An analysis of a modified M/G/1 queue using a martingale technique

1996 ◽  
Vol 33 (1) ◽  
pp. 224-238
Author(s):  
Matthew Roughan
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Service Distribution ◽  
General Service ◽  
Embedded Process ◽  
Stationary Behaviour

We consider a variation of the M/G/1 queue in which, when the system contains more than k customers, it switches from its initial general service distribution to a different general service distribution until the server is cleared, whereupon it switches back to the original service distribution. Using a technique due to Baccelli and Makowski we define a martingale with respect to an embedded process and from this arrive at a relationship between the process and a modified Markov renewal process. Using this an analysis of the stationary behaviour of the queue is possible.

An analysis of a modified M/G/1 queue using a martingale technique

1996 ◽  
Vol 33 (01) ◽  
pp. 224-238
Author(s):  
Matthew Roughan
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Service Distribution ◽  
General Service ◽  
Embedded Process ◽  
Stationary Behaviour

We consider a variation of the M/G/1 queue in which, when the system contains more than k customers, it switches from its initial general service distribution to a different general service distribution until the server is cleared, whereupon it switches back to the original service distribution. Using a technique due to Baccelli and Makowski we define a martingale with respect to an embedded process and from this arrive at a relationship between the process and a modified Markov renewal process. Using this an analysis of the stationary behaviour of the queue is possible.

Bounds for the expected delays in some tandem queues

1980 ◽  
Vol 17 (03) ◽  
pp. 831-838 ◽  
Author(s):  
Shun-Chen Niu
Keyword(s):  
Upper Bound ◽  
Renewal Process ◽  
Arrival Process ◽  
Tandem Queues ◽  
Service Distribution ◽  
Deterministic Service ◽  
Service Times ◽  
Constant Service Times ◽  
General Service

Tandem queues are analyzed. An upper bound for the stationary expected delay in front of the second server is found for a sequence of two queues in tandem where the first server has deterministic service times, the second server has general service distribution, and the arrival process is an arbitrary renewal process. The result is extended to the case of n queues in tandem where all the servers except the last one have constant service times.

Bounds for the expected delays in some tandem queues

1980 ◽  
Vol 17 (3) ◽  
pp. 831-838 ◽  
Author(s):  
Shun-Chen Niu
Keyword(s):  
Upper Bound ◽  
Renewal Process ◽  
Arrival Process ◽  
Tandem Queues ◽  
Service Distribution ◽  
Deterministic Service ◽  
Service Times ◽  
Constant Service Times ◽  
General Service

Tandem queues are analyzed. An upper bound for the stationary expected delay in front of the second server is found for a sequence of two queues in tandem where the first server has deterministic service times, the second server has general service distribution, and the arrival process is an arbitrary renewal process. The result is extended to the case of n queues in tandem where all the servers except the last one have constant service times.

scholarly journals Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 55
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Renewal Process ◽  
Markov Chain Model ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Chain Model ◽  
Markov Renewal Processes ◽  
Risky Bonds ◽  
Markov Structure

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

scholarly journals Theory of semiregenerative phenomena

1988 ◽  
Vol 25 (A) ◽  
pp. 257-274
Author(s):  
N. U. Prabhu
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Renewal Process ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Discrete Time Case ◽  
Markov Renewal Theory

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.

Sign in / Sign up

Export Citation Format

Share Document