A single server queue with Markov modulated service rates and impatient customers

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i. The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i. In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

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A single-server queue with server vacations and a class of non-renewal arrival processes

Advances in Applied Probability ◽

10.2307/1427464 ◽

1990 ◽

Vol 22 (3) ◽

pp. 676-705 ◽

Cited By ~ 307

Author(s):

David M. Lucantoni ◽

Kathleen S. Meier-Hellstern ◽

Marcel F. Neuts

Keyword(s):

Waiting Times ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Arbitrary Time ◽

Markov Modulated Poisson Process ◽

Special Cases ◽

Server Queue ◽

Arrival Processes ◽

Markov Modulated

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes.Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

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A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

Journal of Applied Probability ◽

10.1017/s0021900200103572 ◽

1995 ◽

Vol 32 (04) ◽

pp. 1103-1111 ◽

Cited By ~ 1

Author(s):

Qing Du

Keyword(s):

Poisson Process ◽

Arrival Rate ◽

Loss Probability ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Markov Modulated Poisson Process ◽

Server Queue ◽

Monotonicity Result ◽

Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i . The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i . In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

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The Markov-Modulated Single Server Queue

Stochastic Storage Processes ◽

10.1007/978-1-4612-1742-8_6 ◽

1998 ◽

pp. 155-165

Author(s):

N. U. Prabhu

Keyword(s):

Single Server ◽

Single Server Queue ◽

Server Queue ◽

Markov Modulated

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Slow Retrial Asymptotics for a Single Server Queue with Two-Way Communication and Markov Modulated Poisson Input

Journal of Systems Science and Systems Engineering ◽

10.1007/s11518-018-5404-6 ◽

2019 ◽

Vol 28 (2) ◽

pp. 181-193 ◽

Cited By ~ 1

Author(s):

Anatoly Nazarov ◽

Tuan Phung-Duc ◽

Svetlana Paul

Keyword(s):

Single Server ◽

Single Server Queue ◽

Poisson Input ◽

Server Queue ◽

Markov Modulated

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A Batch Arrival Single Server Queue with Server Providing General Service in Two Fluctuating Modes and Reneging during Vacation and Breakdowns

Journal of Probability and Statistics ◽

10.1155/2014/319318 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12

Author(s):

Monita Baruah ◽

Kailash C. Madan ◽

Tillal Eldabi

Keyword(s):

Repair Process ◽

Batch Arrival ◽

Single Server ◽

Single Server Queue ◽

Numerical Illustration ◽

Random Breakdown ◽

Special Cases ◽

Server Queue ◽

Service Rates ◽

General Service

We study the behavior of a batch arrival queuing system equipped with a single server providing general arbitrary service to customers with different service rates in two fluctuating modes of service. In addition, the server is subject to random breakdown. As soon as the server faces breakdown, the customer whose service is interrupted comes back to the head of the queue. As soon as repair process of the server is complete, the server immediately starts providing service in mode 1. Also customers waiting for service may renege (leave the queue) when there is breakdown or when server takes vacation. The system provides service with complete or reduced efficiency due to the fluctuating rates of service. We derive the steady state queue size distribution. Some special cases are discussed and numerical illustration is provided to see the effect and validity of the results.

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A monotonicity result for the workload in Markov-modulated queues

Journal of Applied Probability ◽

10.1017/s0021900200016387 ◽

1998 ◽

Vol 35 (03) ◽

pp. 741-747 ◽

Cited By ~ 1

Author(s):

Nicole Bäuerle ◽

Tomasz Rolski

Keyword(s):

Poisson Process ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Convex Ordering ◽

Markov Modulated Poisson Process ◽

Server Queue ◽

Monotonicity Result ◽

Markov Modulated ◽

Stationary Workload

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

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