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 10.6
• Last time on NoQ …
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–
–
–
–
•
•
•
•
•
•
•
Kelly-Whittle network
Partial balance
Time reversed process
Quasi reversibility
Queue disciplines, Symmetric queues, BCMP networks
Network of quasi reversible queues
Insensitivity: phase type distributions
Insensitivity: processor sharing queue
Insensitivity: general case – nominal life time
Insensitivity: general case – process description
Insensitivity and partial balance
Summary / Exercises
2
3
Routes, network description
•
•
•
•
•
•
•
Multiclass queueing network, type i=1,..,I
J queues
Customer type identifies route
Poisson arrival rate per type i=1,…,I
Route r(i,1),r(i,2),…,r(i,S(i))
Type i at stage s in queue r(i,s)
S(c,x) set of states in which queue contains one more class c
than in state x
• State X(t)=(x1(t),…,xJ(t))
• Fixed number of visits; cannot use Markov routing
• 1, 2, or 3 visits to queue: use 3 types
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 γj(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
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 γj(k,nj) of this effort directed to job in position k,
(iv) job arriving at queue j moves into position k with prob.
δj(k,nj + 1)
• Examples:
FCFS
LCFS
PS
infinite server queue
• BCMP network
Symmetric queues
• 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 γ j(k,nj) of this effort directed to job in position
k,
(iv) job arriving at queue j moves into position k with prob.
δj(k,nj + 1)
 j (k , n)   j (k , n)
•
•
•
•
Examples: IS, LCFS, PS
Symmetric queue QR (for general service requirement)
Instantaneous attention
Note: FCFS with identical service rate for all types is QR
Network of Quasi-reversible queues
• 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 4
Nelson, sec 10.6
• Last time on NoQ …
–
–
–
–
–
•
•
•
•
•
•
•
Kelly-Whittle network
Partial balance
Time reversed process
Quasi reversibility
Queue disciplines, Symmetric queues, BCMP networks
Network of quasi reversible queues
Insensitivity: phase type distributions
Insensitivity: processor sharing queue
Insensitivity: general case – nominal life time
Insensitivity: general case – process description
Insensitivity and partial balance
Summary / Exercises
8
9
General distribution
k 1
Erl (k , )( x)  1  
• Erlang(k, ν)
j 0
ν
• mean EL = k/ τ
• Hyperexponential
• mean
• CV > 1
ν
(x) j x
e
j!
ν
CV=1/ √k < 1
Hyp ( pi , i , i  1,.., n)  p1Exp( 1 )  ...  pn Exp( n )
pn
p1
EL   ... 
1
n
p1
p2
pn
1
2
n
10
General distribution: phase type distribution


• With probability pk Erlang(k, ν)
k 1
ν
p1
p2
pn
ν
ν
ν
ν
ν

• phase type distribution
• mean

EL  
k 1
k pk

FL ( x)   pk Erl (k , )( x)
k 1
pk  1
11
General distribution: phase type distribution
• With probability pk Erlang(k, ν)
phase 1
p1
pn
p2
ν
phase 2
phase 1
ν
ν
ν
ν
ν
• phase type distribution
• dense in class of distributions with non-negative support

FL ( x)   pk Erl (k , )( x)
k 1
12
General distribution: phase type distribution
• Markov chain that records the remaining number of phases and that
restarts in phase k wp pk each time phase 1 is completed
• state k records number of remaining phases of renewal process
• state space S={1,2,…}
• transition rates q(k,k-1) = ν
q(1,k) = ν pk
• Let H(k) denote equilibrium distribution, then H(k) satisfies global
balance:
H(k) ν = H(1) ν
pk+ H(k+1) ν,
k=1,2,…
• or discrete renewal equation (TK VII-6)
H(k) = H(1) pk + H(k+1),

