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Letr be the ratio of the arrival rate to the service rate in an M/M/1 queueing system, 0 0, the probability that there are k customers in the system, pk, is equal to pork. Prove that the expected number of customers in the system, N, is equal to, that is, N = 2 kpx=1²r крк k=0

Letr be the ratio of the arrival rate to the service rate in an M/M/1 queueing system, 0 0, the probability that there are k customers in the system, pk, is equal to pork. Prove that the expected number of customers in the system, N, is equal to, that is, N = 2 kpx=1²r крк k=0

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Subject: Operating Systems

 

Letr be the ratio of the arrival rate to the service rate in an M/M/1 queueing system, 0<r < 1. Let Po be the probability that the system has no customer. It is shown in class that po is equal to 1-r. Furthermore, for k > 0, the probability that there are k customers in the system, pk, is equal to pork. Prove that the expected number of customers in the system, N, is equal to, that is, - Σkipe - I = k=0 N=kpk T r
Transcribed Image Text:Letr be the ratio of the arrival rate to the service rate in an M/M/1 queueing system, 0<r < 1. Let Po be the probability that the system has no customer. It is shown in class that po is equal to 1-r. Furthermore, for k > 0, the probability that there are k customers in the system, pk, is equal to pork. Prove that the expected number of customers in the system, N, is equal to, that is, - Σkipe - I = k=0 N=kpk T r
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