WUSTL CSE 567M  Analysis of A Single Queue (23 pages)
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Analysis of A Single Queue
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 Washington University in St. Louis
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 Cse 567m  Computer Systems Analysis
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Analysis of A Single Queue Raj Jain Washington University in Saint Louis Saint Louis MO 63130 Jain cse wustl edu These slides are available on line at http www cse wustl edu jain cse567 06 Washington University in St Louis CSE567M 31 1 2006 Raj Jain Overview Birth Death Processes M M 1 Queue M M m Queue M M m B Queue with Finite Buffers Results for other Queueing systems Washington University in St Louis CSE567M 31 2 2006 Raj Jain Birth Death Processes Jobs arrive one at a time and not as a batch State Number of jobs n in the system Arrival of a new job changes the state to n 1 birth Departure of a job changes the system state to n 1 Death State transition diagram 0 0 1 1 1 2 2 2 Washington University in St Louis 1 J 1 3 J CSE567M 31 3 1 J 1 1 2 2006 Raj Jain Birth Death Processes Cont When the system is in state n it has n jobs in it The new arrivals take place at a rate n The service rate is n We assume that both the inter arrival times and service times are exponentially distributed Washington University in St Louis CSE567M 31 4 2006 Raj Jain Theorem State Probability The steady state probability pn of a birth death process being in state n is given by Here p0 is the probability of being in the zero state Washington University in St Louis CSE567M 31 5 2006 Raj Jain Proof Suppose the system is in state j at time t There are j jobs in the system In the next time interval of a very small duration t the system can move to state j 1 or j 1 with the following probabilities Washington University in St Louis CSE567M 31 6 2006 Raj Jain Proof Cont If there are no arrivals or departures the system will stay in state j and thus t small zero probability of two events two arrivals two departure or a arrival and a departure occurring during this interval pj t probability of being in state j at time t Washington University in St Louis CSE567M 31 7 2006 Raj Jain Proof Cont The jth equation above can be written as follows Under steady state pj t approaches a fixed value pj that is
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