Analysis of the Task Assignment based on Guessing Size policy

Abstract

We study the Task Assignment based on Guessing Size (TAGS) policy in a parallel and homogeneous server system. The policy parameters are the number of servers h and a set of cutoffs s1<s2,ldots,sh−1. In this policy, all the incoming jobs are routed to the first server and jobs are run up to s1 time units. If they complete they leave the system, but jobs that do not complete after s1 time units are killed and moved to the end of the queue of the second server, where service starts from scratch. Likewise, jobs that are executed in server i and complete service before si units of time, leave the system, whereas jobs that do not complete are killed and routed to the next server. We first study the stability of such system and provide a precise utilization threshold for the existence of stable parameters for a given job size distribution. We compute the threshold for several families of distributions and provide bounds for others. We show that TAGS is most stable for bounded Pareto distributions with parameter α=1. Besides, we provide tight bounds on the performance of the TAGS policy where the cutoffs are chosen to minimize average waiting time in the asymptotic regime where the largest job size tends to infinity for the Bounded Pareto distribution and the system load smaller than one. In this case, we show that the performance ratio between TAGS and a version called SITA which does require knowledge of job size is at most 2. We then consider more broadly the same asymptotic regime and consider a bound on the average waiting time for any distribution. This is compatible with having a conservative policy which will work well for any job size distribution. We show rather tight upper and lower bounds which again match those of SITA up to a factor of 2. These bounds are considerably lower than the corresponding bounds of competing policies such as Random or Least Work Remaining. We also show that for all these policies the bounded Pareto distribution with α=1 is close to being the worst possible among all distributions with the same job size range. We show using the stability results that in the same regime, if we increase the load the gap between SITA and TAGS grows dramatically and eventually, the TAGS system cannot be stabilized. The conclusion from all our analysis is that TAGS is best used as a conservative policy with minimal requirements on small systems with relatively low utilization and where the range of job sizes is large.