Queueing Theory IE 502: Probabilistic Models Jayendran - - PowerPoint PPT Presentation

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Sep 22, 2022 446 likes •515 views

Queueing Theory IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR Example: Which is better? Suppose at Bank X, the customers arrivals are a Poisson process with rate . All incoming customers join a single queue and are

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Queueing Theory

IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR

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IE502: Probabilistic Models IEOR @ IITBombay

Example: Which is better?

Suppose at Bank X, the customers arrivals are a Poisson

process with rate λ. All incoming customers join a single queue and are served in FIFO order. Now, the customers can be served by either of two clerks available. Service times of each clerk is exponentially distributed with rate μ

Suppose at Bank Y also, the customers arrivals are a

Poisson process with rate λ. All incoming customers join a single queue and are served in FIFO order. Now, the customers are served by a single clerk whose service times are exponentially distributed with rate 2μ.

In which of the above system will the expected time spent

in system be smaller?

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IE502: Probabilistic Models IEOR @ IITBombay

Example: Which is better? (in other words)

System X is a M/M/2 system with arrivals at rate λ

and service at each server at rate μ. Now, consider System Y which is a M/M/1 system having arrival rate λ and service rate 2μ.

Compare W, the expected time customer spends

in systems X and Y. Specifically,

– Is WX > WY? – Is WX < WY? – Is WX = WY?

Is the result intuitive? Does similar result hold for

waiting time in the queues?

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IE502: Probabilistic Models IEOR @ IITBombay

Storage capacity of system is K (one customer in service

and K − 1 customers in the waiting line) and the exceeding customers are refused.

State space representation of M/M/1/K queue
What is the limiting probabilities Pn for M/M/1/K?
Compute expected number in system, L.
Compute expected time a customer spends in system, W.

M/M/1/K

μ λ K

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IE502: Probabilistic Models IEOR @ IITBombay

Example

Customers arrive at a bar at the rate of λ per hour.

The bar provides service at the rate of μ. It costs the bar Rs.cμ per hour to offer service at that rate. Suppose that the bar gets a revenue of Rs.A for every customer served. If the bar has a capacity of K seats, what service rate μ maximizes the total profit?