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# qlec2 - Queueing Systems Lecture 2 Amedeo R Odoni Lecture...

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Queueing Systems: Lecture 2 Amedeo R. Odoni October 13, 2004 Lecture Outline Birth-and-death processes State transition diagrams Steady-state probabilities M/M/1 M/M/m M/M/ Reference: Chapter 4, pp. 194-206

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Birth-and-Death Queueing Systems 1. m parallel, identical servers. 2. Infinite queue capacity (for now). 3. Whenever n users are in system (in queue plus in service) arrivals are Poisson at rate of λ n per unit of time. 4. Whenever n users are in system, service completions are Poisson at rate of µ n per unit of time. 5. FCFS discipline (for now). The Fundamental Relationship 1-( n + n ) t n t n t n +1 users n users n -1 users n users P n ( t ) = Prob [ n users in system at time t ] ] ) ( 1 [ ) ( ) ( ) ( ) ( 1 1 1 1 t t P t t P t t P t t P n n n n n n n n + + + = + + +
The differential equations that determine the state probabilities ] ) ( 1 [ ) ( ) ( ) ( ) ( 1 1 1 1 t t P t t P t t P t t P n n n n n n n n λ µ + + + = + + + After a simple manipulation: ) ( ) ( ) ( ) ( ) ( 1 1 1 1 t P t P t P dt t dP n n n n n n n n + + + + + = (1) (1) applies when n = 1, 2, 3,….; when n = 0, we have: ) ( ) ( ) ( 1 1 0 0 0 t P t P dt t dP + = (2) The system of equations (1) and (2) is known as the Chapman-Kolmogorov equations for a birth-and-death system The “state balance” equations We now consider the situation in which the queueing system has reached “steady state”, i.e., t is large enough to have , independent of t, or Then, (1) and (2) provide the state balance equations: The state balance equations can also be written directly from the state transition diagram n n P t P = ) ( 0 ) ( = dt t dP n ) 4 ( ,.. 3 , 2 , 1 ) ( ) 3 ( 0 1 1 1 1 1 1 0 0 = + = + = = + + n P P P n P P n n n n n n n

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Birth-and-Death System: State Transition Diagram 01 2 mm + 1 λ 0 1 2 m-1 m m+1 µ 3 m m+1 m+2 …… ……
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qlec2 - Queueing Systems Lecture 2 Amedeo R Odoni Lecture...

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