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Unformatted text preview: Queueing Systems Part II J. M. Akinpelu 2 M/M/1/K Queue M/M/1/K queueing system – exponential interarrival distribution – exponential service distribution – 1 server – finite queue (K spaces) – FCFS service discipline 3 M/M/1/K Queue 1 , , 2 , 1 , , 1 , , 2 , 1 , , 1 1 = = = = + + K j P P or K j P P j j j j μ λ μ λ j‒1 j+1 j λ j1 = λ λ j = λ μ j+1 = μ μ j = μ 4 M/M/1/K Queue It follows that Since then, if λ < μ , and . , , 2 , 1 , K j P P j j = = μ λ ( 29 1 1 1 1 + = = = ∑ K K j j P μ λ μ λ μ λ , 1 = ∑ = K j j P ( 29 ( 29 ( 29 . , , 2 , 1 , , 1 1 1 K j P K j j = = + μ λ μ λ μ λ 5 M/M/1/K Queue The utilization of the server is given by The expected number in the system is ( 29 ( 29 . 1 1 1 1 + + = = K K P μ λ μ λ μ λ ρ ( 29 ( 29 [ ] ( 29 ( 29 ( 29 . 1 ) 1 ( 1 1 1 + + = + + = = ∑ K K K K j j K K jP N μ λ λ μ μ λ μ λ λ 6 M/M/1/K Queue We can derive the expected delay using Little’s Theorem if we interpret the arrival rate correctly: or ( 29 W P W N K a = = 1 λ λ λ λ b = λ P K λ a = λ (1P K ) . ) 1 ( K P N W = λ 7 Multiserver Queues • Now let’s assume there are s > 1 servers • We will consider two systems: – The Erlang Loss System – The Erlang Delay System 8 The Erlang Loss System Blocked Calls Cleared: The Erlang Loss System – M/M/ s / s • exponential interarrival time • exponential service time • s servers • no queueing – Customers enter the system if at least one of the servers is free 9 The Erlang Loss System 1 , , 2 , 1 , , ) 1 ( 1 , , 2 , 1 , , ) 1 ( 1 1 = + = = + = + + s j P j P or s j P j P j j j j μ λ μ λ j‒1 j+1 j λ j1 = λ λ j = λ μ j+1 = ( j+1) μ μ j = j μ 10 The Erlang Loss System It follows that and , , , 2 , 1 , ! 1 s j P j P j j = = μ λ , ! 1 1 = = ∑ s j j j P μ λ ( 29 ( 29 . , , 2 , 1 , , ! / ! / s j k j P s k k j j = = ∑ = μ λ μ λ 11 The Erlang Loss System The probability that an arriving customer is lost is given by This is called the ErlangB formula (named after the Danish mathematician A. K. Erlang). ( 29 ( 29 . ! / ! / ∑ = = s k k s s k s P μ λ μ λ 12 The Erlang Loss System The Erlang loss system is used in engineering telephone networks to provide some grade of service. The telephone lines are the “servers”. Let λ = the rate of calls between two exchanges 1/ μ = the mean length of a telephone call p e = the engineered blocking probability s telephone lines 13 The Erlang Loss System We want to determine the number of telephone lines needed to meet the engineered blocking probability p e ....
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 Spring '11
 Subjolly
 Poisson Distribution, Probability theory, probability density function, Queueing theory

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