Bounds for the expected delays in some tandem queues

1980 ◽  
Vol 17 (03) ◽  
pp. 831-838 ◽  
Author(s):  
Shun-Chen Niu
Keyword(s):  
Upper Bound ◽  
Renewal Process ◽  
Arrival Process ◽  
Tandem Queues ◽  
Service Distribution ◽  
Deterministic Service ◽  
Service Times ◽  
Constant Service Times ◽  
General Service

Tandem queues are analyzed. An upper bound for the stationary expected delay in front of the second server is found for a sequence of two queues in tandem where the first server has deterministic service times, the second server has general service distribution, and the arrival process is an arbitrary renewal process. The result is extended to the case of n queues in tandem where all the servers except the last one have constant service times.

Bounds for the expected delays in some tandem queues

1980 ◽  
Vol 17 (3) ◽  
pp. 831-838 ◽  
Author(s):  
Shun-Chen Niu
Keyword(s):  
Upper Bound ◽  
Renewal Process ◽  
Arrival Process ◽  
Tandem Queues ◽  
Service Distribution ◽  
Deterministic Service ◽  
Service Times ◽  
Constant Service Times ◽  
General Service

Tandem queues are analyzed. An upper bound for the stationary expected delay in front of the second server is found for a sequence of two queues in tandem where the first server has deterministic service times, the second server has general service distribution, and the arrival process is an arbitrary renewal process. The result is extended to the case of n queues in tandem where all the servers except the last one have constant service times.

scholarly journals On the independence of sojourn times in tandem queues

1989 ◽  
Vol 21 (2) ◽  
pp. 488-489 ◽  
Author(s):  
Thomas M. Chen
Keyword(s):  
Tandem Queues ◽  
Sojourn Times ◽  
Deterministic Service ◽  
Service Times

Reich (1957) proved that the sojourn times in two tandem queues are independent when the first queue is M/M /1 and the second has exponential service times. When service times in the first queue are not exponential, it has been generally expected that the sojourn times are not independent. A proof for the case of deterministic service times in the first queue is offered here.

The interchangeability of tandem queues with heterogeneous customers and dependent service times

1992 ◽  
Vol 24 (3) ◽  
pp. 727-737 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Service Time ◽  
Finite Size ◽  
Arrival Process ◽  
Tandem Queues ◽  
Departure Process ◽  
Special Cases ◽  
Service Times ◽  
Special Case

Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is cj. Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the pi's. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ·/M/1 queues is a special case of this result. Other special cases provide interesting new results.

An analysis of a modified M/G/1 queue using a martingale technique

1996 ◽  
Vol 33 (01) ◽  
pp. 224-238
Author(s):  
Matthew Roughan
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Service Distribution ◽  
General Service ◽  
Embedded Process ◽  
Stationary Behaviour

We consider a variation of the M/G/1 queue in which, when the system contains more than k customers, it switches from its initial general service distribution to a different general service distribution until the server is cleared, whereupon it switches back to the original service distribution. Using a technique due to Baccelli and Makowski we define a martingale with respect to an embedded process and from this arrive at a relationship between the process and a modified Markov renewal process. Using this an analysis of the stationary behaviour of the queue is possible.

Characterizing pure loss GI/G/1 queues with renewal output

1974 ◽  
Vol 75 (1) ◽  
pp. 103-107 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Poisson Process ◽  
Service Time ◽  
Renewal Process ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arrival Process ◽  
Renewal Function ◽  
Service Distribution ◽  
Necessary And Sufficient

SummaryIn a pure loss GI/G/1 queueing system, necessary and sufficient conditions are given for the output to be a renewal process. These conditions involve dependence between the service distribution and the renewal function of the arrival process: for example, if pr {service time < ξ} = 0 for some ξ > 0, then it is sufficient for the renewal function to be that of a quasi-Poisson process with index ξ.

An analysis of a modified M/G/1 queue using a martingale technique

1996 ◽  
Vol 33 (1) ◽  
pp. 224-238
Author(s):  
Matthew Roughan
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Service Distribution ◽  
General Service ◽  
Embedded Process ◽  
Stationary Behaviour

We consider a variation of the M/G/1 queue in which, when the system contains more than k customers, it switches from its initial general service distribution to a different general service distribution until the server is cleared, whereupon it switches back to the original service distribution. Using a technique due to Baccelli and Makowski we define a martingale with respect to an embedded process and from this arrive at a relationship between the process and a modified Markov renewal process. Using this an analysis of the stationary behaviour of the queue is possible.

The interchangeability of tandem queues with heterogeneous customers and dependent service times

1992 ◽  
Vol 24 (03) ◽  
pp. 727-737 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Service Time ◽  
Finite Size ◽  
Arrival Process ◽  
Tandem Queues ◽  
Departure Process ◽  
Special Cases ◽  
Service Times ◽  
Special Case

Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is cj . Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the pi 's. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ·/M/1 queues is a special case of this result. Other special cases provide interesting new results.

scholarly journals On the independence of sojourn times in tandem queues

1989 ◽  
Vol 21 (02) ◽  
pp. 488-489
Author(s):  
Thomas M. Chen
Keyword(s):  
Tandem Queues ◽  
Sojourn Times ◽  
Deterministic Service ◽  
Service Times

Reich (1957) proved that the sojourn times in two tandem queues are independent when the first queue is M/M /1 and the second has exponential service times. When service times in the first queue are not exponential, it has been generally expected that the sojourn times are not independent. A proof for the case of deterministic service times in the first queue is offered here.

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