Multi-Class Queueing Networks with Infinite Virtual Queues
Standard MCQN
Stability and Performance of
Multi-Class Queueing Networks with
Infinite Virtual Queues
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1
2
5
4
3
YongJiang Guo1, Erjen Lefeber2, Yoni Nazarathy3,
Gideon Weiss4, Hanqin Zhang5
Workshop in honor of Frank Kelly,
Eurandom, Eindhoven, 28-29/4/2011
Qk (t) = Qk (0) + Ak (t) ! S k (Tk (t)) +
%
k '#$
" k ' k (S k ' (Tk ' (t)))
1
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
Multi-Class Queueing Networks with Infinite Virtual Queues
A static production planning problem
MCQN with IVQ
Harrison static planning problem
Two types of classes
Standard Queues
Infinite Virtual Queues
K0 = {1,2,3,5}
minu !
6
!
K ! = {4,6}
1
2
5
4
No exogenous arrivals
Nominal Input Rate
k ! K0 :
k ! K' :
Qk (t) = Qk (0) " S k (Tk (t)) +
Qk (t) = Qk (0) + ( k t
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
%
k '!$
R input output matrix
#" &
Ru = % (
$0'
Cu ) ! 1
u*0
3
!
= (I ! P') µ
"1 k !C(i)
C constituency matrix ci,k = #
else
$0
! < 1 sub-critical, ! > 1 unstable, ! = 1 critical
Static production planning problem
max! ,u
$
k"K #
%! (
Ru = ' *
&0)
# k ' k (S k ' (Tk ' (t))) & 0
Cu + 1
! ,u , 0
" S k (Tk (t))
2
wk! k
w are revenues, ! are optimal production rates
u are optimal service allocations
!i = (Cu)i =
!! i =
$
i"K #
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
K
" ci,k uk
k =1
ci,k
is node i workload
workload excluding IVQ
3
The question:
example: Push pull system – KSRS with IVQ
Static production planning problem
max! ,u
$
k"K #
wk! k
%! (
Ru = ' *
&0)
Cu + 1
Static production planning problem
!i = (Cu)i =
!! i =
$
i"K #
! ,u , 0
K
" ci,k uk
max! ,u w1!1 + w3! 3
is node i workload
k =1
# µ1
0
%
% " µ1 µ2
% 0
0
%
% 0
0
$
0 & # u1 & #! &
(% ( % 1(
0
0 ( %u2 ( % 0 (
=
µ3
0 (( %% u3 (( %! 3 (
% (
" µ3 µ4 (' %$u4 (' %$ 0 ('
# u1 & #1&
% ( % (
# 1 0 1 0 & %u2 ( %1(
%
(% ( ) % (
$0 1 0 1 ' % u3 ( 1
% (
%u ( $1'
$ 4'
ci,k
! = max !i
!! = max !! i
typically, ! = 1
MCQN:
! <1
! =1
!
Q1
µ1
µ2
0
µ4
µ3
!
Q2
! ,u * 0
network is stable under some policies, e.g. max pressure,
but, becomes congested as ! " 1
!3
µ3
µ4
MCQN w IVQ:
! = 1 but !! < 1
µ2
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
example: Push pull system – Stable policies
Case 1: µ2 > µ1, µ4 > µ3
Pull priority, one queue at a time is empty
!
)
(
)
µ µ µ " µ2
!3 = 3 4 1
µ1µ3 " µ2 µ4
!
w
Can it be stabilized ? will it remained uncongested ?
Answers very far away . . . , we discuss some examples . . .
(
µ µ µ " µ4
!1 = 1 2 3
µ1µ3 " µ2 µ4
network is rate stable under some policies, e.g. max pressure
µ1
!1
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
5
Main tools for stability results
Q1
µ1
µ2
µ4
µ3
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
Q2
Case 2: µ1 > µ2 , µ3 < µ4
Threshold policy, don’t let a queue become empty
This “fluid framework” was pioneered and exploited in the 90’s by
Dai, Meyn, Stolyar, Bramson, Williams, Chen . . .
In the memoryless case, probabilities decay geometrically
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
7
Stochastic system vs fluid model
Stability of queueing networks
k ! K0 :
Qk (t) = Qk (0) " S k (Tk (t)) +
k ! K' :
Qk (t) = Qk (0) + ( k t
Tk (0) = 0, Tk !,
Markov process
Fluid limits:
n
Q (t) =
Q
nr
%
k!C(i)
%
k '!$
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)
# k ' k (S k ' (Tk ' (t))) & 0
" S k (Tk (t))
Tk (t) " Tk (s) ) t " s
+ Policy implied equations
X (t) = (Q(t),T (t), more)
Q n (nt)
,
n
T n (t) =
(t, ! ) $ $$" Q(t),
r"#
Theorem (Dai 95, holds for MCQN with IVQ) with technical condition:
“compact sets of states are petite” fluid stability implies MCQN is stable.
Fluid weak stability implies MCQN is rate stable.
T n (nt)
,
n
T
nr
fluid model
Stable if there is t0 so that starting at |q(t)|=1, q(t)=0, t >t0
Weakly stable if starting at q(t)=0 it stays =0.
Theorem (Dai &Lin 2004, Tassiulas, Stolyar) Under maximum pressure,
if " ! 1 then fluid weakly stable, if " < 1 fluid is stable.
(t, ! ) $ $$" T (t)
r"#
Fluid model:
Q(t) = Q(0) ! ( I ! P ')[ µ ]T (t)
T (0) = 0, T! (t) " 0, CT (t) # 1 + Policy implied equations
Observation: Extending definition of maximum pressure to
MCQN with IVQ, theorem continues to hold.