H (k ) 
• solution



i k
pi
k=1,2,…
1  k pk

where  k 1 
13
General distribution: phase type distribution
•

H (k ) 



i k
pi
is distribution that satisfies
• discrete renewal equation
H(k) = H(1) pk + H(k+1),
• Proof
• insert H(k) into equation:


i k
k=1,2,…

pi  pk  pi 
i 1


i  k 1
pi
• show that H(k) is distribution:


k 1
 
H (k )  
 k 1


i k
 
pi  
 i 1
i

k 1
 
pi   ipi  1
 i 1
Networks of Queues: lecture 4
Nelson, sec 10.6
• Last time on NoQ …
–
–
–
–
–
•
•
•
•
•
•
•
Kelly-Whittle network
Partial balance
Time reversed process
Quasi reversibility
Queue disciplines, Symmetric queues, BCMP networks
Network of quasi reversible queues
Insensitivity: phase type distributions
Insensitivity: processor sharing queue
Insensitivity: general case – nominal life time
Insensitivity: general case – process description
Insensitivity and partial balance
Summary / Exercises
14
15
Processor sharing queue
• Poisson arrivals
• Service request L
rate
•
•
•
•
•
n = # calls in queue
S = {0,1,…}
X = {X(t), t≥0}
q(n,n+1)= λ
q(n,n-1)= μ
State
State space
Markov chain
birth rate
death rate
• Equilibrium distribution
λ
mean τ=1/μ
 n  (1   )( ) n n  0,1,2,...
16
Proof: (exponential case)
 n  (1   )( ) n
equilibrium distribution
solution global balance
 n [q(n, n  1)  q(n, n  1)]   n1q(n  1, n)   n1q(n  1, n) 0  n  C
rate out of state n = rate into state n
 n [   ]   n1 1(n  0)   n1
0n
detailed balance
 n    n1 0  n
17
Processor sharing queue: phase type call length
•
•
•
•
•
•
Poisson arrivals
rate λ

FL ( x)   pk Erl (k , )( x)
call length L
mean τ=1/μ
k 1
State
call i has ri remaining phases;
(r1 ,..., rn )
State space
{1,2,...}C
Markov chain
X = {X(t), t≥0}

Transition rates
q(( r1 ,..., rn ), (r1 ,.., ri  1,.., rn ))  1(ri  1)
• Equilibrium distribution
 (r1,...,rn )  G (  )
1
n

q(( r1 ,..., rn ), (r1 ,.., ri 1 , ri 1 , ,.., rn ))  1(ri  1)
n

q(( r1 ,..., rn ), (r1 ,.., ri , r , ri 1 , ,.., rn )) 
pr
n 1
n
n
 H(r )
i
i1

H (k ) 



i k
pi
• H(k) is distribution of the remaining number of phases = remaining call length
18
Erlang loss queue: phase type call length
•
Equilibrium distribution
 (r1 ,..., rn )  G ( )
1

H (k ) 

n
n

H (ri )
i 1
•
•
Proof
global balance
 n [   ]   n1 1(n  0)   n1
n

i 1
n
 (r1 ,..., rn )[  ]    (r1 ,..., ri 1 , ri 1 ,..., rn )

n


i k
pi
0n
pri 1(n  0)

n
   (r1 ,..., ri 1 , ri  1, ri 1 ,..., rn )    (r1 ,..., ri ,1, ri 1 ,..., rn )
n i 0
n 1
i 1
n
1 
[  ]   {H (ri )}
pr 1(n  0)
i 1
n
n
i

n

  {H (ri  1) / H (ri )}   {H (1)}
n i 0
n 1
i 1
H(1)=μ/ν
and use discrete renewal equation H (ri )  H (1) pk  H (ri  1),
ri  1,2,...
19
Processor sharing queue: phase type call length
• Theorem 1
Equilibrium distribution
 (r1 ,..., rn )  G ( )
1
•

H (k ) 

n
n

H (ri )
i 1
where


i k
pi
• moreover, equilibrium distribution of number of calls depends on
call length distribution only through its mean (insensitivity
property):
 n  (1   ) ( ) n
• Proof
n
nof phases
• sum distribution over1 all possible
configurations
n
1
n
n 

ri 1
i 1,..., n
 (r1 ,..., rn ) 

ri 1
i 1,..., n
G ( )

i 1
H (ri )  G ( )

