Tandem Queues with Correlated Service Times and Finite Capacity

1993 ◽  
Vol 18 (4) ◽  
pp. 901-915 ◽  
Author(s):  
Ilze Ziediņš
Keyword(s):  
Tandem Queues ◽  
Finite Capacity ◽  
Service Times

The dependence of sojourn times on service times in tandem queues

1984 ◽  
Vol 21 (3) ◽  
pp. 661-667 ◽  
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.

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.

Tandem queues with subexponential service times and finite buffers

2010 ◽  
Vol 66 (2) ◽  
pp. 195-209 ◽  
Author(s):  
Jung-Kyung Kim ◽  
Hayriye Ayhan
Keyword(s):  
Tandem Queues ◽  
Finite Buffers ◽  
Service Times

A Comparison Between Tandem Queues with Dependent and Independent Service Times

1982 ◽  
Vol 30 (3) ◽  
pp. 464-479 ◽  
Author(s):  
Michael Pinedo ◽  
Ronald W. Wolff
Keyword(s):  
Tandem Queues ◽  
Service Times

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.

scholarly journals The asymptotic variance of departures in critically loaded queues

2011 ◽  
Vol 43 (01) ◽  
pp. 243-263 ◽  
Author(s):  
A. Al Hanbali ◽  
M. Mandjes ◽  
Y. Nazarathy ◽  
W. Whitt
Keyword(s):  
Explicit Expression ◽  
Counting Process ◽  
Asymptotic Variance ◽  
Arrival Rate ◽  
Finite Capacity ◽  
System Load ◽  
Multiserver Queues ◽  
Service Patterns ◽  
Coefficients Of Variation ◽  
Service Times

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies lim t→∞varD(t) / t = λ(1–2/π)(c a 2 + c s 2), where λ is the arrival rate, and c a 2 and c s 2 are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

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