Networks of queues
• Networks of queues
reversibility, output theorem, tandem networks, partial
balance, product-form distribution, blocking, insensitivity,
BCMP networks, mean-value analysis, Norton's theorem,
sojourn times
Richard J. Boucherie
Stochastic Operations Research
department of Applied Mathematics
University of Twente
1
Networks of Queues: lecture 5
Nelson, sec 10.3.5-10.6.6
• Last time on NoQ …
– Kelly-Whittle network
– Partial balance
– Quasi reversibility
• PASTA / MUSTA
• Norton’s theorem
2
Queue disciplines
• Operation of the queue j:
(i) Each job requires exponential(1) amount of service.
(ii) Total service effort supplied at rate φj(nj)
(iii) Proportion Υj(k,nj) of this effort directed to job in position
k, k=1,…, nj ; when this job leaves, his service is completed,
jobs in positions k+1,…, nj move to positions k,…, nj -1.
(iv) When a job arrives at queue j he moves into position k
with probability δj(k,nj + 1),
k=1,…, nj +1; jobs previously in positions k,…, nj move to
positions k+1,…, nj +1.
n

k 1
 j ( k , n)  1
n

k 1
j
(k , n)  1  j (n)  0 if n  0
• FCFS, LCFS, PS, infinite server queue
• BCMP network
4
Network of queues
•
•
•
•
•
•
•
Multiclass queueing network, type or class i=1,..,I
J queues
Customer type identifies route
Poisson arrival rate per type ν(i), i=1,…,I
Route r(i,1),r(i,2),…,r(i,S(i))
Fixed number of visits; cannot use Markov routing
1, 2, or 3 visits to queue: use 3 types
•
•
•
•
•
•
Type i at stage s in queue r(i,s)
tj(l): type of customer in position l in queue j
sj(l): stage of this customer along his route
cj(l)=(tj(l),sj(l)): class of this customer
cj=(cj(1),…, cj(nj)): state of queue j
C=(c1,…, cJ): Markov process representing states of the system
Product form: BCMP network
• Arrival rate class i to network ν(i), so also for each stage,
say
 j (i, s )   (i )
• Transition rates
 j (n j ) j (l , n j )
c pos l in q j is in last stage and leaves net


q(C , C ' )   j (n j ) j (l , n j ) k (m, nk  1) c pos l leaves q j routes to q k in pos m

 (i) k (m, nk  1)
c type i enters in q k in pos m, k  r(i,1)

• Equilibrium distribution queue j
n
j
 j (c j )  b j 
i 1
 j (t j (l ), s j (l ))
 j (l )
• Theorem: equilibrium distribution for open network
J
(closed):
 (C )    j (c j )
j 1
(and departure process from the network is Poisson)
Network of Quasi-reversible queues
• Rates


q k ( xk , xk ' )

q ( x, x ' )  
q j (x j , x j ')

q k ( xk , xk ' )
q j ( x j , x j ' )
 k (i, s  1)

xk ' S k (i,1, xk )
x j  S j (i, S (i ), x j ' )
xk ' S k (i, s  1, xk ) x j  S j (i, s, x j ' )
• Theorem : For an open network of QR queues
(i) the states of individual queues are independent at fixed time
 ( x)   1 ( x1 )... J ( xJ )
(ii) an arriving customer sees the equilibrium distribution
(ii’) the equibrium distribution for a queue is as it would be in
isolation with arrivals forming a Poisson process.
(iii) time-reversal: another open network of QR queues
(iv) system is QR, so departures form Poisson process
Network of Quasi-reversible queues
• Rates


q k ( xk , xk ' )

q ( x, x ' )  
q j (x j , x j ')

q k ( xk , xk ' )
q j ( x j , x j ' )
 k (i, s  1)

