Approximations in Finite Capacity Multi-Server Queues with Poisson Arrivals

1977 ◽  
Author(s):  
Shirley A. Nozaki ◽  
Sheldon M. Ross
Keyword(s):  
Finite Capacity ◽  
Poisson Arrivals ◽  
Multi Server

Approximations in finite-capacity multi-server queues by Poisson arrivals

1978 ◽  
Vol 15 (4) ◽  
pp. 826-834 ◽  
Author(s):  
Shirley A. Nozaki ◽  
Sheldon M. Ross
Keyword(s):  
Service Time ◽  
Joint Distribution ◽  
Finite Capacity ◽  
Queuing Model ◽  
Average Delay ◽  
Service Distribution ◽  
Poisson Arrivals ◽  
Multi Server

An approximation for the average delay in queue of an entering customer is presented for the M/G/K queuing model with finite capacity. The approximation is obtained by means of an approximation relating a joint distribution of remaining service time to the equilibrium service distribution.

Approximations in finite-capacity multi-server queues by Poisson arrivals

1978 ◽  
Vol 15 (04) ◽  
pp. 826-834 ◽  
Author(s):  
Shirley A. Nozaki ◽  
Sheldon M. Ross
Keyword(s):  
Service Time ◽  
Joint Distribution ◽  
Finite Capacity ◽  
Queuing Model ◽  
Average Delay ◽  
Service Distribution ◽  
Poisson Arrivals ◽  
Multi Server

An approximation for the average delay in queue of an entering customer is presented for the M/G/K queuing model with finite capacity. The approximation is obtained by means of an approximation relating a joint distribution of remaining service time to the equilibrium service distribution.

The synchronization of Poisson processes and queueing networks with service and synchronization nodes

2000 ◽  
Vol 32 (03) ◽  
pp. 824-843 ◽  
Author(s):  
Balaji Prabhakar ◽  
Nicholas Bambos ◽  
T. S. Mountford
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Poisson Processes ◽  
Finite Capacity ◽  
Random Input ◽  
Jackson Network ◽  
Limiting Behavior ◽  
Network Output ◽  
Poisson Arrivals ◽  
Buffer Capacities

This paper investigates the dynamics of a synchronization node in isolation, and of networks of service and synchronization nodes. A synchronization node consists of M infinite capacity buffers, where tokens arriving on M distinct random input flows are stored (there is one buffer for each flow). Tokens are held in the buffers until one is available from each flow. When this occurs, a token is drawn from each buffer to form a group-token, which is instantaneously released as a synchronized departure. Under independent Poisson inputs, the output of a synchronization node is shown to converge weakly (and in certain cases strongly) to a Poisson process with rate equal to the minimum rate of the input flows. Hence synchronization preserves the Poisson property, as do superposition, Bernoulli sampling and M/M/1 queueing operations. We then consider networks of synchronization and exponential server nodes with Bernoulli routeing and exogenous Poisson arrivals, extending the standard Jackson network model to include synchronization nodes. It is shown that if the synchronization skeleton of the network is acyclic (i.e. no token visits any synchronization node twice although it may visit a service node repeatedly), then the distribution of the joint queue-length process of only the service nodes is product form (under standard stability conditions) and easily computable. Moreover, the network output flows converge weakly to Poisson processes. Finally, certain results for networks with finite capacity buffers are presented, and the limiting behavior of such networks as the buffer capacities become large is studied.

The synchronization of Poisson processes and queueing networks with service and synchronization nodes

2000 ◽  
Vol 32 (3) ◽  
pp. 824-843 ◽  
Author(s):  
Balaji Prabhakar ◽  
Nicholas Bambos ◽  
T. S. Mountford
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Poisson Processes ◽  
Finite Capacity ◽  
Random Input ◽  
Jackson Network ◽  
Limiting Behavior ◽  
Network Output ◽  
Poisson Arrivals ◽  
Buffer Capacities

This paper investigates the dynamics of a synchronization node in isolation, and of networks of service and synchronization nodes. A synchronization node consists of M infinite capacity buffers, where tokens arriving on M distinct random input flows are stored (there is one buffer for each flow). Tokens are held in the buffers until one is available from each flow. When this occurs, a token is drawn from each buffer to form a group-token, which is instantaneously released as a synchronized departure. Under independent Poisson inputs, the output of a synchronization node is shown to converge weakly (and in certain cases strongly) to a Poisson process with rate equal to the minimum rate of the input flows. Hence synchronization preserves the Poisson property, as do superposition, Bernoulli sampling and M/M/1 queueing operations. We then consider networks of synchronization and exponential server nodes with Bernoulli routeing and exogenous Poisson arrivals, extending the standard Jackson network model to include synchronization nodes. It is shown that if the synchronization skeleton of the network is acyclic (i.e. no token visits any synchronization node twice although it may visit a service node repeatedly), then the distribution of the joint queue-length process of only the service nodes is product form (under standard stability conditions) and easily computable. Moreover, the network output flows converge weakly to Poisson processes. Finally, certain results for networks with finite capacity buffers are presented, and the limiting behavior of such networks as the buffer capacities become large is studied.

Approximations for the steady-state probabilities in the M/G/c queue

1981 ◽  
Vol 13 (1) ◽  
pp. 186-206 ◽  
Author(s):  
H. C. Tijms ◽  
M. H. Van Hoorn ◽  
A. Federgruen
Keyword(s):  
Steady State ◽  
Queue Size ◽  
Numerical Experience ◽  
General Service Times ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean ◽  
Multi Server ◽  
General Service ◽  
Steady State Probabilities

For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.

An upper bound on the performance of queues with returning customers

1987 ◽  
Vol 24 (02) ◽  
pp. 466-475 ◽  
Author(s):  
Betsy S. Greenberg ◽  
Ronald W. Wolff
Keyword(s):  
Steady State ◽  
Upper Bound ◽  
System Performance ◽  
Finite Capacity ◽  
Multiple Channel ◽  
Poisson Arrivals ◽  
Time Averages ◽  
Steady State Probabilities

Multiple channel queues with Poisson arrivals, exponential service distributions, and finite capacity are studied. A customer who finds the system at capacity either leaves the system for ever or may return to try again after an exponentially distributed time. Steady state probabilities are approximated by assuming that the returning customers see time averages. The approximation is shown to result in an upper bound on system performance.

Approximations for the steady-state probabilities in the M/G/c queue

1981 ◽  
Vol 13 (01) ◽  
pp. 186-206 ◽  
Author(s):  
H. C. Tijms ◽  
M. H. Van Hoorn ◽  
A. Federgruen
Keyword(s):  
Steady State ◽  
Queue Size ◽  
Numerical Experience ◽  
General Service Times ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean ◽  
Multi Server ◽  
General Service ◽  
Steady State Probabilities

For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.

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