Download Report

Introduction to Queuing Networks
MATH 35800/M5800
Problem Sheet 5
Autumn 2014
∗∗ Questions 2 and 5 on this sheet counts towards your final mark for Level M students ∗∗
1. Consider an open Jackson network consisting of three linked single server queues Q1 , Q2
and Q3 . The only non-zero system parameters of the network are: η1 = 4; r12 =
3/4, r13 = 1/4; r21 = 1/4, r23 = 1/4, r20 = 1/2; r31 = 1/2, r30 = 1/2. The service
distribution at each queue is Exponential, with parameters µ1 = 7, µ2 = 6, µ3 = 3.
(a) Sketch the network as a series of linked nodes and arcs, marking in the external arrival
rates and the routing probabilities at each node.
(b) Use the trafic equations to find the effective arrival rates λ1 , λ2 and λ3 , and determine
whether or not the network is stable.
(c) Denote the state of the network by n = (n1 , n2 , . . . , nJ ) where nj is the number of
customers in Qj , j = 1, . . . , J, and let X(t) denote the corresponding CTMC. Find
the invariant distribution for this process.
∗2. Consider an open Jackson network consisting of four linked single servers queues Q1 , . . . , Q4 .
Customers arrive to Q1 from outside at rate η1 = 1, and there are no other external arrivals.
Customers departing from Q1 have equal probability of moving to Q2 or Q3 . Customers
departing from Q2 have equal probability of moving to Q3 or Q4 . Customers departing
from Q3 move to Q4 . Customers departing from Q4 have equal probability of moving to
Q1 or departing the network. There are no other external departures. The service rates in
the four queues are µ1 = 3, µ2 = 2, µ3 = 2 and µ4 = 3.
(a) Sketch the network as a series of linked nodes and arcs, marking in the external arrival
rates and the routing probabilities at each node.
(b) For j = 1, . . . , 4 let λj denote the effective arrival rate to queue Qj in steady state.
Find λj , j = 1, . . . , 4, and say (with reasons) whether or not the network is stable.
(c) Find the overall expected number in the network in steady state and the expected time
a customer would take to pass through the network.
d) Find the probability that, in steady state, all four queues Q1 , . . . , Q4 contain the same
number of customers.
3. Consider an open network of J queues Q1 , . . . , QJ , where the service times for each customer at each Qj are independent Exponential(µj ) random variables. Assume the network
behaves like a Jackson network in terms of the external arrival processes and the routing
processes, but assume each queue Qj behaves like an ‘infinte server’ ·/M/∞ queue rather
than a ·/M/1 queue. Denote the state of the network by n = (n1 , n2 , . . . , nJ ) where nj
is the number of customers in Qj , j = 1, . . . , J, and let X(t) denote the corresponding
CTMC, where Xj (t) is the number of customers in Qj at time t. Set ρj = λj /µj .
c
University
of Bristol 2014-15. This material is copyright of the University unless explicitly stated otherwise. It is
provided exclusively for educational purposes and is to be downloaded or copied for your private study only.
(i) Write down the possible transitions and the corresponding transition rates for the X(t)
process (the forward process).
(ii) Write down your guess for the parameters and the transitions rates for the reversed
process in terms of the parameters for the forward process and the effective arrival
rates λ1 , . . . , λJ .
(iii) Use these values in Kelly’s lemma to show that the invariant distribution π(n) for X
is given by
ρn2
ρnJ
ρn1
π(n1 , n2 , . . . , nJ ) = e−ρ1 1 e−ρ2 2 . . . e−ρJ J .
n1 !
n2 !
nJ !
4. Consider a simple open Jackson network with J single server queues Q1 , . . . , QJ . Assume
each queue has the same service rate µ and the network satisfies the standard Markov
arrival and routing properties. For j = 1, . . . , J − 1, assume the arrival rates and routing
parameters are ηj = 1/2; rjj+1 = 3/4; rjj−1 = 1/4. For j = J, ηJ = (J + 3)/4, rJJ−1 =
1/4 and rJ0 = 3/4. All other routing parameters are zero.
(a) Write down the traffic equations for the effective arrival rates λj , j = 1, . . . , J. You
are given that λ1 = 1. Use this fact, and the traffic equation for Q1 , to determine λ2 ,
and then use the traffic equation for λ2 to determine λ3 .
(b) Guess the general form for λj and check that your guess satisfies both the general
equation for λj , j = 1, . . . , J − 1 and the equation for λJ . Hence find a lower bound
on the value of µ for the network as a whole to be stable.
∗5. Consider a simple open Jackson network consisting of J linked single server queues Q1 , . . . , QJ
each with the same service rate µ. Customers arrive to Q1 from outside as a Poisson process, rate η1 = 1, and there are no other external arrivals. For j = 2, . . . , J − 1, customers
departing Qj are routed to Qj+1 with probability rjj+1 = (J − j)/J and to Qj−1 with probability rjj−1 = j/J. Customers departing Q1 are routed to Q2 with probability (J − 1)/J
and leave the network with probability r10 = 1/J. Customers departing QJ are routed
back to QJ−1 with probability rJJ−1 = 1. All other routing probabilities are zero.
(a) Sketch the network as a series of linked nodes and arcs, marking in the external arrival
rates and the routing probabilities at each node.
(b) Considering each queue in turn, write down the traffic equations for the effective
arrival rates λ1 , . . . , λJ . Show that these equations are satisfied by taking
J
λj =
, j = 1, . . . , J.
j
(c) Show that, for J odd, the maximum value of λj occurs at j = (J − 1)/2, and hence
find a lower bound on µ for the network to be stable. Show that for J = 5 this means
the network is stable if and only if µ > 10.
c
University
of Bristol 2014-15. This material is copyright of the University unless explicitly stated otherwise. It is
provided exclusively for educational purposes and is to be downloaded or copied for your private study only.