Insensitivity of processes with interruptions

The theory of insensitivity within generalised semi-Markov processes is extended to cover classes of models in which the generally distributed lifetimes can be terminated prematurely by the deaths of negative exponentially distributed lifetimes. As a consequence of this approach it is shown that there exist classes of processes which are insensitive with respect to characteristics of the general distributions other than the mean. Two examples are given. The first is an analysis of networks of queues in which the generally distributed service times can be interrupted with resulting changes in routing probabilities. The second is a model for the effect of disturbances on the evolution of a vegetation community.

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Insensitive average residence times in generalized semi-Markov processes

Advances in Applied Probability ◽

10.1017/s0001867800036478 ◽

1981 ◽

Vol 13 (04) ◽

pp. 720-735 ◽

Cited By ~ 6

Author(s):

A. D. Barbour ◽

R. Schassberger

Keyword(s):

Steady State ◽

Markov Processes ◽

Broad Class ◽

Residence Times ◽

Steady State Distribution ◽

State Distribution ◽

Lifetime Distributions ◽

Definition Of ◽

Semi Markov Processes ◽

Networks Of Queues

For a broad class of stochastic processes, the generalized semi-Markov processes, conditions are known which imply that the steady state distribution of the process, when it exists, depends only on the means, and not the exact shapes, of certain lifetime distributions entering the definition of the process. It is shown in the present paper that this insensitivity extends to certain average and conditional average residence times. Particularly interesting applications can be found in the field of networks of queues.

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Tight bounds on the sensitivity of generalised semi-Markov processes with a single generally distributed lifetime

Journal of Applied Probability ◽

10.1017/s0021900200102578 ◽

1995 ◽

Vol 32 (01) ◽

pp. 63-73

Author(s):

A. J. Coyle ◽

P. G. Taylor

Keyword(s):

Performance Measures ◽

Markov Processes ◽

Queueing System ◽

Mean Value ◽

Performance Measure ◽

Upper And Lower Bounds ◽

Lifetime Distributions ◽

Actual Distribution ◽

The Mean ◽

Semi Markov Processes

There are some generalised semi-Markov processes (GSMP) which are insensitive, that is the value of some performance measures for the system depend only on the mean value of lifetimes and not on their actual distribution. In most cases this is not true and a performance measure can take on a number of values depending on the lifetime distributions. In this paper we present a method for finding tight bounds on the sensitivity of performance measures for the class of GSMPs with a single generally distributed lifetime. Using this method we can find upper and lower bounds for the value of a function of the stationary distribution as the distribution of the general lifetime ranges over a set of distributions with fixed mean. The method is applied to find bounds on the average queue length of the Engset queue and the time congestion in theGI/M/n/nqueueing system.

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Tight bounds on the sensitivity of generalised semi-Markov processes with a single generally distributed lifetime

Journal of Applied Probability ◽

10.2307/3214921 ◽

1995 ◽

Vol 32 (1) ◽

pp. 63-73 ◽

Cited By ~ 2

Author(s):

A. J. Coyle ◽

P. G. Taylor

Keyword(s):

Performance Measures ◽

Markov Processes ◽

Queueing System ◽

Mean Value ◽

Performance Measure ◽

Upper And Lower Bounds ◽

Lifetime Distributions ◽

Actual Distribution ◽

The Mean ◽

Semi Markov Processes

There are some generalised semi-Markov processes (GSMP) which are insensitive, that is the value of some performance measures for the system depend only on the mean value of lifetimes and not on their actual distribution. In most cases this is not true and a performance measure can take on a number of values depending on the lifetime distributions. In this paper we present a method for finding tight bounds on the sensitivity of performance measures for the class of GSMPs with a single generally distributed lifetime. Using this method we can find upper and lower bounds for the value of a function of the stationary distribution as the distribution of the general lifetime ranges over a set of distributions with fixed mean. The method is applied to find bounds on the average queue length of the Engset queue and the time congestion in the GI/M/n/n queueing system.

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Insensitive average residence times in generalized semi-Markov processes

Advances in Applied Probability ◽

10.2307/1426969 ◽

1981 ◽

Vol 13 (4) ◽

pp. 720-735 ◽

Cited By ~ 22

Author(s):

A. D. Barbour ◽

R. Schassberger

Keyword(s):

Steady State ◽

Markov Processes ◽

Broad Class ◽

Residence Times ◽

Steady State Distribution ◽

State Distribution ◽

Lifetime Distributions ◽

Definition Of ◽

Semi Markov Processes ◽

Networks Of Queues

For a broad class of stochastic processes, the generalized semi-Markov processes, conditions are known which imply that the steady state distribution of the process, when it exists, depends only on the means, and not the exact shapes, of certain lifetime distributions entering the definition of the process. It is shown in the present paper that this insensitivity extends to certain average and conditional average residence times. Particularly interesting applications can be found in the field of networks of queues.

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Maximum spacing estimation for continuous time Markov chains and semi-Markov processes

Statistical Inference for Stochastic Processes ◽

10.1007/s11203-021-09238-4 ◽

2021 ◽

Author(s):

Kristi Kuljus ◽

Bo Ranneby

Keyword(s):

Markov Chains ◽

Markov Processes ◽

Continuous Time ◽

Continuous Time Markov Chains ◽

Semi Markov Processes

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The dependence of sojourn times on service times in tandem queues

Journal of Applied Probability ◽

10.2307/3213628 ◽

1984 ◽

Vol 21 (3) ◽

pp. 661-667 ◽

Cited By ~ 2

Author(s):

Xi-Ren Cao

Keyword(s):

Sojourn Time ◽

Past History ◽

Interarrival Time ◽

Conditional Probabilities ◽

Tandem Queues ◽

Sojourn Times ◽

Distributed Service ◽

Service Times

In this paper we study a series of servers with exponentially distributed service times. We find that the sojourn time of a customer at any server depends on the customer's past history only through the customer's interarrival time to that server. A method of calculating the conditional probabilities of sojourn times is developed.

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