Performance evaluation of nonpreemptive response - ratio

The schedulers of nonpreemptive batch processing systems are often based on the first-come first-serve (FCFS) or shortest-job-first (SJF) algorithms. The former is least discriminating in the sense that it does not base its scheduling decisions on the service requirements of jobs. In the SJF scheme, the service requirement of a job (or at least an estimate for it) must be known at the time of arrival. It is used to assure good response for short jobs by discriminating against long ones. Overall, the system response for SJF is better than that for FCFS, but the penalty imposed on long jobs is excessive. Brinch Hansen l suggested an alternate scheme which improves the overall response characteristics of FCFS without excessively delaying long jobs. In this scheme, the highest response-ratio next (HRN) scheme, the priority of an arriving job is set to zero. It then increases linearly in time at a rate inversely proportional to the service requirement of the particular job. When a job departs, the priorities of all jobs in the queue are evaluated and the one with the highest priority starts to execute. The expected waiting time (the time between arrival and start of execution, also called the delay) of ajob conditioned on its service requirement is a common measure for the performance of a scheduling algorithm. We call this conditional expectation the delay function W(x), where the parameter x denotes the service requirement. The response time of a job is the sum of its delay in the queue and its service requirement, and the response-ratio is defined as the response time divided by the service requirement. Note that the response-ratio is equal to one plus the delay divided by the service requirement. Overall, the rate at which a job attains service is equal to its service requirement divided by the response time. Thus, its processing rate is equal to the inverse of the response-ratio. By selecting the job with the highest response-ratio for service, the HRN discipline attempts to service competing jobs at equal rates. To quantify the behavior of the HRN scheme, Brinch Hansen derived an approximate expression for its delay function which is