The saturated Multiserver Job Queuing Model with two classes of jobs: Exact and approximate results

We consider a multiserver queue where jobs request for a varying number of servers for a random service time. The requested number of servers is assigned to each job following a First-In First-Out (FIFO) order. When the number of free servers is not sufficient to accommodate the next job in line, that job and any subsequent jobs in the queue are forced to wait. As a result, not all available servers are allocated to jobs if the next job requires more servers than are currently free. This queuing system is often called a Multiserver Job Queuing Model (MJQM). In this paper, we study the behavior of a MJQM under saturation, i.e., when the waiting line always contains jobs to be served. We categorize jobs into two classes: the first class consists of jobs that only require one server, while the second class includes jobs that require a larger number of servers. We obtain the system utilization and the throughput of the two job classes for the case in which the number of servers requested by jobs in the second class is equal to the number of available servers, using a simple approach that allows for a general distribution of the service time of jobs in the second class. Hence, we derive the stability condition of the non-saturated MJQM under these assumptions. Additionally, we develop an approximate analysis for the case in which the jobs of the second class require a fraction of the available servers. Based on analytical and numerical results, we highlight interesting system properties and insights.