Tandem of Infinite-Server Queues with Markovian Arrival Process

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

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Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals

Journal of Applied Probability ◽

10.2307/3213039 ◽

1980 ◽

Vol 17 (2) ◽

pp. 498-514 ◽

Cited By ~ 17

Author(s):

V. Ramaswami ◽

Marcel F. Neuts

Keyword(s):

Queue Length ◽

Arrival Process ◽

Infinite Server ◽

Highly Sensitive ◽

Phase Type ◽

Random Variability ◽

Infinite Server Queues ◽

Linear Differential ◽

Mean Queue Length ◽

Transient And Steady State

This paper discusses infinite-server queues with phase-type input. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations, provided with as many as ten phases, show that although (even the time-dependent) mean queue length is very insensitive to substantial random variability in the arrival process, the higher moments of the queue length are highly sensitive. These examples indicate that considerable caution should be exercised in using robustness results for such stochastic models.

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Infinite-server queues with Hawkes input

Journal of Applied Probability ◽

10.1017/jpr.2018.58 ◽

2018 ◽

Vol 55 (3) ◽

pp. 920-943 ◽

Cited By ~ 8

Author(s):

D. T. Koops ◽

M. Saxena ◽

O. J. Boxma ◽

M. Mandjes

Keyword(s):

Heavy Traffic ◽

Probability Generating Function ◽

Arrival Process ◽

Hawkes Process ◽

Asymptotic Results ◽

Recursive Procedure ◽

Infinite Server ◽

Infinite Server Queues ◽

Heavy Tailed ◽

Number Of Customers

Abstract In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and nonexponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy tailed, and we consider a heavy-traffic regime. We conclude by discussing how our results can be used computationally and by verifying the numerical results via simulations.

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The infinite server queue with semi-Markovian arrivals and negative exponential services

Journal of Applied Probability ◽

10.1017/s002190020009478x ◽

1972 ◽

Vol 9 (01) ◽

pp. 178-184 ◽

Cited By ~ 3

Author(s):

Marcel F. Neuts ◽

Shun-Zer Chen

Keyword(s):

Discrete Time ◽

Infinite Number ◽

Queue Length ◽

Markovian Arrival Process ◽

Arrival Process ◽

Infinite Server ◽

Server Queue ◽

Service Times ◽

Negative Exponential ◽

Stationary Probabilities

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

Download Full-text

Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals

Journal of Applied Probability ◽

10.1017/s002190020004732x ◽

1980 ◽

Vol 17 (02) ◽

pp. 498-514 ◽

Cited By ~ 4

Author(s):

V. Ramaswami ◽

Marcel F. Neuts

Keyword(s):

Queue Length ◽

Arrival Process ◽

Infinite Server ◽

Highly Sensitive ◽

Phase Type ◽

Random Variability ◽

Infinite Server Queues ◽

Linear Differential ◽

Mean Queue Length ◽

Transient And Steady State

This paper discusses infinite-server queues with phase-type input. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations, provided with as many as ten phases, show that although (even the time-dependent) mean queue length is very insensitive to substantial random variability in the arrival process, the higher moments of the queue length are highly sensitive. These examples indicate that considerable caution should be exercised in using robustness results for such stochastic models.

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Stationary Poisson departure processes from non-stationary queues

Journal of Applied Probability ◽

10.2307/3214138 ◽

1986 ◽

Vol 23 (1) ◽

pp. 256-260 ◽

Cited By ~ 5

Author(s):

Robert D. Foley

Keyword(s):

Poisson Process ◽

Service Time ◽

Queueing Systems ◽

Arrival Rate ◽

Distribution Functions ◽

Arrival Process ◽

Service Time Distribution ◽

Departure Process ◽

Infinite Server ◽

Departure Processes

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

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