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
Push-pull fluid stability
Re-entrant line
Case 1: µ2 > µ1, µ4 > µ3
Lyapunov function:
!
Q2
!1
µ2
µ1
f (t) = Q2 (t) + Q4 (t)
f! (t) < ! < 0,
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
!3
t>0
Case 2: µ1 > µ2 , µ3 < µ4
Piecewise linear Lyapunov function:
µ4
µ3
!
For "<1, random exogenous input, stable under: LBFS, FBFS, Max-Press
Assume "=1 but !! < 1 hence:
node 1 is bottleneck
1
2
3
!
Theorem: Under LBFS system is
stable.
Q4
Cannot use f (t) =
5
4
K
! Qk (t)
k =2
Q4 (t)
6
as Lyapunov function
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8
9
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
f (t) =
f! (t) < ! < 0,
t>0
Q2 (t)
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
11
Re-entrant line
Two Re-entrant lines
Further one can see:
• System is not stable under
Max pressure,
it is only rate stable (by simulation)
!
1
2
6
• Low priority to the IVQ and
Max pressure for all other buffers
is unstable.
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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.
!
3
1,3
4
1,2
G2-Pull
2,5
Policy: priority to G2 over G3, and G4 over G1
LBFS within each group.
7
2,4
G4-Pull
2,2
2,3
G3-Push
2,1
8
• Low priority to the IVQ and FBFS
is stable under some necessary and
sufficient conditions on the parameters
9
!
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
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
Two Re-entrant lines
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
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A ring system
Theorem: Fluid stable under this policy.
Modes: T1(t) time on line 1 only
T2(t) time on line 2 only
T1&2(t) sharaing both lines
Define: !1(t), !2 (t) average allocation
of machine 1,2 to group G1,G3
1,1
G1-Push
!
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
1,1
G1-Push
1,3
1,2
G2-Pull
using pull priority we have
2,5
2,4
G4-Pull
2,2
Output inequalities - lower bound:
2,3
G3-Push
2,1
D1,K (t) ! TL1(t) + "1(t)T1&2 (t)
!
1
D2,K (t) ! TL2 (t) + " 2 (t)T1&2 (t)
2
(
)
D1,K (t) + D2,K (t) ! TL1(t)TL2 (t) + ("1(t) + " 2 (t))T1&2 (t) (1 + # )
1
2
Input inequalities - upper bound:
more chalk,
Hence: system emty by t0 and stays empty.
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
15
A ring system M=3, pull priority
Stability of the stochastic systems – petiteness of compacts
Stability of the fluid implies stability of the stochastic system,
Graph of modes according to
non-empty queues
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
fluid trajectory cycles in iff $<0
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
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Fluid and Diffusion approximation
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
Fluid approximation
For simplicity, consider MCQN with deterministic routing
Fluid scaling:
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
Ti,k (n) (t) =
Ti,k (nt)
n
,
Qi,k (n) (t) =
Qi,k (nt)
n
,
Di,k (n) (t) =
Di,k (nt)
n
,
static production planning
Production on line i at rate !i
all steps are at rate !i
departure processes at rate !I
I
max!
Assume under some policy we have full utilization and stable queues.
" wi! i
i=1
I
"
We obtain fluid and diffusion approximation to the cucmulative allocated
processing times and the departure processes
(i ',k )#C(i)
mi ',k! i ' $ 1, i = 1,…, I
Queues are stable
! %0
Fluid approximation
Ti,k (t) = mi,k! i t, Qi,k (t) = 0, Di,K (t) = ! it
i
The actual system is approximated by this deterministic fluid
however stochastic deviation accumulate at a rate of n
we approximate the deviations by diffusion scaling
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
19
Diffusion approximation
Diffusion scaling:
Tˆi,k (n) (t) =
D̂i,k (n) (t) =
Diffusion approximation
Ti,k (nt) ! nTi,k (t)
n
,
Di,k (nt) ! nDi,k (t)
n
Qi,k (nt)
n!
=
,
r! n ! r !
n
(
,
)
Âi (t) =
MCQN with IVQ:
Ai (nt) ! nAi (t)
If ! = 1 but queues are stable, time allocation to IVQ absorbs the variability
n
Di,k (t) = Si,k (Ti,k (t)),
Standard MCQN:
if "<1 and queues are stable, Q̂i,k (n) (t) ! 0
and Âi(n) (t) ! Âi (t) ~ BM independent, Âi (t) = D̂i,1(t) = ! = D̂i,K
#
(i ',k )"C(i)
Ti ',k (t) = t
(n)
ˆ (n)
D̂ (n)
i,k (t) = Ŝ i,k (Ti,k (t)) + µi,k T i,k (t),
i
(n)
(n)
Q̂ (n)
i,k (t) = D̂ i,k !1(t) ! D̂ i,k (t),
Tˆ (n)(t) = !
$ Tˆ (n) (t)
the processing times have no effect on output - just copies input
i,1
If ! " 1 queues are unstable, and output depends on input and service
Âi(n) (t) ! Âi (t) ~ BM ,
Qi,k (t) = Di,k !1(t) ! Di,k (t),
Q̂i,k (n) (t) ! RBM ,
Q̂ (n)
i,k (t) ! 0,
D̂i,k ! combination (Iglehart Whitt, 71)
(i ',k )"C(i)
(i ',k )#(i,1)
i ',k
Tˆ (n)
i,1(t) ! BM ,
D̂ (n)
i,K (t) ! BM
i
we get exact expressions for the covariances - typically negative correlations
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
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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 !! < 1
• 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
Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
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Gideon Weiss, University of Haifa, FCFS Multi-Type, ©2010
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