Stability and Performance ofMulti-Class Queueing Networks with
Infinite Virtual Queues
YongJiang Guo1, Erjen Lefeber2, Yoni Nazarathy3,
Gideon Weiss4, Hanqin Zhang5
Workshop in honor of Frank Kelly,
Eurandom, Eindhoven, 28-29/4/2011
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 1
Multi-Class Queueing Networks with Infinite Virtual Queues
Standard MCQN
1 2 3
45
6
Qk (t) = Qk (0) + Ak (t) ! Sk (Tk (t)) + "k 'k (Sk ' (Tk ' (t)))
k '#$
%
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 2
Multi-Class Queueing Networks with Infinite Virtual Queues
MCQN with IVQ
1 2 3
45
6
!
!
k !K0 : Qk (t) = Qk (0) " Sk (Tk (t)) + #k 'k (Sk ' (Tk ' (t)))
k '!$
% & 0
k !K'
: Qk (t) = Qk (0) + (k t " Sk (Tk (t))
K0 = {1,2,3,5}
K!= {4,6}
Two types of classes
Standard Queues
Infinite Virtual Queues
No exogenous arrivals
Nominal Input Rate
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 3
A static production planning problem
Harrison static planning problem
minu!
Ru ="
0
#
$%
&
'(
Cu ) !1
u * 0
R input output matrix = (I ! P ')µ
C constituency matrix
ci,k =
1 k !C(i)
0 else
"#$
! < 1sub-critical, ! > 1 unstable, ! = 1 critical
Static production planning problem
max! ,uw
k!
k
k"K#
$
Ru =!
0
%
&'
(
)*
Cu + 1
! ,u , 0
w are revenues, ! are optimal production rates
u are optimal service allocations
!!i= c
i,k
i"K#
$
is node i workload
!i= (Cu)
i= c
i,kk=1
K
" uk
workload excluding IVQ
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 4
The question:
Static production planning problem
max! ,uw
k!
k
k"K#
$
Ru =!
0
%
&'
(
)*
Cu + 1
! ,u , 0
!!i= c
i,k
i"K#
$
is node i workload
!i= (Cu)
i= c
i,kk=1
K
" uk
! = max !
i!! = max !!
i
typically, ! = 1
! = 1 network is rate stable under some policies, e.g. max pressure
! < 1 network is stable under some policies, e.g. max pressure,
but, becomes congested as !" 1
MCQN:
MCQN w IVQ:
! = 1 but
!! < 1 Can it be stabilized ? will it remained uncongested ?
Answers very far away . . . , we discuss some examples . . .
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 5
example: Push pull system – KSRS with IVQ
max! ,uw
1!
1+ w
3!
3
µ1
0 0 0
"µ1
µ2
0 0
0 0 µ3
0
0 0 "µ3
µ4
#
$
%%%%%
&
'
(((((
u1
u2
u3
u4
#
$
%%%%%
&
'
(((((
=
!1
0
!3
0
#
$
%%%%%
&
'
(((((
1 0 1 0
0 1 0 1
#
$%
&
'(
u1
u2
u3
u4
#
$
%%%%%
&
'
(((((
)
1
1
1
1
#
$
%%%%
&
'
((((
! ,u * 0
!1
!3
µ2 µ1
µ4
µ3
!w
Static production planning problem
!1=µ
1µ
2µ
3" µ
4( )µ
1µ
3" µ
2µ
4
!3=µ
3µ
4µ
1" µ
2( )µ
1µ
3" µ
2µ
4
µ2µ1
µ4 µ3 !
!
Q1
Q2
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 6
example: Push pull system – Stable policies
Case 1:
Pull priority, one queue at a time is empty µ
2> µ
1, µ
4> µ
3
Case 2:
Threshold policy, don’t let a queue become empty µ
1> µ
2, µ
3< µ
4
In the memoryless case, probabilities decay geometrically
µ2µ1
µ4 µ3 !
!
Q1
Q2
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 7
Main tools for stability results
Establish that an “associated” deterministic fluid system is “stable”
The “framework” then implies the stochastic system is “stable”
Nice, since stability of deterministic system is easier to establish
This “fluid framework” was pioneered and exploited in the 90’s by
Dai, Meyn, Stolyar, Bramson, Williams, Chen . . .
