Performance and Stability Analysis of the Task Assignment Based on Guessing Size Routing Policy

Abstract

In a system formed by parallel servers and one dispatcher, we study the Task Assignment based on Guessing Size (TAGS) policy, an open loop task assignment policy where jobs are non-preemptive, servers are First-Come-First-Served and the size of incoming jobs is not known. This policy works as follows: all the incoming jobs are routed to the first server and jobs that complete service before s 1 units of time leave the system, but jobs that do not complete service before s 1 are killed and they are routed to the second server, where the service starts from scratch. Likewise, jobs that are executed in server i, if they complete service before s i units of time, leave the system, whereas jobs that do not complete service before s i units of time are killed and routed to the next server. For an arbitrary job size distribution, we provide a necessary and sufficient condition for the stability of a system operating under the TAGS policy. We also analyze the performance of the optimal TAGS policy, i.e., when the cutoffs s 1, s 2, . . . are chosen to minimize the waiting time of jobs for an arbitrary job size distribution and we show that it is lower bounded by the performance of the TAGS policy where the maximum queue length is minimized divided by the number of servers minus one. For Bounded Pareto distributed job sizes, we consider the asymptotic regime where the largest job size tends to infinity and we show that, when the system load is less than one, the performance of the optimal TAGS policy is, at most, two times worst than the performance of the optimal SITA policy, which a routing policy where the size of jobs is known. This result shows that the penalty caused by not knowing the size of incoming jobs is upper bounded by a factor of 2. For a higher system load, we show that the order of magnitude of the performance of the optimal TAGS policy in the asymptotic regime depends on the number of spare servers, i.e., the difference between the number of servers in the system and the minimum number of servers to stabilize the system. According to our numerical experiments, when the largest job size is finite, the difference on the performance between the TAGS policy and the SITA policy can be extremely large when the system load is higher than one, whereas it is small when the system load is less than one.