Batch sojourn time in the processor sharing queue with geometric batch size

2021 ◽  
pp. 1-120 ◽  
Author(s):  
Fabrice Guillemin ◽  
Alain Simonian ◽  
Ridha Nasri ◽  
Veronica Quintuna Rodriguez
Keyword(s):  
Sojourn Time ◽  
Batch Size ◽  
Processor Sharing

scholarly journals Processor-sharing and random-service queues with semi-Markovian arrivals

2005 ◽  
Vol 42 (02) ◽  
pp. 478-490
Author(s):  
De-An Wu ◽  
Hideaki Takagi
Keyword(s):  
Waiting Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Interarrival Time ◽  
Size Distributions ◽  
Processor Sharing ◽  
Queue Size ◽  
Mean Waiting Time ◽  
The Mean

We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-lcustomer who, upon his arrival, meetskcustomers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-lcustomer who, upon his arrival, meetsk+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

Sojourn-time approximations for a multi-server processor sharing system with priorities

2003 ◽  
Vol 54 (3) ◽  
pp. 249-261 ◽  
Author(s):  
R.D. van der Mei ◽  
J.L. van den Berg ◽  
R. Vranken ◽  
B.M.M. Gijsen
Keyword(s):  
Sojourn Time ◽  
Processor Sharing ◽  
Multi Server

ON THE VARIANCES OF SYSTEM SIZE AND SOJOURN TIME IN A DISCRETE-TIME DAR(1)/D/1 QUEUE

2011 ◽  
Vol 25 (4) ◽  
pp. 519-535 ◽  
Author(s):  
Daniel Wei-Chung Miao ◽  
Hung Chen
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Performance Measures ◽  
Discrete Time ◽  
Sojourn Time ◽  
Blow Up ◽  
System Size ◽  
Batch Size ◽  
Numerical Examples ◽  
Cross Terms

We consider a discrete-time DAR(1)/D/1 queue and provide an analysis on the variances of both its system size and sojourn time. Our approach is simple, but the results are nice, as these variances are found in closed form. We first establish the relation between these variances, based on which we then use the conditioning technique to analyze the expected cross terms that come from its system recurrence relation. The closed-form results allow us to explicitly examine the effect from the batch size distribution and the autocorrelation parameter p. It is observed that as p grows toward 1, the standard deviations of the two performance measures will blow up in same asymptotic order of O(1/(1−p)) as their means. These are demonstrated through numerical examples.

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