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 packetswitched 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 productform limiting distribution. Solve your
network for the limiting distribution and show that the limiting distribution is in fact not productform,
e.g., prove that the number of jobs at different servers are not
independent.
2
More networks with productform [24 points]
(a)
Statedependent service rates:
Today, the world is shifting towards energyefficient system design.
For some time now, Adam has been actively involved in research related to ‘poweraware’ 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 statedependent, the limiting distribution still has a product form.
1
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(i) To build intuition, let us start by analyzing a singleserver 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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 Fall '09
 Probability theory, Trigraph, Jackson Networks, PH distributions

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