Convex ordering of sojourn times in single-server queues: extremal properties of FIFO and LIFO service disciplines

1987 ◽  
Vol 24 (03) ◽  
pp. 737-748 ◽  
Author(s):  
J. George Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Recent Result ◽  
Service Time ◽  
Single Server ◽  
Convex Ordering ◽  
Sojourn Times ◽  
Single Server Queues ◽  
Preemptive Resume ◽  
Service Disciplines ◽  
Extremal Properties

In this paper, the extremal properties of the ergodic sojourn times in G/G/1queues under various service disciplines are studied in terms of the convex ordering. It is shown that among work-conserving non-preemptive service disciplines that are service time independent, the FIFO and the LIFO service disciplines provide the minima and the maxima, respectively, of the ergodic sojourn times for any G/G/1 queue. For G/M/1 queues, this class of work-conserving service disciplines is extended to include preemptive/resume disciplines. In this case the FIFO and LIFO-P (preemptive/resume LIFO) service disciplines attain the minima and maxima, respectively. These extend results of Durr (1971), Kingman (1962) and a recent result of Ramaswami (1984). Further results are obtained for G/Em/1 and G/D/1 queues.

Convex ordering of sojourn times in single-server queues: extremal properties of FIFO and LIFO service disciplines

1987 ◽  
Vol 24 (3) ◽  
pp. 737-748 ◽  
Author(s):  
J. George Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Recent Result ◽  
Service Time ◽  
Single Server ◽  
Convex Ordering ◽  
Sojourn Times ◽  
Single Server Queues ◽  
Preemptive Resume ◽  
Service Disciplines ◽  
Extremal Properties

In this paper, the extremal properties of the ergodic sojourn times in G/G/1queues under various service disciplines are studied in terms of the convex ordering. It is shown that among work-conserving non-preemptive service disciplines that are service time independent, the FIFO and the LIFO service disciplines provide the minima and the maxima, respectively, of the ergodic sojourn times for any G/G/1 queue. For G/M/1 queues, this class of work-conserving service disciplines is extended to include preemptive/resume disciplines. In this case the FIFO and LIFO-P (preemptive/resume LIFO) service disciplines attain the minima and maxima, respectively. These extend results of Durr (1971), Kingman (1962) and a recent result of Ramaswami (1984). Further results are obtained for G/Em/1 and G/D/1 queues.

Convex ordering of the attained waiting times in single-server queues and related problems

1991 ◽  
Vol 28 (02) ◽  
pp. 433-445 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Service Discipline ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Convex Order ◽  
Convex Ordering ◽  
Single Server Queues ◽  
Service Disciplines

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

Smoothed Perturbation Analysis for Single-Server Queues by Some General Service Disciplines

1997 ◽  
Vol 29 (2) ◽  
pp. 545-566 ◽  
Author(s):  
Naoto Miyoshi ◽  
Toshiharu Hasegawa
Keyword(s):  
Perturbation Analysis ◽  
Service Time ◽  
Single Server ◽  
Single Server Queues ◽  
The Family ◽  
Service Disciplines ◽  
Preemptive Resume Priority ◽  
Queueing Processes ◽  
Strongly Consistent ◽  
General Service

We consider some single-server queues with general service disciplines, where the family of the queueing processes are parameterized by the service time distributions. Through the smoothed perturbation analysis (SPA) technique, we present under some mild conditions a unified approach to give the strongly consistent estimator for the gradient of the steady-state mean sojourn time with respect to the parameter of service time distributions, provided that it exists. Although the implementation of the SPA requires the additional sub-paths in general, the derived estimator is given as suitable for single-run computation. Simulation results are presented for queues with non-preemptive and preemptive-resume priority disciplines which demonstrate the performance of our estimators.

Smoothed Perturbation Analysis for Single-Server Queues by Some General Service Disciplines

1997 ◽  
Vol 29 (02) ◽  
pp. 545-566
Author(s):  
Naoto Miyoshi ◽  
Toshiharu Hasegawa
Keyword(s):  
Perturbation Analysis ◽  
Service Time ◽  
Single Server ◽  
Single Server Queues ◽  
The Family ◽  
Service Disciplines ◽  
Preemptive Resume Priority ◽  
Queueing Processes ◽  
Strongly Consistent ◽  
General Service

We consider some single-server queues with general service disciplines, where the family of the queueing processes are parameterized by the service time distributions. Through the smoothed perturbation analysis (SPA) technique, we present under some mild conditions a unified approach to give the strongly consistent estimator for the gradient of the steady-state mean sojourn time with respect to the parameter of service time distributions, provided that it exists. Although the implementation of the SPA requires the additional sub-paths in general, the derived estimator is given as suitable for single-run computation. Simulation results are presented for queues with non-preemptive and preemptive-resume priority disciplines which demonstrate the performance of our estimators.

