We study a nonpreemptive scheduling on two parallel identical machines with a dedicated loading server and a dedicated unloading server. Each job has to be loaded by the loading server before being processed on one of the machines and unloaded immediately by the unloading server after its processing. The loading and unloading times are both equal to one unit of time. The goal is to minimize the makespan. Since the problem is NP-hard, we apply the classical list scheduling and largest processing time heuristics, and show that they have worst-case ratios, $8/5$ and $6/5$, respectively.

We study the problem of preemptive scheduling of jobs in a two-machine open shop. Jobs require processing on both machines, but the order does not matter. We define the D-LERPT (double longest expected remaining processing time) policy as the policy that first processes jobs that have not yet been processed by either machine (double jobs), in decreasing order of expected remaining processing times, and then processes jobs that require processing on only one machine in any order. We show that D-LERPT stochastically minimizes the makespan when preemption is not permitted and jobs (but not machines) are stochastically identical, and that D-LERPT minimizes the makespan in the increasing convex sense when preemption is permitted and the machines are stochastically identical and processing times are exponential or geometric with a job dependent rate.

We consider preemptive scheduling on parallel machines where the number of available machines may be an arbitrary, possibly random, function of time. Processing times of jobs are from a family of DLR (decreasing likelihood ratio) distributions, and jobs may arrive at random agreeable times. We give a constructive coupling proof to show that LEPT stochastically minimizes the makespan, and that it minimizes the expected cost when the cost function satisfies certain agreeability conditions.

A single machine is available to process a collection of stochastic jobs. Processing is preemptive and so (for example) the machine is allowed to switch away from a job before completion, should that prove advantageous. The jobs are deteriorating in the sense that their processing requirements grow (at job-specific rates) as they await processing. This phenomenon might be expected to enhance the status of non-preemptive policies. The primary objective of the paper is to find conditions which are sufficient to ensure the existence of a permutation policy to minimise the expected makespan. We also derive results for a weighted flowtime criterion. Applications of such models to the control of queues and to communication systems have been cited by other authors.

We consider scheduling problems with m machines in parallel and n jobs. The machines are subject to breakdown and repair. Jobs have exponentially distributed processing times and possibly random release dates. For cost functions that only depend on the set of uncompleted jobs at time t we provide necessary and sufficient conditions for the LEPT rule to minimize the expected cost at all t within the class of preemptive policies. This encompasses results that are known for makespan, and provides new results for the work remaining at time t. An application is that if the cµ rule has the same priority assignment as the LEPT rule then it minimizes the expected weighted number of jobs in the system for all t. Given appropriate conditions, we also show that the cµ rule minimizes the expected value of other objective functions, such as weighted sum of job completion times, weighted number of late jobs, or weighted sum of job tardinesses, when jobs have a common random due date.

We consider a system where jobs are processed by parallel machines. The processing times are exponentially distributed. An essential feature is that the assignment of the jobs to the machines is decided before the system starts to work. We consider both the flow time and the makespan. In the case of the flow time we allow both the machines and the jobs to be non-homogeneous. The optimization is by minimizing the flow time in the sense of stochastic order and the optimal assignment is obtained for this case. The case of the makespan is harder. We consider the expected makespan and as a partial solution we prove an optimality result for the case where there are two non-homogeneous machines and the jobs are homogeneous. It turns out that the optimal assignment can be expressed by using a quantile of a binomial distribution.

Consider n jobs and m machines. The m machines are identical and set up in parallel. All n jobs are available at t = 0 and each job has to be processed on one of the machines; any one can do. The processing time of job j is Xj, a random variable with distribution Fj. The sequence in which the jobs start with their processing is predetermined and preemptions are not allowed. We investigate the effect of the variability of the processing times on the expected makespan and the expected time to first idleness. Bounds are presented for these quantities in case the distributions of the processing times of the jobs are new better (worse) than used.

A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters.