10.1 Queueing Networks - Notes

# 10.1 Queueing Networks - Notes - Queueing Networks Systems...

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Queueing Networks Systems modeled by queueing networks can roughly be grouped into four categories 1. Open networks - Customers arrive from outside the system are served and then depart. Example: Packet switched data network. 1 g 1 m 2 m 3 m 2 g 3 g 2. Networks with population constraints - Customers arrive from outside the system if there is room in the queues. They enter, served and then depart. Example: queues sharing a common buffer pool. g 1 m 2 m 3 m 3. Closed networks - Fixed number of customers ( K ) are trapped in the system and circulate among the queues. Example: CPU job scheduling problem 1 m 2 m k m 4. Mixed network - Any combination of the types above. Example: simple model of virtual circuit that is window flow controlled.

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1 m 2 m k m Several features can occur in queueing networks that do not occur in single queue. 1. Jocking - Customers moving among parallel queues. 2. Blocking - Customer waiting depart a server and join next queue is unable to due to limited waiting space, and therefore stays in server (blocking it.) 3. Forking - Customer leaving a queue clones into multiple customers going along different routes. 4. Joining - Multiple streams of customers are combined into a single stream. Forking and joining are used in models of parallel processing systems. i. Open networks The simplest type of network is an open network with the following assumptions (called Jackson Network) Assume arbitrary network of M queues Service time of queue i is exponentially distributed with rate i m . Arrivals from outside the network to queue i are a Poisson process with mean rate i g . Let j i r , - routing probability that a customer completing service at queue i goes to queue j . 1 , + m i r - routing probability that a customer completing service at queue i leaves the network. Note 1 1 1 , = + = m j j i r Let i l be the total mean customer arrival rate to queue i .
i m i r 1 1 l i r 2 2 l mi m r l i g i l i l 1 i r 2 i r im r ) 1 ( + m i g ii r and i i i m l r = We can see that at each queue i + = + = 1 1 m j j i j i i r l g l Note that the flow conservation equation holds in any open network regardless of arrival and service distributions. This equation can easily be solved for i l . Let [ ] m i i l l l l , , , K = [ ] m i i g g g g , , , K = [ ] m j m i r R j i £ £ £ £ = 1 1 R does not include 1 + m i r The flow conservation equation can be written in matrix vector form as R l g l + = sloving g l = - ) ( R I 1 ) ( - - = R I g l Now consider queue i in the Jackson network, from the analysis of the single M/M/1 queue we know 1. Merging of independent Poisson processes is Poisson with rate equal to the sum of the individual rates. That is Poisson process.

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1 l 2 l n l n l l l l + + + = K 2 1 2. The departure process of an M/M/1 queue is Poisson with rate equal to input rate of queue. 3.
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10.1 Queueing Networks - Notes - Queueing Networks Systems...

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