scholarly journals Optimal Multiserver Stochastic Scheduling of two Interconnected Priority Queues

1994 ◽  
Vol 26 (1) ◽  
pp. 258-279 ◽  
Author(s):  
Dimitrios G. Pandelis ◽  
Demosthenis Teneketzis
Keyword(s):  
Sufficient Conditions ◽  
Stochastic Scheduling ◽  
Random Variable ◽  
General Distribution ◽  
Single Server ◽  
Preemptive Scheduling ◽  
Discounted Cost ◽  
Holding Costs ◽  
Holding Cost ◽  
Server Problem

A number of jobs on two interconnected queues are to be processed by m identical servers. The servers operate in parallel, so that every server can process any job. Jobs in queue i, i = 1, 2, incur an instantaneous holding cost Ci during the time they remain in the system. The service time for jobs in queue i, denoted by Xi, is a random variable with a general distribution. The interconnection process is independent of the service process. We establish sufficient conditions on the service times, the holding costs and the interconnection process under which the non-preemptive scheduling strategy that gives priority to queue 1 minimizes the total expected α -discounted cost. We call this strategy P1. We present counterexamples showing that if any of the sufficient conditions is not satisfied P1 may not be optimal, and that the optimal policy for the single-server problem is not necessarily optimal for the multiserver problem.

scholarly journals Optimal Multiserver Stochastic Scheduling of two Interconnected Priority Queues

1994 ◽  
Vol 26 (01) ◽  
pp. 258-279 ◽  
Author(s):  
Dimitrios G. Pandelis ◽  
Demosthenis Teneketzis
Keyword(s):  
Sufficient Conditions ◽  
Stochastic Scheduling ◽  
Random Variable ◽  
General Distribution ◽  
Single Server ◽  
Preemptive Scheduling ◽  
Discounted Cost ◽  
Holding Costs ◽  
Holding Cost ◽  
Server Problem

A number of jobs on two interconnected queues are to be processed by m identical servers. The servers operate in parallel, so that every server can process any job. Jobs in queue i, i = 1, 2, incur an instantaneous holding cost Ci during the time they remain in the system. The service time for jobs in queue i, denoted by Xi , is a random variable with a general distribution. The interconnection process is independent of the service process. We establish sufficient conditions on the service times, the holding costs and the interconnection process under which the non-preemptive scheduling strategy that gives priority to queue 1 minimizes the total expected α -discounted cost. We call this strategy P1. We present counterexamples showing that if any of the sufficient conditions is not satisfied P1 may not be optimal, and that the optimal policy for the single-server problem is not necessarily optimal for the multiserver problem.

scholarly journals Optimal stochastic scheduling of forest networks with switching penalties

1994 ◽  
Vol 26 (02) ◽  
pp. 474-497 ◽  
Author(s):  
Mark P. Van Oyen ◽  
Demosthenis Teneketzis
Keyword(s):  
Structural Properties ◽  
Optimal Policy ◽  
Switching Costs ◽  
Stochastic Scheduling ◽  
Single Server ◽  
Reward Rate ◽  
Holding Costs ◽  
Lump Sum ◽  
Optimal Policies ◽  
Set Up

We present structural properties of optimal policies for the problem of scheduling a single server in a forest network of N queues (without arrivals) subject to switching penalties. In addition to linear holding costs, we impose either lump sum switching costs or batch set-up delays which are incurred at each instant the server processes a job in a queue different from the previous one. We use reward rate notions to unearth conditions on the holding costs and service distributions for which an exhaustive policy is optimal. For the case of two nodes connected probabilistically in tandem, we explicitly define an optimal policy under similar conditions.

scholarly journals Optimal stochastic scheduling of forest networks with switching penalties

1994 ◽  
Vol 26 (2) ◽  
pp. 474-497 ◽  
Author(s):  
Mark P. Van Oyen ◽  
Demosthenis Teneketzis
Keyword(s):  
Structural Properties ◽  
Optimal Policy ◽  
Switching Costs ◽  
Stochastic Scheduling ◽  
Single Server ◽  
Reward Rate ◽  
Holding Costs ◽  
Lump Sum ◽  
Optimal Policies ◽  
Set Up

We present structural properties of optimal policies for the problem of scheduling a single server in a forest network of N queues (without arrivals) subject to switching penalties. In addition to linear holding costs, we impose either lump sum switching costs or batch set-up delays which are incurred at each instant the server processes a job in a queue different from the previous one. We use reward rate notions to unearth conditions on the holding costs and service distributions for which an exhaustive policy is optimal. For the case of two nodes connected probabilistically in tandem, we explicitly define an optimal policy under similar conditions.

