

pp. 97-107 ◽

Cited By ~ 11

Author(s):

R.L. Tweedie

Keyword(s):

Markov Chain ◽

Waiting Time ◽

Busy Period ◽

Hitting Times ◽

Arrival Process ◽

General State ◽

State Dependent ◽

General State Space ◽

The Mean ◽

Number Of Customers

We present in this note a useful extension of the criteria given in a recent paper [Advances in Appl. Probability 8 (1976), 737–771] for the finiteness of hitting times and mean hitting times of a Markov chain on sets in its (general) state space. We illustrate our results by giving conditions for the finiteness of the mean number of customers in the busy period of a queue in which both the service-times and the arrival process may depend on the waiting time in the queue. Such conditions also suffice for the embedded waiting time chain to have a unique stationary distribution.

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LOWER BOUNDS FOR LRD/GI/1 QUEUES WITH SUBEXPONENTIAL SERVICE TIMES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964804181060 ◽

2004 ◽

Vol 18 (1) ◽

pp. 87-101 ◽

Cited By ~ 1

Author(s):

Cathy H. Xia ◽

Zhen Liu ◽

Mark S. Squillante ◽

Li Zhang

Keyword(s):

Long Range ◽

Lower Bounds ◽

Waiting Time ◽

Waiting Times ◽

Arrival Process ◽

Performance Impact ◽

Tail Distribution ◽

Service Times ◽

Stationary Waiting Time ◽

Bounded Below

We investigate the tail distribution of the virtual waiting times in a LRD/GI/1 queue where the arrival process is long-range dependent (LRD) and the service times are independent and identically distributed (i.i.d.) random variables. We present two lower bounds on the stationary waiting time tail asymptotics, which illustrate the different dominating components that influence server performance under various conditions. In particular, we show that the tail distribution of the stationary waiting time is bounded below by that of the associated LRD/D/1 queues resulting from replacing all random service times by the mean. This shows the performance impact purely due to the long-range dependency of the arrival process. On the other hand, when the service times are subexponential, we show that the tail distribution of the stationary waiting time is bounded below by that of the corresponding D/GI/1 queue by replacing the dependent arrival process with its associated independent version. This shows the minimum performance impact due to the tail distribution of the service times. The above two lower bounds indicate that the performance of LRD/GI/1 queues will be dominated by the heavier tail of the corresponding LRD/D/1 and D/GI/1 queues. These features are further illustrated and quantified through examples and via numerous simulation experiments.

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