Performance Evaluation of Prioritized Limited Processor-Sharing System

We propose a novel prioritized limited processor-sharing (PS) rule and a simulation algorithm for the performance evaluation of this rule. The performance measures of practical interest are evaluated using this algorithm. Suppose that there are two classes and that an arriving (class-1 or class-2) request encounters n1 class-1 and n2 class-2 requests (including the arriving one) in a single-server system. According to the proposed rule, class-1 requests individually and simultaneously receive m / (m * n1+ n2) of the service-facility capacity, whereas class-2 requests receive 1 / (m * n1 + n2) of it, if m * n1 + n2 ≤ C. Otherwise (m * n1 + n2 > C), the arriving request will be queued in the corresponding class waiting room or rejected. Here, m ( 1) denotes the priority ratio, and C ( ∞), the service-facility capacity. In this rule, when a request arrives at [or departs from] the system, the extension [shortening] of the remaining sojourn time of each request receiving service can be calculated using the number of requests of each class and the priority ratio. Employing a simulation program to execute these events and calculations enables us to analyze the performance of the proposed prioritized limited PS rule, which is realistic in a time-sharing system (TSS) with a sufficiently small time slot. Moreover, this simulation algorithm is expanded for the evaluation of the prioritized limited PS system with N 3 priority classes. Keywords—PS rule, priority ratio, service-facility capacity, simulation algorithm, sojourn time, performance measures HE Processor-Sharing (PS) discipline has gained an important role in evaluating the performance of a variety of a resource allocation mechanism. Under processor-sharing (PS) discipline, if there are n (> 0) requests in a single server system, then each request receives 1 / n of the service-facility capacity (called the service ratio for individual request). No arriving request has to wait for service because it will be served promptly, even if the service rate becomes slow [1][4]. The PS paradigm emerged as an idealization of Round-Robin scheduling algorithms in time-shared computer system. In such a PS paradigm, with an increase in the number of arriving requests, the service ratio for individual request decreases. Therefore, in theory, the sojourn time of each request increases to infinity with an increase in the number of arriving requests. In order to prevent an increase in the sojourn time of each request in such a PS paradigm and to realize a realistic model of sharing, a method for limiting the number of requests receiving service has been proposed [5]. In addition, a PS discipline with a priority structure has been proposed, wherein a larger service ratio is allocated to the high-priority request than that for a low-priority request [1]. Prof. Dr. Y. Shikata is with the Faculty of Informatics for Arts, Shobi University, Kawagoe, Saitama 350-1153, Japan (Corresponding author:phone:81-49-246-5251;e-mail:[email protected]). Mr. W. Katagiri is with the Faculty of Informatics for Arts, Shobi University, Kawagoe, Saitama 350-1153, Japan Prof. Dr. Y. Takahashi is with the Faculty of Commerce, Waseda University, Shinjuku, Tokyo 169-8050, Japan. In order to prevent excessive increase in the sojourn time of each request in such a PS discipline with a priority structure, we propose a prioritized limited PS system. In the proposed system, a high-priority request is allocated the service ratio that is m ( 1, called the priority ratio) times that for a low-priority request. Moreover, the sum of the number of the requests receiving service is kept below a fixed value. The arriving request which cannot receive service will be queued or rejected. Performance measures of practical interest, e.g., the loss probability, waiting time in queue, and mean sojourn time in the server are evaluated using simulation programs. Under the PS rule, when a request either arrives at or departs from the system, the remaining sojourn time of the other requests will be extended or reduced, respectively. In our priority system, this extension or reduction of the sojourn time is calculated using the number of requests of each class and the priority ratio. Employing a simulation program to execute these events and calculations enables us to analyze the performance of the proposed prioritized limited PS rule, which is realistic in a time-sharing system (TSS) with a sufficiently small time slot. Moreover, this simulation algorithm is expanded for the analysis of the prioritized limited PS model with N 3 priority classes. II. PROCESSOR SHARING A. An approximate formula for the mean number of requests An approximate formula for the mean number of requests in the GI/G/1 (PS) system has been obtained as shown below it [6]-[8]. The request-arrival process forms a renewal process with independent and identically distributed (iid) inter-arrival time, A, with arrival rate λ , and squared coefficient of variation as E A 1/λ , C V A /E A (1) The requested service time of an arriving request is iid with mean and squared coefficient of variation as E B 1/μ , C V B /E B (2) The traffic intensity, ρ is then given by ρ γ μ ⁄ (3) which is assumed to be less than unity (ρ < 1) for the system stability. An approximate formula for the mean number of requests, E(L), in the GI/G/1 (PS) system is obtained by E L ρ C C g ρ, C , C ρ 1 ρ 1 C /{(1-ρ 1 C } where g ρ, x, y exp 2 1 ρ 1 x /3ρ x y if x 1 exp 1 ρ x 1 / x 4y if x'1 (4) Performance Evaluation of Prioritized Limited Processor-Sharing System Yoshiaki Shikata, Wataru Katagiri, Yoshitaka Takahashi T