Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

1991 ◽  
Vol 23 (1) ◽  
pp. 210-228 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Xiu Li Chao ◽  
Michael Pinedo
Keyword(s):  
Queueing Systems ◽  
Loss Probability ◽  
Service Time Distribution ◽  
Doubly Stochastic ◽  
Virtual Waiting Time ◽  
Two Systems ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Monotonicity Results

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V(i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q(i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

1991 ◽  
Vol 23 (01) ◽  
pp. 210-228 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Xiu Li Chao ◽  
Michael Pinedo
Keyword(s):  
Queueing Systems ◽  
Loss Probability ◽  
Service Time Distribution ◽  
Doubly Stochastic ◽  
Virtual Waiting Time ◽  
Two Systems ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Monotonicity Results

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V (i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q (i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

Tollbooth tandem queues with infinite homogeneous servers

2015 ◽  
Vol 52 (4) ◽  
pp. 941-961 ◽  
Author(s):  
Xiuli Chao ◽  
Qi-Ming He ◽  
Sheldon Ross
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Stochastic Ordering ◽  
Tandem Queues ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Steady State Solutions ◽  
Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Comparing multi-server queues with finite waiting rooms, II: Different numbers of servers

1979 ◽  
Vol 11 (02) ◽  
pp. 448-455 ◽  
Author(s):  
David Sonderman
Keyword(s):  
Probability Space ◽  
Queueing Systems ◽  
Stochastic Order ◽  
Sample Path ◽  
System Size ◽  
Waiting Room ◽  
Virtual Waiting Time ◽  
Number Of Customers ◽  
Multi Server ◽  
Quantities Of Interest

We compare two queueing systems with identical general arrival streams, but different numbers of servers, different waiting room capacities, and stochastically ordered service time distributions. Under appropriate conditions, it is possible to construct two new systems on the same probability space so that the new systems are probabilistically equivalent to the original systems and each sample path of the stochastic process representing system size in one system lies entirely below the corresponding sample path in the other system. This construction implies stochastic order for these processes and many associated quantities of interest, such as a busy period, the number of customers lost in any interval, and the virtual waiting time.

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.

On extremal service disciplines in single-stage queueing systems

1990 ◽  
Vol 27 (2) ◽  
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.

On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions

1983 ◽  
Vol 20 (04) ◽  
pp. 860-871 ◽  
Author(s):  
Dieter König ◽  
Masakiyo Miyazawa ◽  
Volker Schmidt
Keyword(s):  
Stationary State ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Arrival Process ◽  
Loss Probabilities ◽  
Poisson Arrivals ◽  
Number Of Customers

For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.

Steady state solution of a single-server queue with linear repeated requests

1997 ◽  
Vol 34 (01) ◽  
pp. 223-233 ◽  
Author(s):  
J. R. Artalejo ◽  
A. Gomez-Corral
Keyword(s):  
Steady State ◽  
Hypergeometric Functions ◽  
Queueing Systems ◽  
General Setting ◽  
Steady State Solution ◽  
Single Server ◽  
Service Time Distribution ◽  
Steady State Distribution ◽  
Server Queue ◽  
Number Of Customers

Queueing systems with repeated requests have many useful applications in communications and computer systems modeling. In the majority of previous work the repeat requests are made individually by each unsatisfied customer. However, there is in the literature another type of queueing situation, in which the time between two successive repeated attempts is independent of the number of customers applying for service. This paper deals with the M/G/1 queue with repeated orders in its most general setting, allowing the simultaneous presence of both types of repeat requests. We first study the steady state distribution and the partial generating functions. When the service time distribution is exponential we show that the performance characteristics can be expressed in terms of hypergeometric functions.

On a decomposition result for a class of vacation queueing systems

1990 ◽  
Vol 27 (1) ◽  
pp. 227-231 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Steady State ◽  
Time Distribution ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Random Size ◽  
Poisson Arrivals ◽  
General Service

We consider a single-server infinite-capacity queueing sysem with Poisson arrivals of customer groups of random size and a general service time distribution, the server of which applies a general exhaustive service vacation policy. We are concerned with the steady-state distribution of the actual waiting time of a customer arriving while the server is active.

On a decomposition result for a class of vacation queueing systems

1990 ◽  
Vol 27 (01) ◽  
pp. 227-231 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Steady State ◽  
Time Distribution ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Random Size ◽  
Poisson Arrivals ◽  
General Service

We consider a single-server infinite-capacity queueing sysem with Poisson arrivals of customer groups of random size and a general service time distribution, the server of which applies a general exhaustive service vacation policy. We are concerned with the steady-state distribution of the actual waiting time of a customer arriving while the server is active.

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