This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full Document
 Spring '11
 Subjolly
 Poisson Distribution, Probability theory, probability density function, Queueing theory

Click to edit the document details