DYNAMIC CONTROL OF A SINGLE-SERVER SYSTEM WHEN JOBS CHANGE STATUS

2017 ◽  
Vol 32 (3) ◽  
pp. 353-395
Author(s):  
Gabriel Zayas-Cabán ◽  
Hyun-Soo Ahn
Keyword(s):  
Queueing Systems ◽  
Dynamic Control ◽  
Sufficient Conditions ◽  
Single Server ◽  
Priority Rules ◽  
Holding Costs ◽  
Class 1 ◽  
Markov Decision ◽  
Service Rates ◽  
Customer Model

From health care to maintenance shops, many systems must contend with allocating resources to customers or jobs whose initial service requirements or costs change when they wait too long. We present a new queueing model for this scenario and use a Markov decision process formulation to analyze assignment policies that minimize holding costs. We show that the classic cμ rule is generally not optimal when service or cost requirements can change. Even for a two-class customer model where a class 1 task becomes a class 2 task upon waiting, we show that additional orderings of the service rates are needed to ensure the optimality of simple priority rules. We then show that seemingly-intuitive switching curve structures are also not optimal in general. We study these scenarios and provide conditions under which they do hold. Lastly, we show that results from the two-class model do not extend to when there are n≥3 customer classes. More broadly, we find that simple priority rules are not optimal. We provide sufficient conditions under which a simple priority rule holds. In short, allowing service and/or cost requirements to change fundamentally changes the structure of the optimal policy for resource allocation in queueing systems.

scholarly journals Sequential scheduling of priority queues and Arm-acquiring bandits

1987 ◽  
Vol 1 (2) ◽  
pp. 115-135 ◽  
Author(s):  
Chuanshu Ji
Keyword(s):  
Optimal Policy ◽  
Average Cost ◽  
Queueing Network ◽  
Single Server ◽  
Index Policy ◽  
Discounted Cost ◽  
Poisson Arrival ◽  
Long Run ◽  
Holding Cost ◽  
Arrival Processes

In a queueing network with a single server and r service nodes, a non-preemptive non-idling policy chooses a node to service at each service completion epoch. Under the assumptions of independent Poisson arrival processes, fixed routing probabilities, and linear holding cost rates, we apply Whistle's method for Arm-acquiring bandits to show that for minimizing discounted cost or long-run average cost the optimal policy is an index policy. We also give explicit expressions for those priority indices.

Optimal control of the service rate in an M/G/1 queueing system

1978 ◽  
Vol 10 (3) ◽  
pp. 682-701 ◽  
Author(s):  
Bharat T. Doshi
Keyword(s):  
Optimal Control ◽  
Average Cost ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Service Cost ◽  
Stationary Policy ◽  
Discounted Cost ◽  
Cost Rate ◽  
Holding Cost

We consider an M/G/1 queue in which the service rate is subject to control. The control is exercised continuously and is based on the observations of the residual workload process. For both the discounted cost and the average cost criteria we obtain conditions which are sufficient for a stationary policy to be optimal. When the service cost rate and the holding cost rates are non-decreasing and convex it is shown that these sufficient conditions are satisfied by a monotonic policy, thus showing its optimality.

Optimal control of the service rate in an M/G/1 queueing system

1978 ◽  
Vol 10 (03) ◽  
pp. 682-701 ◽  
Author(s):  
Bharat T. Doshi
Keyword(s):  
Optimal Control ◽  
Average Cost ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Service Cost ◽  
Stationary Policy ◽  
Discounted Cost ◽  
Cost Rate ◽  
Holding Cost

We consider an M/G/1 queue in which the service rate is subject to control. The control is exercised continuously and is based on the observations of the residual workload process. For both the discounted cost and the average cost criteria we obtain conditions which are sufficient for a stationary policy to be optimal. When the service cost rate and the holding cost rates are non-decreasing and convex it is shown that these sufficient conditions are satisfied by a monotonic policy, thus showing its optimality.

STOCHASTIC SCHEDULING WITH ASYMMETRIC EARLINESS AND TARDINESS PENALTIES UNDER RANDOM MACHINE BREAKDOWNS

2006 ◽  
Vol 20 (4) ◽  
pp. 635-654 ◽  
Author(s):  
Xiaoqiang Cai ◽  
Xian Zhou
Keyword(s):  
Processing Time ◽  
Sufficient Conditions ◽  
Stochastic Scheduling ◽  
General Distribution ◽  
Optimal Sequence ◽  
Total Cost ◽  
Earliness And Tardiness ◽  
Special Cases ◽  
Expected Total Cost ◽  
Stochastic Breakdowns

We study a stochastic scheduling problem of processing a set of jobs on a single machine. Each job has a random processing time Pi and a random due date Di, which are independently and exponentially distributed. The machine is subject to stochastic breakdowns in either preempt-resume or preempt-repeat patterns, with the uptimes following an exponential distribution and the downtimes (repair times) following a general distribution. The problem is to determine an optimal sequence for the machine to process all jobs so as to minimize the expected total cost comprising asymmetric earliness and tardiness penalties, in the form of E[[sum ]αi max{0,Di − Ci} + βi max{0,Ci − Di}]. We find sufficient conditions for the optimal sequences to be V-shaped with respect to {E(Pi)/αi} and {E(Pi)/βi}, respectively, which cover previous results in the literature as special cases. We also find conditions under which optimal sequences can be derived analytically. An algorithm is provided that can compute the best V-shaped sequence.

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.

scholarly journals Minimizing spectral risk measures applied to Markov decision processes

2021 ◽  
Author(s):  
Nicole Bäuerle ◽  
Alexander Glauner
Keyword(s):  
Minimization Problem ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Planning Horizon ◽  
Insurance Company ◽  
Existence Of A Solution ◽  
Discounted Cost ◽  
Spectral Risk Measures ◽  
Markov Decision

AbstractWe study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in Bäuerle and Ott (Math Methods Oper Res 74(3):361–379, 2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital.

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