xk ' S k (i,1, xk )
x j  S j (i, S (i ), x j ' )
xk ' S k (i, s  1, xk ) x j  S j (i, s, x j ' )
• Theorem : For an open network of QR queues
(i) the states of individual queues are independent at fixed time
 ( x)   1 ( x1 )... J ( xJ )
(ii) an arriving customer sees the equilibrium distribution
(ii’) the equibrium distribution for a queue is as it would be in
isolation with arrivals forming a Poisson process.
(iii) time-reversal: another open network of QR queues
(iv) system is QR, so departures form Poisson process
Network of Quasi-reversible queues
• Rates


q k ( xk , xk ' )

q ( x, x ' )  
q j (x j , x j ')

q k ( xk , xk ' )
q j ( x j , x j ' )
 k (i, s  1)

xk ' S k (i,1, xk )
x j  S j (i, S (i ), x j ' )
xk ' S k (i, s  1, xk ) x j  S j (i, s, x j ' )
• Theorem : For an open network of QR queues
(i) the states of individual queues are independent at fixed time
 ( x)   1 ( x1 )... J ( xJ )
(ii) an arriving customer sees the equilibrium distribution
(ii’) the equibrium distribution for a queue is as it would be in
isolation with arrivals forming a Poisson process.
(iii) time-reversal: another open network of QR queues
(iv) system is QR, so departures form Poisson process
Network of Quasi-reversible queues
• Rates


q k ( xk , xk ' )

q ( x, x ' )  
q j (x j , x j ')

q k ( xk , xk ' )
q j ( x j , x j ' )
 k (i, s  1)

xk ' S k (i,1, xk )
x j  S j (i, S (i ), x j ' )
xk ' S k (i, s  1, xk ) x j  S j (i, s, x j ' )
• Theorem : For an open network of QR queues
(i) the states of individual queues are independent at fixed time
 ( x)   1 ( x1 )... J ( xJ )
(ii) an arriving customer sees the equilibrium distribution
(ii’) the equibrium distribution for a queue is as it would be in
isolation with arrivals forming a Poisson process.
(iii) time-reversal: another open network of QR queues
(iv) system is QR, so departures form Poisson process
Networks of Queues: lecture 6
Nelson, sec 10.3.5-10.6.6
• Last time on NoQ …
– Kelly-Whittle network
– Partial balance
– Quasi reversibility
• PASTA / MUSTA
• Norton’s theorem
10
11
PASTA: Poisson Arrivals See Time Averages
•
Pn ',n (t ) fraction of time system in state n
• probability outside observer sees n customers at time t
• P0 (t ) probability that arriving customer sees n customers at time t
n ', n
(just before arrival at time t there are n customers in the system)
• in general P (t )  P 0 (t )
n ', n
n ', n
PASTA: Poisson Arrivals See Time Averages 12
• Let C(t,t+h) event customer arrives in (t,t+h)
Pn0',n (t )  lim Pr{ X (t )  n | C (t , t  h), X (0)  n'}
h0
 lim
h0
Pr{C (t , t  h) | X (t )  n, X (0)  n'} Pr{ X (t )  n | X (0)  n'}


Pr{C (t , t  h) | X (t )  k , X (0)  n'} Pr{ X (t )  k | X (0)  n'}
k 0
 lim
h0
[q(n, n  1)h  o(h)]Pn ',n (t )


k 0
[q(k , k  1)h  o(h)]Pn ',k (t )

q(n, n  1) Pn ',n (t )


k 0
q(k , k  1) Pn ',k (t )
• For Poisson arrivals q(n,n+1)=λ so that Pn',n (t )  Pn0',n (t )
• Alternative explanation; PASTA holds in general!
PASTA: Poisson Arrivals See Time Averages 13
• Transient
P (t ) 
0
n ', n
q (n, n  1) Pn ',n (t )


k 0
• In equilibrium
P 
0
n

q (k , k  1) Pn ',k (t )
q (n, n  1) Pn

k 0
q (k , k  1) Pk
14
MUSTA: Moving Units See Time Averages
•
Palm probabilities:
Each type of transition nn’ for Markov chain associated with subset H of SxS \diag(SxS)
•
Example:transition in which customer queue i  queue j
H ij   (m  ei , m  e j ), m  ei , m  e j  S 
m
•
Transition in which customer leaves queue i
H iout   H ij
j
•
Transition in which customer enters queue j
H inj   H ij
i
15
MUSTA: Moving Units See Time Averages
•
NH process counting the H-transitions
•
Palm probability PH (C) of event C given that H occurs:
PH