i 1
ri 1
H (ri )  G 1 ( ) n
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) queue length distribution is insensitive to the distribution of
the holding time except for its mean.
Networks of Queues: lecture 4
Nelson, sec 10.6
• Last time on NoQ …
–
–
–
–
–
•
•
•
•
•
•
•
Kelly-Whittle network
Partial balance
Time reversed process
Quasi reversibility
Queue disciplines, Symmetric queues, BCMP networks
Network of quasi reversible queues
Insensitivity: phase type distributions
Insensitivity: processor sharing queue
Insensitivity: general case – nominal life time
Insensitivity: general case – process description
Insensitivity and partial balance
Summary / Exercises
21
22
Assumptions, Notation, Partial balance
• General Markov proces, states x, State space S,
transition rates q(x,x’)
• Assume that
sup
q( x, x' )  
x

x ' x
• this implies regularity
• Global balance
{ ( x' )q( x' , x)   ( x)q( x, x' )}  0
x'
• Partial balance over A
{ ( x' )q( x' , x)   ( x)q( x, x' )}  0
x 'A
23
Nominal life time
• At time t amount of work T has to be done, T random.
T worked off at rate ρ(u): work completed at time s such that
s

 (u)du  T
t
• Also holds if ρ(u) is itself a random variable
• T nominal lifetime
• Now suppose event completion at time u has intensity ρ(u)
• Equivalent to unit intensity at unit workrate, but workrate ρ(u)
• Otherwise expressed:
T is exponentially distributed with unit mean,
and workrate is ρ(u)
24
Nominal sojourn time
• A arbitrary subset of S of Markov process
• Intensity out of A when current state is x
q(x, A) 
 q(x, x')
x' A
• Completion of sojourn time in A, or
• Completion of nominal sojourn time T in A, where T
exponentially distributed with unit mean and worked off at rate

 (u )  q( x(u ), A)
• is random, since x(u), the state of the MC, is random
• abbreviate
q( x, A)
Networks of Queues: lecture 4
Nelson, sec 10.6
• Last time on NoQ …
–
–
–
–
–
•
•
•
•
•
•
•
Kelly-Whittle network
Partial balance
Time reversed process
Quasi reversibility
Queue disciplines, Symmetric queues, BCMP networks
Network of quasi reversible queues
Insensitivity: phase type distributions
Insensitivity: processor sharing queue
Insensitivity: general case – nominal life time
Insensitivity: general case – process description
Insensitivity and partial balance
Summary / Exercises
25
26
Semi Markov process type description
Modify dynamics of Markov process within A
• Nominal sojourn time within A arbitrary distribution of unit
mean worked off at rate q( x, A)
• Before completion of the sojourn time transitions in A have
intensities q(x,x’) (x,x’εA), transitions out of A are forbidden
• Upon completion of sojourn time transition out of A
immediate, with transition probability
q ( x, x ' )
p ( x, x ' ) 
q( x, A)
( x  A, x' A)
• If equilibrium distribution π(x) unaffected by modification,
whatever distribution of nominal sojourn time, we say that
π(x) is insensitive to nominal sojourn time in A
• Theorem: The equilibrium distribution is insensitive to
nominal sojourn time in A if and only if the Markov process
shows partial balance in A.
27
Phase type distribution
q (0, j )   j
• Subsidiary Markov process
• State space J, states j=1,2,…
q ( j , k )   jk
• Transition rates (0 is outside)
q ( j ,0 )   j
with Σνj =1
• T(j) sojourn time in j up to departure out of J
• Total sojourn time in J is T=UjT(j)
• Expected sojourn times αj=ET(j)  j (  j    jk )   j    k kj
k
• So that
1    j    j j
k
  j 1
• And ET=1 implies
• T is passage time, any distribution with non-negative
support can be appr. arbitrary closely by phase type distr.
28
Supplementary variables
• Modify original Markov process on S by supplementing state
x to (x,j) when x in A, with the following transition rules
• First entry in A, at x say, adopt state (x,j) w.p. νj
• Transition (x,j)(x,k) intensity
• Transition (x,j)(x’,j) intensity
• Transition (x,j)x’ insensity
q( x, A) jk
( x  A)
q ( x, x ' )
( x, x' A)
q ( x, x ' )  j
( x  A, x' A)
• Nominal sojourn time in A is passage time through J for
auxiliary process
29
Partial balance for supplemented process
• Equilibrium distribution modified process
 ( x)
( x  A)
 ( x, j ) ( x  A)
• Global balance
  ( x' )q( x' , x)     ( x' , j ) q( x' , x)   ( x)q( x, S )
x ' A
j
x ' A
j
  ( x' )q( x' , x)    ( x' , j )q( x' , x)    ( x, k )
j
x ' A
x ' A
kj
q ( x, A)
k


