We consider a system of parallel queues where tasks are assigned (dispatched) to one of the available servers upon arrival. The dispatching decision is based on the full state information, i.e., on the sizes of the new and existing jobs. We are interested in minimizing the so-called mean slowdown criterion corresponding to the mean of the sojourn time divided by the processing time. Assuming no new jobs arrive, the shortest-processing-time-product (SPTP) schedule is known to minimize the slowdown of the existing jobs. The main contribution of this paper is three-fold: 1) To show the optimality of SPTP with respect to slowdown in a single server queue under Poisson arrivals; 2) to derive the so-called size-aware value functions for M G 1-FIFO LIFO SPTP SPT SRPT with general holding costs of which the slowdown criterion is a special case; and 3) to utilize the value functions to derive efficient dispatching policies so as to minimize the mean slowdown in a heterogeneous server system. The derived policies offer a significantly better performance than e.g., the size-aware-task-assignment with equal load (SITA-E) and least-work-left (LWL) policies.
@cite_24: As jobs arrive they have to be routed to one of two similar exponential servers. It is shown that if the queue lengths at both servers are observed then the Optimal decision is to route jobs to the shorter queue, whereas if the queue lenths are not observed then it is best to alternate between queues, provided the initial distribution of the two queue sizes is the same. The optimality of these routing strategies is independent of the statistics of the job arrivals.
@cite_29: We consider a queuing system consisting of a finite number of identical exponential servers. Each server has its own queue, and upon arrival each customer must be assigned to some server's queue. Under the assumption that no jockeying between queues is permitted, it is shown that the intuitively satisfying rule of assigning each arrival to the shortest line maximizes, with respect to stochastic order, the discounted number of customers to complete their service in any time t. QUEUING; SHORTEST LINE; STOCHASTIC ORDER; MARKOV DECISION PROCESSES
@cite_25: Join the Shortest Queue (JSQ) is a popular routing policy for server farms. However, until now all analysis of JSQ has been limited to First-Come-First-Serve (FCFS) server farms, whereas it is known that web server farms are better modeled as Processor Sharing (PS) server farms. We provide the first approximate analysis of JSQ in the PS server farm model for general job-size distributions, obtaining the distribution of queue length at each queue. To do this, we approximate the queue length of each queue in the server farm by a one-dimensional Markov chain, in a novel fashion. We also discover some interesting insensitivity properties of PS server farms with JSQ routing, and discuss the near-optimality of JSQ.
In the context of dispatching problems, FIFO has been studied extensively in the literature since the early work by Winston @cite_29 , @cite_24 , and others. Often the number of tasks per server is assumed to be known, cf., e.g., (JSQ) dispatching policy @cite_29 . Even though FIFO queues have received the most of the attention, also other scheduling disciplines have been studied. For example, Gupta et. al consider JSQ with (PS) scheduling discipline in @cite_25 .