Stochastic monotonicity properties of multiserver queues with impatient customers

1991 ◽  
Vol 28 (3) ◽  
pp. 673-682 ◽  
Author(s):  
Partha P. Bhattacharya ◽  
Anthony Ephremides
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Time Interval ◽  
Impatient Customers ◽  
Monotone Functions ◽  
Stochastic Monotonicity ◽  
Multiserver Queues ◽  
Monotonicity Properties ◽  
Number Of Customers

We consider multiserver queues in which a customer is lost whenever its waiting time is larger than its (possibly random) deadline. For such systems, the number of (successful) departures and the number of customers lost over a time interval are the performance measures of interest. We show that these quantities are (stochastically) monotone functions of the arrival, service and deadline processes.

Stochastic monotonicity properties of multiserver queues with impatient customers

1991 ◽  
Vol 28 (03) ◽  
pp. 673-682
Author(s):  
Partha P. Bhattachary ◽  
Anthony Ephremides
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Time Interval ◽  
Impatient Customers ◽  
Monotone Functions ◽  
Stochastic Monotonicity ◽  
Multiserver Queues ◽  
Monotonicity Properties ◽  
Number Of Customers

We consider multiserver queues in which a customer is lost whenever its waiting time is larger than its (possibly random) deadline. For such systems, the number of (successful) departures and the number of customers lost over a time interval are the performance measures of interest. We show that these quantities are (stochastically) monotone functions of the arrival, service and deadline processes.

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)).

Waiting Time in M/G/1 Queues with Impolite Arrival Disciplines

1995 ◽  
Vol 9 (2) ◽  
pp. 255-267 ◽  
Author(s):  
Süleyman Òzekici ◽  
Jingwen Li ◽  
Fee Seng Chou
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Time Distribution ◽  
Iterative Procedure ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Special Model ◽  
Average Waiting Time ◽  
Random Position ◽  
Number Of Customers

We consider a queueing system where arriving customers join the queue at some random position. This constitutes an impolite arrival discipline because customers do not necessarily go to the end of the line upon arrival. Although mean performance measures like the average waiting time and average number of customers in the queue are the same for all such disciplines, we show that the variance of the waiting time increases as the arrival discipline becomes more impolite, in the sense that a customer is more likely to choose a position closer to the server. For the M/G/1 model, we also provide an iterative procedure for computing the moments of the waiting time distribution. Explicit computational formulas are derived for an interesting special model where a customer joins the queue either at the head or at the end of the line.

Correlation functions in queueing theory

1972 ◽  
Vol 9 (3) ◽  
pp. 604-616 ◽  
Author(s):  
S. K. Srinivasan ◽  
R. Subramanian ◽  
R. Vasudevan
Keyword(s):  
Waiting Time ◽  
Correlation Functions ◽  
Queueing Theory ◽  
Product Space ◽  
Point Processes ◽  
Time Interval ◽  
Important Criterion ◽  
Time Parameters ◽  
Set Up ◽  
Number Of Customers

The object of this paper is to study the actual waiting time of a customer in a GI/G/1 queue. This is an important criterion from the viewpoint of both the customers and the efficient functioning of the counter. Suitable point processes in the product space of load and time parameters for any general inter-arrival and service time distributions are defined and integral equations governing the correlation functions are set up. Solutions of these equations are obtained and with the help of these, explicit expressions for the first two moments of the number of customers who have waited for a time longer than w in a given time interval (0, T) are calculated.

Correlation functions in queueing theory

1972 ◽  
Vol 9 (03) ◽  
pp. 604-616 ◽  
Author(s):  
S. K. Srinivasan ◽  
R. Subramanian ◽  
R. Vasudevan
Keyword(s):  
Waiting Time ◽  
Correlation Functions ◽  
Queueing Theory ◽  
Product Space ◽  
Point Processes ◽  
Time Interval ◽  
Important Criterion ◽  
Time Parameters ◽  
Set Up ◽  
Number Of Customers

The object of this paper is to study the actual waiting time of a customer in a GI/G/1 queue. This is an important criterion from the viewpoint of both the customers and the efficient functioning of the counter. Suitable point processes in the product space of load and time parameters for any general inter-arrival and service time distributions are defined and integral equations governing the correlation functions are set up. Solutions of these equations are obtained and with the help of these, explicit expressions for the first two moments of the number of customers who have waited for a time longer than w in a given time interval (0, T) are calculated.

A relation between stationary queue and waiting time distributions

1971 ◽  
Vol 8 (03) ◽  
pp. 617-620 ◽  
Author(s):  
Rasoul Haji ◽  
Gordon F. Newell
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Arrival Process ◽  
Time Interval ◽  
Queue Size ◽  
Random Time Interval ◽  
Queue Discipline ◽  
Stationary Waiting Time ◽  
Number Of Customers ◽  
First In First Out

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

scholarly journals On the GI/M/1 Queue with Vacations and Multiple Service Phases

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Jianjun Li ◽  
Liwei Liu
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Solution Method ◽  
System Size ◽  
Mean Waiting Time ◽  
The Matrix ◽  
Matrix Geometric Solution ◽  
Semi Markov Process ◽  
Stationary Waiting Time ◽  
Number Of Customers

This paper considers a GI/M/1 queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase 0. If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase 0 to some service phase i with probability qi,  i=1,2,…,N. Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the type-i cycle, and mean number of customers in the system at the end of phase 0. Finally, some numerical examples are presented.

A relation between stationary queue and waiting time distributions

1971 ◽  
Vol 8 (3) ◽  
pp. 617-620 ◽  
Author(s):  
Rasoul Haji ◽  
Gordon F. Newell
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Arrival Process ◽  
Time Interval ◽  
Queue Size ◽  
Random Time Interval ◽  
Queue Discipline ◽  
Stationary Waiting Time ◽  
Number Of Customers ◽  
First In First Out

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

scholarly journals Neutrosophic Event-Based Queueing Model

2020 ◽  
pp. 48-55
Author(s):  
Mohamed Bisher Zeina ◽  
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Communication Systems ◽  
Queueing Systems ◽  
Real Life ◽  
Main Tool ◽  
Expected Number ◽  
Mean Waiting Time ◽  
Event Based ◽  
Number Of Customers

In this paper we have defined the concept of neutrosophic queueing systems and defined its neutrosophic performance measures. An important application of neutrosophic logic in queueing systems we face in real life were discussed, that is the neutrosophic events accuring times, because of its wide applications in networking and simulating communication systems specialy when probability distribution is not known, and because it’s more realistic to consider and to not ignore the imprecise events times. Event-based table of a neutrosophic queueing system was presented and its neutrosophic performance measures were derived, i.e. neutrosophic mean waiting time in queue, neutrosophic mean waiting time in system, neutrosophic expected number of customers in queue and neutrosophic expected number of customers in system. Neutrosophic Little’s Formulas (NLF) were also defined which is a main tool in queueing systems problems to make it easier finding performance measures from each other.

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