Gideon Weiss

TIME-REVERSIBILITY OF LINEAR STOCHASTIC PROCESSES

Time-reversibility is defined for a process X(t) as the property that {X(t), - - -, X(t.)} and {X(- t), - -, X(- t.)} have the same joint probability distribution. It is shown that, for discrete mixed autoregressive moving-average processes, this is a unique property of Gaussian processes. TIME-REVERSIBILITY; SHOT NOISE; CHARACTERISATIONS OF THE NORMAL DISTRIBUTION; TIME SERIES; STOCHASTIC PROCESSES

Stability and instability of fluid models for reentrant lines

Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai Dai, J. G. 1995. On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid models. Ann. Appl. Probab.5 49--77. which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served FBFS and Last-Buffer-First-Served LBFS disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer reentrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such networks; however, all work-conserving policies are stable in a ring network.

Scheduling tasks with exponential service times on non-identical processors to minimize various cost functions

Abstract : We consider preemptive scheduling of N tasks on m processors; processors have different speeds, tasks require amounts of work which are exponentially distributed, with different parameters. The policies of assigning at every moment the task with shortest (longest) expected processing time among those not yet completed to the fastest processor available, 2nd shortest (longest) to the 2nd fastest etc., are examined, and shown to minimize expected values of various cost functions. As special cases we obtain policies which minimize expected flowtime, expected makespan and expected lifetime of a series system with m component locations and N spares. (Author)

Stochastic scheduling problems I — General strategies

The paper contains an introduction to recent developments in the theory of non-preemptive stochastic scheduling problems. The topics covered are: arbitrary joint distributions of activity durations, arbitrary regular measures of performance and arbitrary precedence and resource constraints. The possible instability of the problem is demonstrated and hints are given on stable classes of strategies available, including the combinatorial vs. analytical characterization of such classes. Given this background, the main emphasis of the paper is on the monotonicity behaviour of the model and on the existence of optimal strategies. Existing results are presented and generalized, in particular w.r.t. the cases of lower semicontinuous performance measures or joint duration distributions having a Lebesgue density.

Stochastic scheduling problems II-set strategies-

The paper introduces the finite class of set strategies for stochastic scheduling problems. It is shown that the knownstable classes of strategies such as ES and MES strategies are of this type, as arelist-scheduling strategies such as LEPT and SEPT and other, more complicatedpriority-type strategies. Roughly speaking, set strategies are characterized by the fact that the decision as to which jobs should be started at timet depends only on the knowledge of the two sets of jobs finished up to timet and being processed at timet. Contrary to list scheduling strategies, set strategies may involve deliberate idleness of machines, i.e. may not be greedy and can therefore not generally be induced by priority rules. It is demonstrated that set strategies have useful properties. They are e.g.λn-almost everywhere continuous and therefore show satisfactorystability behaviour w.r.t. weak convergence of the joint distribution of job durations. Furthermore, the optimum w.r.t.all strategies is already attained on this class if job durations are independent and exponentially distributed and the performance measure fulfills a certainshift condition. This shift property is a quite natural concept and generalizes aspects of the notion ofadditivity in semi-Markov decision theory and stochastic dynamic optimization. Its complete analytical characterization is a major object of this paper. Typical additive cost criteria such as makespan and flowtime are of course covered, which yields simultaneously a first step towards generalization of optimality of LEPT and SEPT rules, as known for special cases. In fact, in view of the obtained optimality result, the question of when deliberate idleness of machines can be avoided, gains considerable interest, as it characterizes stochastic environments in whichpriority strategies are optimal. This provides a major link with current research on the analysis of networks of queues in the context of computer systems.

A simplex based algorithm to solve separated continuous linear programs

We consider the separated continuous linear programming problem with linear data. We characterize the form of its optimal solution, and present an algorithm which solves it in a finite number of steps, using an analog of the simplex method, in the space of bounded measurable functions.

Multiserver Stochastic Scheduling

m Parallel machines are available for the processing of n jobs. The jobs require random amounts of processing. When processing times are exponentially distributed, SEPT (shortest expected processing time first) minimizes the flowtime, LEPT (longest expected processing time first) minimizes the makespan and maximizes the time to first machine idleness. For m = 2, various other problems can be optimized by different rules. Optimality of preemptive SEPT and LEPT also holds when processing times are drawn from a common MHR (monotone hazard rate) distribution.

Approximation results in parallel machines stochastic scheduling

We consider scheduling a batch of jobs with stochastic processing times on parallel machines. We derive various new formulae for the expected flowtime and weighted flowtime under general scheduling rules. Smiths Rule, which orders job starts by decreasing ratio of weight to expected processing time provides a natural heuristic for this problem. We obtain a bound on the worst case difference between the expected weighted flow time under Smiths Rule and under an optimal policy. For a wide class of processing time distributions, this bound is of oderO(1) and does not increase with the number of jobs.

Turnpike optimality of Smith's Rule in parallel machines stochastic scheduling

Consider scheduling a batch of jobs with stochastic processing times on parallel machines, with minimization of expected weighted flowtime as objective. Smiths Rule, which orders job starts by decreasing ratio of weight to expected processing time, provides a natural heuristic for this problem. In a previous paper we have found an upper bound for the difference between the expected objective under Smiths Rule and under the optimal strategy. In this paper we find an upper bound on the expected number of times that Smiths rule differs from the optimal decision. Under some mild and reasonable assumptions these bounds remain constant when the number of jobs increases. Hence, under these conditions Smiths Rule is asymptotically optimal, and it has a Turnpike Optimality property, that is, Smiths Rule is optimal most of the time.

Multiple feedback at a single-server station

A single server facility is equipped to perform a collection of operations. The service rendered to a customer is a branching process of operations. While the performance of an operation may not be interrupted before its completion, once completed, the required follow-up work may be delayed, at a cost per unit time of waiting that depends on the type and load of work being delayed. Under some probabilistic assumptions on the nature of the required service and on the stream of customers, the problem is to find service schedules that minimize expected costs. The authors generalize results of Bruno [2], Chazan, Konheim and B. Weiss [4], Harrison [8], Klimov [10], Konheim [11], and Meilijson and G. Weiss [13], using a dynamic programming approach.