Some distributional approximations in Markovian queueing networks

1982 ◽  
Vol 14 (3) ◽  
pp. 654-671 ◽  
Author(s):  
T. C. Brown ◽  
P. K. Pollett
Keyword(s):  
Queueing Networks ◽  
Poisson Processes ◽  
Single Class ◽  
Open Network ◽  
State Dependent ◽  
Service Rates ◽  
Distributional Approximations ◽  
Closed Network ◽  
Poisson Approximations ◽  
Markovian Queueing Networks

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

Some distributional approximations in Markovian queueing networks

1982 ◽  
Vol 14 (03) ◽  
pp. 654-671 ◽  
Author(s):  
T. C. Brown ◽  
P. K. Pollett
Keyword(s):  
Queueing Networks ◽  
Poisson Processes ◽  
Single Class ◽  
Open Network ◽  
State Dependent ◽  
Service Rates ◽  
Distributional Approximations ◽  
Closed Network ◽  
Poisson Approximations ◽  
Markovian Queueing Networks

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

scholarly journals Monotonicity of throughput in non-Markovian networks

1989 ◽  
Vol 26 (01) ◽  
pp. 134-141 ◽  
Author(s):  
Pantelis Tsoucas ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Buffer Size ◽  
Waiting Room ◽  
Finite Buffers ◽  
Closed Network ◽  
Markovian Queueing Networks

Monotonicity of throughput is established in some non-Markovian queueing networks by means of pathwise comparisons. In a series of · /GI/s/N queues with loss at the first node it is proved that increasing the waiting room and/or the number of servers increases the throughput. For a closed network of · /GI/s queues it is shown that the throughput increases as the total number of jobs increases. The technique used for these results does not apply to blocking systems with finite buffers and feedback. Using a stronger coupling argument we prove throughput monotonicity as a function of buffer size for a series of two ·/M/1/N queues with loss and feedback from the second to the first node.

Realization probability and throughput sensitivity in a closed jackson network

1989 ◽  
Vol 26 (03) ◽  
pp. 615-624
Author(s):  
Xi-Ren Cao
Keyword(s):  
Perturbation Analysis ◽  
Queuing Networks ◽  
Single Class ◽  
Jackson Network ◽  
State Dependent ◽  
Closed Queuing Networks ◽  
Service Rates

Realization probability is a new concept pertaining to perturbation analysis of closed queuing networks. The sensitivities of throughputs in a closed single-class Jackson network can be expressed in terms of realization probabilities. In this paper, based on a discussion of perturbation analysis for networks with state-dependent service rates, we derive some new formulas for sensitivities of throughputs using realization probability.

Realization factors and sensitivity analysis of queueing networks with state-dependent service rates

1990 ◽  
Vol 22 (1) ◽  
pp. 178-210 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Service Rate ◽  
Steady State Probability ◽  
State Dependent ◽  
Service Rates ◽  
Realization Factors

The paper studies the sensitivity of the throughput with respect to a mean service rate in a closed queueing network with exponentially distributed service requirements and state-dependent service rates. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced. The properties of realization factors are discussed, and a set of equations specifying the realization factors are derived. The elasticity of the steady state throughput with respect to a mean service rate equals the product of the steady state probability and the corresponding realization factor. This elasticity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. The sample path elasticity of the throughput with respect to a mean service rate converges with probability 1 to the elasticity of the steady state throughput. The theory provides an analytical method of calculating the throughput sensitivity and justifies the application of perturbation analysis.

Realization probability and throughput sensitivity in a closed jackson network

1989 ◽  
Vol 26 (3) ◽  
pp. 615-624 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Perturbation Analysis ◽  
Queuing Networks ◽  
Single Class ◽  
Jackson Network ◽  
State Dependent ◽  
Closed Queuing Networks ◽  
Service Rates

Realization probability is a new concept pertaining to perturbation analysis of closed queuing networks. The sensitivities of throughputs in a closed single-class Jackson network can be expressed in terms of realization probabilities. In this paper, based on a discussion of perturbation analysis for networks with state-dependent service rates, we derive some new formulas for sensitivities of throughputs using realization probability.

scholarly journals Monotonicity of throughput in non-Markovian networks

1989 ◽  
Vol 26 (1) ◽  
pp. 134-141 ◽  
Author(s):  
Pantelis Tsoucas ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Buffer Size ◽  
Waiting Room ◽  
Finite Buffers ◽  
Closed Network ◽  
Markovian Queueing Networks

Monotonicity of throughput is established in some non-Markovian queueing networks by means of pathwise comparisons. In a series of · /GI/s/N queues with loss at the first node it is proved that increasing the waiting room and/or the number of servers increases the throughput. For a closed network of · /GI/s queues it is shown that the throughput increases as the total number of jobs increases. The technique used for these results does not apply to blocking systems with finite buffers and feedback. Using a stronger coupling argument we prove throughput monotonicity as a function of buffer size for a series of two ·/M/1/N queues with loss and feedback from the second to the first node.

Realization factors and sensitivity analysis of queueing networks with state-dependent service rates

1990 ◽  
Vol 22 (01) ◽  
pp. 178-210 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Service Rate ◽  
Steady State Probability ◽  
State Dependent ◽  
Service Rates ◽  
Realization Factors

The paper studies the sensitivity of the throughput with respect to a mean service rate in a closed queueing network with exponentially distributed service requirements and state-dependent service rates. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced. The properties of realization factors are discussed, and a set of equations specifying the realization factors are derived. The elasticity of the steady state throughput with respect to a mean service rate equals the product of the steady state probability and the corresponding realization factor. This elasticity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. The sample path elasticity of the throughput with respect to a mean service rate converges with probability 1 to the elasticity of the steady state throughput. The theory provides an analytical method of calculating the throughput sensitivity and justifies the application of perturbation analysis.

Sign in / Sign up

Export Citation Format

Share Document