(C) 

(n,n' )C
(n,n' ) H
•
 (n)q(n,n')
 (n)q(n,n')
, CH
Probability customer queue i  queue j sees state m
 (m  ei )q(m  ei ,m  e j )
Pij (m)  PH ((m  ei ,m  e j )) 
,
( n,n' ) H  (n)q(n,n')
ij
•
Probability customer arriving to queue j sees state m
P j (m)  PH in
j
ij
 (m  e )q(m  e ,m  e )

(U (m  e ,m  e )) 
,
 (n)q(n,n')
 
i
i
i
i
j
i
(n,n' ) H ij
i
j
Last time on NoQ :Kelly Whittle network
16
 (n  e j )
q (n, n  e j  ek ) 
 j p jk
 ( n)
 (n  e j )
q ( n, n  e j ) 
 j p j0
 ( n)
 ( n)
q (n, n  ek ) 
k
 ( n)
Theorem: The equilibrium distribution for the Kelly Whittle
network is
J
 (n)  B (n)  j
where
nj
j   j / j nS
j 1
 j   j    k pkj
k
and π satisfies partial balance
J
  (n)q(n, n  e
k 0
J
j
 ek )    (n  e j  ek )q(n  e j  ek , n)
k 0
17
MUSTA :Kelly Whittle network
J
 (n)  B (n)  j n  j   j /  j n  S
j
j 1
 (n  e j )
q (n, n  e j  ek ) 
 j p jk
 ( n)
 (n  e j )
q ( n, n  e j ) 
 j p j0
 ( n)
 ( n)
q (n, n  ek ) 
k
 ( n)
Theorem: The distribution seen by a customer
moving from queue i tot queue j is
J
Pij (m)  Bij (m) pij   k
mk
mS
mk
mSd
d
k 1
Entering queue j is
J
Pj (m)  B j (m)  k
k 1
where
S d  m : i, j : m  ei , m  e j  S 

,
 ( n ) q ( n, n ' )
 
 (m  e )q(m  e , m  e )


   (m  e )q(m  e , m  e )
 (m  e )q(m  e , m  e )


   (m  e )q(m  e , m  e )
Pj (m) 
i
 (m  ei )q (m  ei , m  e j )
( n , n ')H ij
i
i
i
i
j
k
k 1

 (m)  k m
k
k 1
J

 (m)  k m
k
k 1

J
m
 (m)  k m
k 1
i
j
J
m
j
j
m
 (m)  k m

i
j
J

j
i
m
i
i
i
k
i
 j p ji

i
 j p ji
i
MUSTA
18
Networks of Queues: lecture 5
Nelson, sec 10.3.5-10.6.6
• Last time on NoQ …
– Kelly-Whittle network
– Partial balance
– Quasi reversibility
• PASTA / MUSTA
• Norton’s theorem
19
Insensitivity and partial balance, Norton’s theorem
20
• Theorem: Insensitivity
Suppose subsets Ai of S are such that there is no single
transition of positive intensity in which more than one Ai
is vacated or more than one Ai entered. Then the
equilibrium distribution π(x) is insensitive to the nominal
sojourn times in the Ai if and only if the Markov process
shows partial balance in all the Ai.
• Norton’s theorem: state aggregation, flow equivalent
servers, Nelson, sec 10.6.14-15
• Consider network of subnetworks, each subnetwork
represented by auxiliary process. Then we may lump
subnet into single node if and only if partial balance over
the subnets