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.

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (02) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network. This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

Stability condition for a single-server retrial queue

1993 ◽  
Vol 25 (03) ◽  
pp. 690-701 ◽  
Author(s):  
Huei-Mei Liang ◽  
V. G. Kulkarni
Keyword(s):  
Stability Condition ◽  
Service Time ◽  
Retrial Queue ◽  
Arrival Rate ◽  
Retrial Queues ◽  
Single Server ◽  
Sufficient Stability Condition ◽  
Sufficient Stability ◽  
The Mean ◽  
The Stability

A single-server retrial queue consists of a primary queue, an orbit and a single server. Assume the primary queue capacity is 1 and the orbit capacity is infinite. Customers can arrive at the primary queue either from outside the system or from the orbit. If the server is busy, the arriving customer joins the orbit and conducts a retrial later. Otherwise, he receives service and leaves the system. We investigate the stability condition for a single-server retrial queue. Let λ be the arrival rate and 1/μ be the mean service time. It has been proved that λ / μ < 1 is a sufficient stability condition for the M/G /1/1 retrial queue with exponential retrial times. We give a counterexample to show that this stability condition is not valid for general single-server retrial queues. Next we show that λ /μ < 1 is a sufficient stability condition for the stability of a single-server retrial queue when the interarrival times and retrial times are finite mixtures of Erlangs.

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (2) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network.This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

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