Simple relationships among moments of queue lengths in product form queueing networks

1988 ◽  
Vol 37 (9) ◽  
pp. 1125-1129 ◽  
Author(s):  
E. de Souza e Silva ◽  
R.R. Muntz
Keyword(s):  
Queueing Networks ◽  
Product Form ◽  
Queue Lengths ◽  
Product Form Queueing Networks

A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks

1989 ◽  
Vol 26 (01) ◽  
pp. 121-133 ◽  
Author(s):  
James McKenna
Keyword(s):  
Sojourn Time ◽  
Queueing Networks ◽  
Waiting Times ◽  
Arrival Rate ◽  
Product Form ◽  
Single Server ◽  
Queue Lengths ◽  
The Mean ◽  
Expected Sojourn Time ◽  
Product Form Queueing Networks

Little's theorem states that under very general conditions L = λW, where L is the time average number in the system, W is the expected sojourn time in the system, and λ is the mean arrival rate to the system. For certain systems it is known that relations of the form E((L) l ) = λ lE((W) l ) are also true, where (L) l = L(L – 1)· ·· (L – l + 1). It is shown in this paper that closely analogous relations hold in closed, product-form queueing networks. Similar expressions relate Nji and Sji, where Nji is the total number of class j jobs at center i and Sji is the total sojourn time of a class j job at center i, when center i is a single-server, FCFS center. When center i is a c-server, FCFS center, Qji and Wji are related this way, where Qji is the number of class j jobs queued, but not in service at center i and Wji is the waiting time in queue of a class j job at center i. More remarkably, generalizations of these results to joint moments of queue lengths and sojourn times along overtake-free paths are shown to hold.

A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks

1989 ◽  
Vol 26 (1) ◽  
pp. 121-133 ◽  
Author(s):  
James McKenna
Keyword(s):  
Sojourn Time ◽  
Queueing Networks ◽  
Waiting Times ◽  
Arrival Rate ◽  
Product Form ◽  
Single Server ◽  
Queue Lengths ◽  
The Mean ◽  
Expected Sojourn Time ◽  
Product Form Queueing Networks

Little's theorem states that under very general conditions L = λW, where L is the time average number in the system, W is the expected sojourn time in the system, and λ is the mean arrival rate to the system. For certain systems it is known that relations of the form E((L)l) = λ lE((W)l) are also true, where (L)l = L(L – 1)· ·· (L – l + 1). It is shown in this paper that closely analogous relations hold in closed, product-form queueing networks. Similar expressions relate Nji and Sji, where Nji is the total number of class j jobs at center i and Sji is the total sojourn time of a class j job at center i, when center i is a single-server, FCFS center. When center i is a c-server, FCFS center, Qji and Wji are related this way, where Qji is the number of class j jobs queued, but not in service at center i and Wji is the waiting time in queue of a class j job at center i. More remarkably, generalizations of these results to joint moments of queue lengths and sojourn times along overtake-free paths are shown to hold.

Queueing networks with instantaneous movements: a unified approach by quasi-reversibility

2000 ◽  
Vol 32 (01) ◽  
pp. 284-313 ◽  
Author(s):  
Xiuli Chao ◽  
Masakiyo Miyazawa
Keyword(s):  
Queueing Networks ◽  
Product Form ◽  
Unified Approach ◽  
Stationary Distributions ◽  
Batch Arrivals ◽  
Unified View ◽  
Batch Services ◽  
Product Form Queueing Networks

In this paper we extend the notion of quasi-reversibility and apply it to the study of queueing networks with instantaneous movements and signals. The signals treated here are considerably more general than those in the existing literature. The approach not only provides a unified view for queueing networks with tractable stationary distributions, it also enables us to find several new classes of product form queueing networks, including networks with positive and negative signals that instantly add or remove customers from a sequence of nodes, networks with batch arrivals, batch services and assembly-transfer features, and models with concurrent batch additions and batch deletions along a fixed or a random route of the network.

Mean value analysis by chain of product form queueing networks

1989 ◽  
Vol 38 (3) ◽  
pp. 432-442 ◽  
Author(s):  
A.E. Conway ◽  
E. de Souza e Silva ◽  
S.S. Lavenberg
Keyword(s):  
Queueing Networks ◽  
Mean Value ◽  
Product Form ◽  
Value Analysis ◽  
Mean Value Analysis ◽  
Product Form Queueing Networks
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