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.

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.

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.

A multivariate IFR class

1985 ◽  
Vol 22 (01) ◽  
pp. 197-204 ◽  
Author(s):  
Thomas H. Savits
Keyword(s):  
Exponential Distribution ◽  
Failure Rate ◽  
Random Vector ◽  
Increasing Failure Rate ◽  
Rate Distribution ◽  
Weak Limits ◽  
Multivariate Exponential Distribution ◽  
Multivariate Exponential

A non-negative random vector T is said to have a multivariate increasing failure rate distribution (MIFR) if and only if E[h(x, T)] is log concave in x for all functions h(x, t) which are log concave in (x, t) and are non-decreasing and continuous in t for each fixed x. This class of distributions is closed under deletion, conjunction, convolution and weak limits. It contains the multivariate exponential distribution of Marshall and Olkin and those distributions having a log concave density. Also, it follows that if T is MIFR and ψ is non-decreasing, non-negative and concave then ψ (T) is IFR.

Optimal policies for machine repairmen problems

1993 ◽  
Vol 30 (3) ◽  
pp. 703-715 ◽  
Author(s):  
Esther Frostig
Keyword(s):  
Failure Rate ◽  
Optimal Policy ◽  
Repair Time ◽  
Increasing Failure Rate ◽  
Rate Distribution ◽  
Unreliable Machines ◽  
Optimal Policies

n unreliable machines are maintained by m repairmen. Assuming exponentially distributed up-time and repair time we find the optimal policy to allocate the repairmen to the failed machines in order to stochastically minimize the time until all machines work. Considering only one repairman, we find the optimal policy to maximize the expected total discount time that machines work. We find the optimal policy for the cases where the up-time and repair time are exponentially distributed or identically arbitrarily distributed up-times and increasing failure rate distribution repair times.

A multivariate IFR class

1985 ◽  
Vol 22 (1) ◽  
pp. 197-204 ◽  
Author(s):  
Thomas H. Savits
Keyword(s):  
Exponential Distribution ◽  
Failure Rate ◽  
Random Vector ◽  
Increasing Failure Rate ◽  
Rate Distribution ◽  
Weak Limits ◽  
Multivariate Exponential Distribution ◽  
Multivariate Exponential

A non-negative random vector T is said to have a multivariate increasing failure rate distribution (MIFR) if and only if E[h(x, T)] is log concave in x for all functions h(x, t) which are log concave in (x, t) and are non-decreasing and continuous in t for each fixed x. This class of distributions is closed under deletion, conjunction, convolution and weak limits. It contains the multivariate exponential distribution of Marshall and Olkin and those distributions having a log concave density. Also, it follows that if T is MIFR and ψ is non-decreasing, non-negative and concave then ψ (T) is IFR.

Optimal policies for machine repairmen problems

1993 ◽  
Vol 30 (03) ◽  
pp. 703-715 ◽  
Author(s):  
Esther Frostig
Keyword(s):  
Failure Rate ◽  
Optimal Policy ◽  
Repair Time ◽  
Increasing Failure Rate ◽  
Rate Distribution ◽  
Unreliable Machines ◽  
Optimal Policies

n unreliable machines are maintained by m repairmen. Assuming exponentially distributed up-time and repair time we find the optimal policy to allocate the repairmen to the failed machines in order to stochastically minimize the time until all machines work. Considering only one repairman, we find the optimal policy to maximize the expected total discount time that machines work. We find the optimal policy for the cases where the up-time and repair time are exponentially distributed or identically arbitrarily distributed up-times and increasing failure rate distribution repair times.

Some bounds on reliability of coherent systems of IFRA components

1991 ◽  
Vol 28 (3) ◽  
pp. 709-714 ◽  
Author(s):  
Gopal Chaudhuri ◽  
Jayant V. Deshpande ◽  
Avinash D. Dharmadhikari
Keyword(s):  
Failure Rate ◽  
Random Variables ◽  
Bridge Structure ◽  
Survival Functions ◽  
Coherent Systems ◽  
Increasing Failure Rate ◽  
Independent Components ◽  
Life Distributions

In this paper we find new bounds for the reliability of coherent systems of independent components with increasing failure rate average (IFRA) lifetimes. These bounds are based on certain bounds available for the survival functions of IFRA random variables and the fact that the IFRA class of life distributions is closed under the formation of coherent systems. These bounds are compared with other applicable bounds in this case. An illustration of explicit computations of the bounds is provided for the bridge structure with components having independent gamma life distributions.

Some bounds on reliability of coherent systems of IFRA components

1991 ◽  
Vol 28 (03) ◽  
pp. 709-714 ◽  
Author(s):  
Gopal Chaudhuri ◽  
Jayant V. Deshpande ◽  
Avinash D. Dharmadhikari
Keyword(s):  
Failure Rate ◽  
Random Variables ◽  
Bridge Structure ◽  
Survival Functions ◽  
Coherent Systems ◽  
Increasing Failure Rate ◽  
Independent Components ◽  
Life Distributions

In this paper we find new bounds for the reliability of coherent systems of independent components with increasing failure rate average (IFRA) lifetimes. These bounds are based on certain bounds available for the survival functions of IFRA random variables and the fact that the IFRA class of life distributions is closed under the formation of coherent systems. These bounds are compared with other applicable bounds in this case. An illustration of explicit computations of the bounds is provided for the bridge structure with components having independent gamma life distributions.

When Is a Renewal Process Convexly Parameterized in Its Mean Parameterization?

1997 ◽  
Vol 11 (1) ◽  
pp. 43-48
Author(s):  
James D. Lynch
Keyword(s):  
Failure Rate ◽  
Renewal Process ◽  
Survival Functions ◽  
Increasing Failure Rate ◽  
Rate Distribution

The convex (concave) parameterization of a generalized renewal process is considered in this paper. It is shown that if the interrenewal times have log concave distributions or have log concave survival functions (i.e., an increasing failure rate distribution), then the renewal process is convexly (concavely) parameterized in its mean parameterization.

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