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 8
Stochastic system vs fluid model
k !K0 : Qk (t) = Qk (0) " Sk (Tk (t)) + #k 'k (Sk ' (Tk ' (t)))
k '!$
% & 0
k !K'
: Qk (t) = Qk (0) + (k t " Sk (Tk (t))
Tk (0) = 0, Tk !, Tk (t) " Tk (s) ) t " s
k!C(i)
% + Policy implied equations
Markov process X (t) = (Q(t),T (t),more)
Fluid limits:
Qn (t) =
Qn (nt)
n, T
n (t) =T
n (nt)
n,
Qn
r (t,! )r"#
$ "$$ Q(t), Tn
r (t,! )r"#
$ "$$ T (t)
Q(t) = Q(0) ! (I ! P ')[µ]T (t)
T (0) = 0, !T (t) " 0, CT (t) # 1 + Policy implied equations
Fluid model:
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 9
Stability of queueing networks
Stochastic system: (MCQN as well as MCQN with IVQ)
Rate stable if Q(t)/t --> 0 as t --> !
Stable if X(t) positive Harris recurrent (has stationary distribution)
fluid model
Stable if there is t0 so that starting at |q(t)|=1, q(t)=0, t >t0
Weakly stable if starting at q(t)=0 it stays =0.
Theorem (Dai 95, holds for MCQN with IVQ) with technical condition:
“compact sets of states are petite” fluid stability implies MCQN is stable.
Fluid weak stability implies MCQN is rate stable.
Theorem (Dai &Lin 2004, Tassiulas, Stolyar) Under maximum pressure,
if " ! 1 then fluid weakly stable, if " < 1 fluid is stable.
Observation: Extending definition of maximum pressure to
MCQN with IVQ, theorem continues to hold.
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 10
Push-pull fluid stability
Case 1:
Lyapunov function:
µ
2> µ
1, µ
4> µ
3
f (t) = Q2(t) + Q4(t)
!f (t) < ! < 0, t > 0
!
1
!
3
µ2µ1
µ4 µ3 !
!
Q2
Q4
Case 2:
Piecewise linear Lyapunov function: µ
1> µ
2, µ
3< µ
4
f (t) =
Q4(t)
Q2(t)
!f (t) < ! < 0, t > 0
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 11
Re-entrant line
1
6
8
2
5
3
4
7
9
!
For "<1, random exogenous input, stable under: LBFS, FBFS, Max-Press
Assume "=1 but hence:
node 1 is bottleneck !! < 1
Theorem: Under LBFS system is
stable.
f (t) = Qk (t)
k=2
K
!Cannot use
as Lyapunov function
Argument for proof:
assume total 1 unit initial fluid, processing time by node 1 is 1 per unit fluid
(1) while not empty, output at rate "1 + # >1 (by LBFS)
(2) by time 1 all original fluid cleared (service is head of the line)
(3) all output after time 1 requires 1 time unit per unit processing at node 1
(4) if not empty by T, output "(1 + # )T, input ! µ1-1 +T-1
(5) must be empty by some t0 and stay empty
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 12
Re-entrant line
Further one can see:
• System is not stable under
Max pressure,
it is only rate stable (by simulation)
• Low priority to the IVQ and
Max pressure for all other buffers
is unstable.
• Low priority to the IVQ and FBFS
is stable under some necessary and
sufficient conditions on the parameters
1
6
8
2
5
3
4
7
9
!
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 13
Two Re-entrant lines
2 Re-entrant lines, starting with IVQ at 2 machines,
Buffers grouped as G1,G2,G3,G4.
Assume total work at G1,G3 is 1 per unit fluid,
total work at G2,G4 is r2 , r4 <1 per unit fluid.
! 1,1
1,2
!
1,3
2,5
2,4
2,3
2,2
2,1
G1-Push
G2-Pull
G4-Pull
G3-Push
Policy: priority to G2 over G3, and G4 over G1
LBFS within each group.
Modes:
Both G2,G4 not empty, transient.
G4 empty, G2 not empty - line 2 frozen, work on line 1 only,
G2 empty, G3 not empty - line 1 frozen, work on line 2 only,
G2, G4 empty, G1+G3 not empty: work on both lines,
All empty: stays empty
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 14
Two Re-entrant lines
Theorem: Fluid stable under this policy.
!1(t),!2(t)
Modes: T1(t) time on line 1 only
T2(t) time on line 2 only
T1&2(t) sharaing both lines
Define: average allocation
of machine 1,2 to group G1,G3
! 1,1
1,2
!
1,3
2,5
2,4
2,3
2,2
2,1
G1-Push
G2-Pull
G4-Pull
G3-Push
Input inequalities - upper bound:
more chalk,
Hence: system emty by t0 and stays empty.
