scholarly journals A note on the processor-sharing queue in a quasi-reversible network

1983 ◽  
Vol 15 (02) ◽  
pp. 468-469
Author(s):  
Pantelis Tsoucas ◽  
Jean Walrand
Keyword(s):  
Service Time ◽  
Sojourn Time ◽  
Processor Sharing ◽  
Expected Sojourn Time

Consider a processor-sharing queue placed in a quasi-reversible network in equilibrium. This note explains why the expected sojourn time of a customer in such a queue is proportional to his service time.

scholarly journals A note on the processor-sharing queue in a quasi-reversible network

1983 ◽  
Vol 15 (2) ◽  
pp. 468-469 ◽  
Author(s):  
Pantelis Tsoucas ◽  
Jean Walrand
Keyword(s):  
Service Time ◽  
Sojourn Time ◽  
Processor Sharing ◽  
Expected Sojourn Time

Consider a processor-sharing queue placed in a quasi-reversible network in equilibrium. This note explains why the expected sojourn time of a customer in such a queue is proportional to his service time.

INDIVIDUAL EQUILIBRIUM DYNAMIC ROUTING IN A MULTIPLE SERVER RETRIAL QUEUE

2000 ◽  
Vol 14 (1) ◽  
pp. 9-26 ◽  
Author(s):  
Anthony C. Brooms
Keyword(s):  
Nash Equilibrium ◽  
Poisson Process ◽  
Sojourn Time ◽  
Service System ◽  
Transmission Rate ◽  
Retrial Queue ◽  
Dynamic Learning ◽  
Arrival Times ◽  
The Public ◽  
Expected Sojourn Time

Customers arrive sequentially to a service system where the arrival times form a Poisson process of rate λ. The system offers a choice between a private channel and a public set of channels. The transmission rate at each of the public channels is faster than that of the private one; however, if all of the public channels are occupied, then a customer who commits itself to using one of them attempts to connect after exponential periods of time with mean μ−1. Once connection to a public channel has been made, service is completed after an exponential period of time, with mean ν−1. Each customer chooses one of the two service options, basing its decision on the number of busy channels and reapplying customers, with the aim of minimizing its own expected sojourn time. The best action for an individual customer depends on the actions taken by subsequent arriving customers. We establish the existence of a unique symmetric Nash equilibrium policy and show that its structure is characterized by a set of threshold-type strategies; we discuss the relevance of this concept in the context of a dynamic learning scenario.

Convexity results for single-server queues and for multiserver queues with constant service times

1990 ◽  
Vol 27 (02) ◽  
pp. 465-468 ◽  
Author(s):  
Arie Harel
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Sojourn Time ◽  
Interarrival Time ◽  
Service Rate ◽  
Single Server ◽  
Multiserver Queues ◽  
Service Rates ◽  
Constant Service Times ◽  
Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates. These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

Convexity results for single-server queues and for multiserver queues with constant service times

1990 ◽  
Vol 27 (2) ◽  
pp. 465-468 ◽  
Author(s):  
Arie Harel
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Sojourn Time ◽  
Interarrival Time ◽  
Service Rate ◽  
Single Server ◽  
Multiserver Queues ◽  
Service Rates ◽  
Constant Service Times ◽  
Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

On extremal service disciplines in single-stage queueing systems

1990 ◽  
Vol 27 (02) ◽  
pp. 409-416 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar ◽  
Genji Yamazaki
Keyword(s):  
Service Time ◽  
Sojourn Time ◽  
Queueing System ◽  
Residual Life ◽  
Queueing Systems ◽  
Mean Residual Life ◽  
Service Discipline ◽  
Service Time Distribution ◽  
Service Disciplines ◽  
Number Of Customers

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.

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.

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