Convex ordering of the attained waiting times in single-server queues and related problems

1991 ◽  
Vol 28 (2) ◽  
pp. 433-445
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Service Discipline ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Convex Order ◽  
Convex Ordering ◽  
Single Server Queues ◽  
Service Disciplines

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (4) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

Weak convergence theorems for priority queues: preemptive-resume discipline

1971 ◽  
Vol 8 (1) ◽  
pp. 74-94 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Weak Convergence ◽  
Service Time ◽  
Priority Queues ◽  
Single Server ◽  
Single Server Queue ◽  
Convergence Theorems ◽  
Lower Priority ◽  
Preemptive Resume ◽  
Server Queue ◽  
Customer Returns

We shall consider a single-server queue with r priority classes of customers and a preemptive-resume discipline. In this system customers are served in order of their priority while customers of the same priority are served in order of their arrival. Higher priority customers, immediately upon arrival, replace lower priority customers at the server, while customers displaced in this way return to the server before any other customers of the same priority receive service. When a displaced customer returns to the server, his remaining service time is the uncompleted portion of his original service time (cf. Jaiswal (1968)).

Weak convergence theorems for priority queues: preemptive-resume discipline

1971 ◽  
Vol 8 (01) ◽  
pp. 74-94 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Weak Convergence ◽  
Service Time ◽  
Priority Queues ◽  
Single Server ◽  
Single Server Queue ◽  
Convergence Theorems ◽  
Lower Priority ◽  
Preemptive Resume ◽  
Server Queue ◽  
Customer Returns

We shall consider a single-server queue with r priority classes of customers and a preemptive-resume discipline. In this system customers are served in order of their priority while customers of the same priority are served in order of their arrival. Higher priority customers, immediately upon arrival, replace lower priority customers at the server, while customers displaced in this way return to the server before any other customers of the same priority receive service. When a displaced customer returns to the server, his remaining service time is the uncompleted portion of his original service time (cf. Jaiswal (1968)).

scholarly journals A Unifying Conservation Law for Single-Server Queues

2007 ◽  
Vol 44 (04) ◽  
pp. 1078-1087 ◽  
Author(s):  
Urtzi Ayesta
Keyword(s):  
Conservation Laws ◽  
Service Time ◽  
Conservation Law ◽  
Single Server ◽  
Processor Sharing ◽  
Time Sharing ◽  
Single Class ◽  
Class Time ◽  
Single Server Queues ◽  
General Service

We develop a conservation law for a multi-class GI/GI/1 queue operating under a general work-conserving scheduling discipline. For single-class single-server queues, conservation laws have been obtained for both nonanticipating and anticipating disciplines with general service time distributions. For multi-class single-server queues, conservation laws have been obtained for (i) nonanticipating disciplines with exponential service time distributions and (ii) nonpreemptive nonanticipating disciplines with general service time distributions. The unifying conservation law we develop generalizes already existing conservation laws. In addition, it covers popular nonanticipating multi-class time-sharing disciplines such as discriminatory processor sharing (DPS) and generalized processor sharing (GPS) with general service time distributions. As an application, we show that the unifying conservation law can be used to compare the expected unconditional response time under two scheduling disciplines.

Asymptotic transient behaviour of the bulk service queue

1963 ◽  
Vol 3 (4) ◽  
pp. 503-512 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Service Time ◽  
Single Server ◽  
Transient Behaviour ◽  
Single Server Queues ◽  
Poisson Input ◽  
Bulk Service ◽  
Service Times

Various authors have studied the transient behaviour of single-server queues. Notably, Takacs [13], [14] has analysed a queue with recurrent input and exponential service time distributions, Keilson and Kooharian [9], [10] and Finch [5] have considered a queue with general independent input and service times, Finch [6] has analysed a queue with non-recurrent input and Erlang service, and Jaiswal [8] has considered the bulk-service queue with Poisson input and Erlang service.

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