Esther Frostig

Single machine flow-time scheduling with a single breakdown

SummaryWe consider the problem of scheduling tasks on a single machine to minimize the flowtime. The machine is subject to breakdowns during the processing of the tasks. The breakdowns occur at a random times and the machine is unavailable until it is repaired. The times for repair are random and independent of each other and of the breakdown process. A task that is preempted due to a breakdown must be restarted and otherwise preemptions are not allowed. We show in the case of a single breakdown that if the distribution function of the time to breakdown is concave then Shortest Processing Time (SPT) first scheduling stochastically minimizes the flowtime. For the case of multiple breakdowns we show that SPT minimizes the expected flowtime when the times to breakdown are exponentially distributed. If the time for a single breakdown is known before scheduling begins, and the processing times of the tasks are also known, then we show that the problem of deciding whether there is a schedule with flowtime less than or equal to a given value is NP-complete. Finally, we bound the performance of SPT scheduling in the deterministic case when there is a single breakdown.

Association and heterogeneity of insured lifetimes in the Lee-Carter framework

This paper is devoted to the study of some unexpected consequences of the Lee–Carter model for mortality projection. The fact that survival probabilities are governed by a stochastic process induces some positive dependence between insured lifetimes (namely, association). This, in turn, has an impact on solvency capital (as measured by distortion risk measures, for instance). Failing to take this dependence into account, by assuming falsely that the lifetimes are independent, leads to systematic underestimations of the risk capital. The heterogeneity between the policy benefits and the insured lifetimes is also studied (with the help of majorisation, Schur-increasingness and a frailty model).

Scheduling on a single machine with a single breakdown to minimize stochastically the number of tardy jobs

Jobs with known processing times and due dates have to be processed on a machine which is subject to a single breakdown. The moment of breakdown and the repair time are independent random variables. Two cases are distinguished with reference to the processing time preempted by the breakdown (no other preemptions are allowed): (i) resumption without time losses and (ii) restart from the beginning. Under certain compatible conditions, we find the policies which minimize stochastically the number of tardy jobs.

FOUR PROOFS OF GITTINS' MULTIARMED BANDIT THEOREM

We study four proofs that the Gittins index priority rule is optimal for alternative bandit processes. These include Gittins’ original exchange argument, Weber’s prevailing charge argument, Whittle’s Lagrangian dual approach, and Bertsimas and Niño-Mora’s proof based on the achievable region approach and generalized conservation laws. We extend the achievable region proof to infinite countable state spaces, by using infinite dimensional linear programming theory.

A Note on Stochastic Scheduling on a Single Machine Subject to Breakdown–The Preemptive Repeat Model

This paper considers scheduling n jobs on one machine to minimize the expected weighted flowtime and the number of late jobs. The processing times of the jobs are independent random variables. The machine is subject to failure and repair where the uptimes are exponentially distributed. We find the optimal policies for the preemptive repeat model.

Availability of inspected systems subject to shocks - A matrix algorithmic approach

We examine the limiting average availability of a maintained system that deteriorates due to random shock process and as a response to its usage (wear out). Systems failures are not self-announcing, hence, failures must be detected via inspection. We consider randomly occurring shocks that arrive according to a Poisson process and cumulatively damage the system. Two models are considered: in Model 1 the shock and wear out processes are independent of the external environment and in Model 2, the shocks arrival rate, the shock magnitudes and the wear out rate are governed by a random environment which evolves as a Markov process. We obtain the systems availability for both models.

Optimal Pricing of a Heterogeneous Portfolio for a Given Risk Level

Consider a portfolio containing heterogeneous risks, where the policyholders’ premiums to the insurance company might not cover the claim payments. This risk has to be taken into consideration in the premium pricing. On the other hand, the premium that the insureds pay has to be fair. This fairness is measured by the distance between the risk and the premium paid. We apply a non-linear programming formulation to find the optimal premium for each class so that the risk is below a given level and the weighted distance between the risk and the premium is minimized. We consider also the dual problem: minimizing the risk level for a given weighted distance between risks and premium.

Ordering ruin probabilities for dependent claim streams

Abstract Claims arriving to an office can be of several types. Due to different types of catastrophes or shocks, different types of claims might arrive at the same time. This phenomena imposes dependence among claims arrival streams of different types. We show that when the dependence among arrival streams increases, the ruin probability also increases. We consider the compound Poisson model. We extend and generalize results obtained by Cossette and Marceau [Math. Econ. 26 (2000) 133].

Upper bounds on the expected time to ruin and on the expected recovery time

It is shown that the time to ruin and the recovery time in a risk process have the same distribution as the busy period in a certain queueing system. Similarly, the deficit at the time of ruin is distributed as the idle period in a single-server queueing system. These duality results are exploited to derive upper bounds for the expected time to ruin and the expected recovery time as defined by Egídio dos Reis (2000). When the claim size is generally distributed, Lordens inequality is applied to derive the bounds. When the claim-size distribution is of phase type, tighter upper bounds are derived.

Jointly optimal allocation of a repairman and optimal control of service rate for machine repairman problem

Abstract Consider a production system with m unreliable machines, which are maintained by a single repairman. The time until failure of machine i and its repair time are exponentially distributed random variables with rates λ i and μ , respectively. Machine i earns at rate r i while it is working. The service rate can be controlled, and a cost c ( μ ) is charged when the service rate is μ . We assume the following compatibility condition: λ i λ j implies that r i ⩾ r j . We consider both the optimal assignment of the repairman to the failed machines, and the optimal service rate. We demonstrate some monotone properties of the optimal policy.