Queueing Systems Part II - Queueing Systems Part II J. M....

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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 j1 j+1 j j-1 = j = j+1 = j = 4 M/M/1/K Queue It follows that Since then, if < , and . , , 2 , 1 , K j P P j j = = ( 29 1 1 1 1 +- =-- = = K K j j P , 1 = = K j j P ( 29 ( 29 ( 29 . , , 2 , 1 , , 1 1 1 K j P K j j =-- = + 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 ( 29 ( 29 [ ] ( 29 ( 29 ( 29 . 1 ) 1 ( 1 1 1 + + =-- +- + = = K K K K j j K K jP N 6 M/M/1/K Queue We can derive the expected delay using Littles Theorem if we interpret the arrival rate correctly: or ( 29 W P W N K a- = = 1 b = P K a = (1-P K ) . ) 1 ( K P N W- = 7 Multiserver Queues Now lets 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 j1 j+1 j j-1 = j = j+1 = ( j+1) j = j 10 The Erlang Loss System It follows that and , , , 2 , 1 , ! 1 s j P j P j j = = , ! 1 1- = = s j j j P ( 29 ( 29 . , , 2 , 1 , , ! / ! / s j k j P s k k j j = = = 11 The Erlang Loss System The probability that an arriving customer is lost is given by This is called the Erlang-B formula (named after the Danish mathematician A. K. Erlang). ( 29 ( 29 . ! / ! / = = s k k s s k s P 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 = the rate of calls between two exchanges 1/ = 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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Queueing Systems Part II - Queueing Systems Part II J. M....

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