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# hw5 - CS/EE 147 Assigned HW 5 Queueing networks and PH...

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CS/EE 147 HW 5: Queueing networks and PH distributions Guru: Raga Assigned: 05/06/10 Due: 05/14/10, Raga’s mailbox, 1pm We encourage you to discuss these problems with others, but you need to write up the actual solutions alone. At the top of your homework sheet, list all the people with whom you discussed. Crediting help from other classmates will not take away any credit from you. Start early and come to office hours with your questions! Note: This homework is your chance to make up for lost points! All extra credit questions are fairly straightforward, and not much harder than the normal questions. 1 Warmup: Queueing networks [12 points] (a) A packet-switched Jackson network routes packets among two routers, according to the routing prob- abilities shown below (Figure 1). Notice that there are two points at which packets enter the network, and two points at which they can depart. Figure 1: A simple Jackson network. (i) What is the maximum allowable rate for r 1 that the network can tolerate? Call this r max 1 . (ii) Set r 1 = 0 . 9 r max 1 . What is the mean response time for a packet entering at the router 1 queue? (b) Give an example of a network which does not have a product-form limiting distribution. Solve your network for the limiting distribution and show that the limiting distribution is in fact not product-form, e.g., prove that the number of jobs at different servers are not independent. 2 More networks with product-form [24 points] (a) State-dependent service rates: Today, the world is shifting towards energy-efficient system design. For some time now, Adam has been actively involved in research related to ‘power-aware’ scheduling. One effective method for reducing energy consumption is dynamic speed scaling, which adapts the processing speed to the current queue state (queue length). Here, you will show that in Jackson networks where the service rates are state-dependent, the limiting distribution still has a product form. 1

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(i) To build intuition, let us start by analyzing a single-server setting (assume FCFS scheduling). Jobs arrive according to a Poisson process with rate λ . The service rate at the server depends on the number of jobs in the system – when there are n jobs in the system, the job in the server is served with rate µ ( n ) . Determine the limiting probability, π i , of there being i jobs in the system.
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