D1,K1(t) ! T
L1(t) +"1(t)T1&2(t)
D2,K2(t) ! T
L2(t) +"2(t)T1&2(t)
D1,K1(t) + D2,K2
(t) ! TL1(t)T
L2(t) + ("1(t) +"2(t))T1&2(t)( )(1+ # )
Output inequalities - lower bound:
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 15
A ring system
M machines, M product lines each
through two successive machines.
processing rate at the IVQ is 1 (w.l.g),
at the second queue it is "i
using pull priority we have
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 16
A ring system M=3, pull priority
Graph of modes according to
non-empty queues
fluid trajectory cycles in iff $<0
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 17
Stability of the stochastic systems – petiteness of compacts
Stability of the fluid implies stability of the stochastic system,
However: a technical condition is required:
In the Markov process X(t)=(Q(t),T(t),more)
Compact sets of states are petite.
This needs some sufficient conditions:
for standard MCQN: work conservation
+ input interarrivals have unbounded support and are spread out
For MCQN with IVQ:
weak pull priority - if not all empty, always work on some non-IVQ
+ input interarrivals have unbounded support and are spread out
For our threshold policy in the push-pull system we don’t know conditions
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 18
Fluid and Diffusion approximation
For simplicity, consider MCQN with deterministic routing
Assume i=1, . . ., I product lines, each starting with IVQ at node i,
with steps (i,1), . . . , (i,Ki),
processing times have mean mik and squared c.o.v dik2
Assume under some policy we have full utilization and stable queues.
We obtain fluid and diffusion approximation to the cucmulative allocated
processing times and the departure processes
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 19
Fluid approximation
Fluid scaling: Ti,k
(n)(t) =Ti,k (nt)
n, Qi,k
(n)(t) =Qi,k (nt)
n, Di,k
(n)(t) =Di,k (nt)
n,
Production on line i at rate !i
all steps are at rate !i
departure processes at rate !I
Queues are stable
max!
wi!
i
i=1
I
"
mi ',k! i '
(i ',k )#C(i)
I
" $ 1, i = 1,…, I
! % 0
static production planning
Fluid approximation Ti,k (t) = mi,k! i t, Qi,k (t) = 0, Di,Ki
(t) = ! it
The actual system is approximated by this deterministic fluid
however stochastic deviation accumulate at a rate of
we approximate the deviations by diffusion scaling n
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 20
Diffusion approximation
Diffusion scaling:
Ti,k(n)(t) =
Ti,k (nt) ! nTi,k (t)
n,
n!
r! n ! r( )!=
Qi,k (nt)
n,
Di,k(n)(t) =
Di,k (nt) ! nDi,k (t)
n, Ai(t) =
Ai(nt) ! nAi(t)
n
Standard MCQN:
if "<1 and queues are stable,
and
the processing times have no effect on output - just copies input
Qi,k
(n)(t) ! 0
Ai
(n)(t) ! Ai(t) ~ BM independent, Ai(t) = Di,1(t) =! = Di,Ki
If queues are unstable, and output depends on input and service
Ai(n)(t) ! Ai(t) ~ BM , Qi,k
(n)(t) ! RBM ,
Di,k ! combination (Iglehart Whitt, 71)
! " 1
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 21
Diffusion approximation
MCQN with IVQ:
If but queues are stable, time allocation to IVQ absorbs the variability
Di,k (t) = Si,k (Ti,k (t)), Qi,k (t) = Di,k!1(t) ! Di,k (t), Ti ',k (t) = t
(i ',k )"C(i)
#
! = 1
D(n)i,k (t) = S (n)
i,k (Ti,k (t)) + µi,kT (n)i,k (t),
Q(n)i,k (t) = D(n)
i,k!1(t) ! D(n)i,k (t),
T (n)i,1(t) = ! T (n)
i ',k (t)
(i ',k )"C(i)(i ',k )#(i,1)
$
Q(n)
i,k (t) ! 0, T (n)i,1(t) ! BM , D(n)
i,Ki(t) ! BM
we get exact expressions for the covariances - typically negative correlations
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010 22
In summary:
• We formulated a new Static Production Planning problem
• Solution will typically imply "=1, hence MCQN congested
• For MCQN with IVQ, we may have "=1 but
• Question: Can this be stable, and remain uncongested
• Old example - KSRS network with IVQ
• Extensions: we show stability under pull priority for:
- re-entrant line, 2 re-entrant lines, ring network
• We derive diffusion approximation,
• New difficulties in ensuring that compact sets are petite
!! < 1