  ( x, j )q ( x, A)   jk   j    ( x, j )q ( x, A)
k

( x  A)
( x  A)
Networks of Queues: lecture 4
Nelson, sec 10.6
• Last time on NoQ …
–
–
–
–
–
•
•
•
•
•
•
•
Kelly-Whittle network
Partial balance
Time reversed process
Quasi reversibility
Queue disciplines, Symmetric queues, BCMP networks
Network of quasi reversible queues
Insensitivity: phase type distributions
Insensitivity: processor sharing queue
Insensitivity: general case – nominal life time
Insensitivity: general case – process description
Insensitivity and partial balance
Summary / Exercises
30
31
Semi Markov process type description
Modify dynamics of Markov process within A
• Nominal sojourn time within A arbitrary distribution of unit
mean worked off at rate q( x, A)
• Before completion of the sojourn time transitions in A have
intensities q(x,x’) (x,x’εA), transitions out of A are forbidden
• Upon completion of sojourn time transition out of A
immediate, with transition probability
q ( x, x ' )
p ( x, x ' ) 
q( x, A)
( x  A, x' A)
• If equilibrium distribution π(x) unaffected by modification,
whatever distribution of nominal sojourn time, we say that
π(x) is insensitive to nominal sojourn time in A
• Theorem: The equilibrium distribution is insensitive to
nominal sojourn time in A if and only if the Markov process
shows partial balance in A.
32
Proof of Theorem
• Insert distribution
 ( x)   ( x)
( x  A)
 ( x, j )   ( x) j
( x  A)
• Into global balance
Satisfied
  ( x' )q( x' , x)     ( x' , j ) q( x' , x)   ( x)q( x, S )
x ' A
j
x ' A
j


 j   j { ( x' )q( x' , x)   ( x)q( x, x' )}  0
x 'A

Reduces to
  ( x' )q( x' , x)    ( x' , j )q( x' , x)    ( x, k )
j
x ' A
x ' A
kj


  ( x, j )q ( x, A)   jk   j    ( x, j )q ( x, A)
k

( x  A, j  J )
q ( x, A)
k
( x  A)
( x  A)
33
Proof of Theorem
• Partial balance sufficient for insensitivity
• Necessity
• Insensitivity implies:

j
• Summing GB for xεA:

 ( x, j )   ( x)
 j  ( x, j )   ( x)
j
• For J containing two states, μ1 ≠μ2: two eqn two unknown
• We must have form
• And thus must have
• Done if
( x  A)
 ( x, j )   ( x) j
( x  A)





{

(
x
'
)
q
(
x
'
,
x
)


(
x
)
q
(
x
,
x
'
)}
0
j
j 
x 'A

possible,

 j j
 ( x)   ( x)
( x  A, j  J )
 j  1 / 2,  jk  0   j  1 / 2 j
Insensitivity and partial balance
• Lemma
Suppose distribution π(x) shows partial balance over each
of the subsets Ai (i=1,2,…,r) of S, and that there is no
single transition of positive probability in which more
than one Ai is vacated or more than one Ai entered. Then
π(x) shows partial balance over the intersection of any
selection of the Ai.
• 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.
34
Networks of Queues: lecture 4
Nelson, sec 10.6
• Last time on NoQ …
–
–
–
–
–
•
•
•
•
•
•
•
Kelly-Whittle network
Partial balance
Time reversed process
Quasi reversibility
Queue disciplines, Symmetric queues, BCMP networks
Network of quasi reversible queues
Insensitivity: phase type distributions
Insensitivity: processor sharing queue
Insensitivity: general case – nominal life time
Insensitivity: general case – process description
Insensitivity and partial balance
Summary / next / Exercises
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Insensitivity and partial balance, Norton’s theorem
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• 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
Examples
• Semi Markov process
• Network of symmetric queues
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Summary / next / exercises:
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Jackson network
Kelly Whittle network
Partial balance
Quasi reversibility
All customers identical
Quasi reversibility, customer types
BCMP networks
Insensitivity: general service times for symmetric queues
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Nelson, sec 10.6
Aggregation / decomposition
PASTA
MUSTA
• Exercises: provided next time for tutorial next week.
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