Stochastic comparisons for queueing models via random sums and intervals

1992 ◽  
Vol 24 (04) ◽  
pp. 960-985 ◽  
Author(s):  
Alain Jean-Marie ◽  
Zhen Liu
Keyword(s):  
Point Process ◽  
Response Times ◽  
Waiting Times ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Partial Sums ◽  
Polling Systems ◽  
Convex Ordering ◽  
Random Intervals ◽  
Increasing Convex Ordering

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered. These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

Stochastic comparisons for queueing models via random sums and intervals

1992 ◽  
Vol 24 (4) ◽  
pp. 960-985 ◽  
Author(s):  
Alain Jean-Marie ◽  
Zhen Liu
Keyword(s):  
Point Process ◽  
Response Times ◽  
Waiting Times ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Partial Sums ◽  
Polling Systems ◽  
Convex Ordering ◽  
Random Intervals ◽  
Increasing Convex Ordering

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered.These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

Optimality of the round-robin routing policy

1994 ◽  
Vol 31 (2) ◽  
pp. 466-475 ◽  
Author(s):  
Zhen Liu ◽  
Don Towsley
Keyword(s):  
Failure Rate ◽  
Response Times ◽  
Random Variables ◽  
Round Robin ◽  
State Information ◽  
Convex Ordering ◽  
Increasing Failure Rate ◽  
Rate Distribution ◽  
Routing Policy ◽  
Increasing Convex Ordering

In this paper we consider the problem of routing customers to identical servers, each with its own infinite-capacity queue. Under the assumptions that (i) the service times form a sequence of independent and identically distributed random variables with increasing failure rate distribution and (ii) state information is not available, we establish that the round-robin policy minimizes, in the sense of a separable increasing convex ordering, the customer response times and the numbers of customers in the queues.

Optimality of the round-robin routing policy

1994 ◽  
Vol 31 (02) ◽  
pp. 466-475 ◽  
Author(s):  
Zhen Liu ◽  
Don Towsley
Keyword(s):  
Failure Rate ◽  
Response Times ◽  
Random Variables ◽  
Round Robin ◽  
State Information ◽  
Convex Ordering ◽  
Increasing Failure Rate ◽  
Rate Distribution ◽  
Routing Policy ◽  
Increasing Convex Ordering

In this paper we consider the problem of routing customers to identical servers, each with its own infinite-capacity queue. Under the assumptions that (i) the service times form a sequence of independent and identically distributed random variables with increasing failure rate distribution and (ii) state information is not available, we establish that the round-robin policy minimizes, in the sense of a separable increasing convex ordering, the customer response times and the numbers of customers in the queues.

A stochastic ordering of partial sums of independent random variables and of some random processes

1992 ◽  
Vol 29 (03) ◽  
pp. 645-654 ◽  
Author(s):  
Philip J. Boland ◽  
Frank Proschan ◽  
Y. L. Tong
Keyword(s):  
Random Variables ◽  
Stochastic Ordering ◽  
Random Processes ◽  
Reliability Theory ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Similar Argument ◽  
First Passage ◽  
Wiener Processes ◽  
Likelihood Ratio Ordering

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.

A stochastic ordering of partial sums of independent random variables and of some random processes

1992 ◽  
Vol 29 (3) ◽  
pp. 645-654 ◽  
Author(s):  
Philip J. Boland ◽  
Frank Proschan ◽  
Y. L. Tong
Keyword(s):  
Random Variables ◽  
Stochastic Ordering ◽  
Random Processes ◽  
Reliability Theory ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Similar Argument ◽  
First Passage ◽  
Wiener Processes ◽  
Likelihood Ratio Ordering

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

1976 ◽  
Vol 13 (2) ◽  
pp. 361-364 ◽  
Author(s):  
M. E. Solari ◽  
J. E. A. Dunnage
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Partial Sums ◽  
Dependent Random Variables ◽  
Finite Segment ◽  
Exchangeable Random Variables ◽  
Partial Sum

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

Stochastic ordering of random processes with an imbedded point process

1991 ◽  
Vol 28 (3) ◽  
pp. 553-567 ◽  
Author(s):  
François Baccelli
Keyword(s):  
Point Process ◽  
Continuous Time ◽  
Random Sequence ◽  
Stochastic Ordering ◽  
Random Processes ◽  
Time Process ◽  
Continuous Time Process ◽  
Partial Orderings ◽  
Stationary Probabilities

We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.

Sign in / Sign up

Export Citation Format

Share Document