RESTART Simulation of Non-Markovian Queueing
Networks
Manuel Villén-Altamirano, José Villén-Altamirano, Enrique
Vázquez-Gallo
Universidad Politécnica de Madrid
1
CONTENTS

Description of RESTART and previous results

Effective Importance Functions

Simulation results

Conclusions
2
Description of RESTART (I)
(t)
L
B3
T3
T2
T1 B1
A
P  Pr  A = Pr    L
B3
D 3 D 3 D3
B2
D2
Pr Ci  = Pr    Ti 
Ri :Number of trials at Bi
D 2 D2
i
D1 D 1
D 1 B1
D1
ri   R j
j 1
t (time)
C1  C2  C3  … CM  A
P A = P C1

P C2 / C1
Pˆ 
NA
rM  N
 …
P A/CM
N : No. of simulated events ( retrials not included )
N : No. of events A ( retrials included )
A
3
Description of RESTART (II)
Q2
P  Pr  A =Pr Q 2  L
T3
Pr Ci  =Pr   Ti 
L
T2
ln  2

Q1  Q2
ln 1
T1
Q1
M
A  Ci i  1,..., M then: P  A   P Ci P  A / Ci 
i 1
M
N
Pˆ   Ai
i 1 ri  N
N : No. of simulated events ( retrials not included )
N : No. of events A in retrials from sets Ci
Ai
4
Gain Obtained with RESTART
Gain 
1
1
fV f 0 f R f T
P   ln P  1
2
Factors f  1 reflect inefficiency due to:
f T - not optimal thresholds f 0 - algorithm overhead
f R - not optimal Ri
f V - variance at Bi
5
Factor fR

Optimal values of ri
r 
i

1
P
i 1 1 i 0
P
f
R
1
Rounding Algorithm
R1 = r1 rounded to an integer number, R2 = r2/ R1 rounded
to an integer number, . . . , Ri = ri / R1 · . . . ·Ri-1 rounded
to an integer number.
6
Factor fT

The thresholds must be set as close as possible
P
min

Min Pi i 1 
1  P 

P ln P 
2
fT
min
2
min
min
Pmin
fT
1
1
0.5
1.04
0.1
1.5
0.01
4.6
0.001
20.9
7
Factor fO

f 0  Max  yi 

Affects to computational time, not to number of events

ye = overhead per event: evaluate  , compare with Ti , …



yri = overhead per retrial: restore state at Bi , re-schedule, ...
y0 = y e
yi = ye yri
This factor usually takes low values with exponential times.
However the rescheduling of Hyperexponential or Erlang times
is more time consuming.
8
Factor fV (I): Rescheduling



It is convenient to reschedule at Bi, for each retrial, the scheduled arrival times and the
scheduled end of service times. Otherwise, there would be high correlation between retrials.
If these times are exponentially distributed, the rescheduling is straightforward, due to the
memory-less property of this distribution.
For other distributions we use the following procedures: we obtain a random value of the
whole e.g., service time of a customer. If the end of service time is greater than the value of
the clock at the current time (Bi), the residual lifetime is obtained as the difference between
the two amounts. Otherwise a new random value is obtained and so on.
If after 50 attempts the new end of service time is lower than the value of the clock at the
current time (Bi), it is not rescheduled.
Start of
service
Bi
Scheduled
end of service
9
CONTENTS

Description of RESTART and previous results

Effective Importance Functions

Simulation results

Conclusions
10
Factor fV (II)
f v  Max si 
V ( PA*/ X i ) 
V ( PA*/ X i )
ai 
si 
i  1
 K 'i 
K A 
( PA*/ i ) 2 
( PA*/ i ) 2
X i1
Ci
PA X
Ci
i1
A
X i2
PA X
i2
PA X
Xi


i3
3
F  Ti
Xi : system state at Bi
*
P A X i: importance of state Xi
*
Ai

P

i
: expected value of
*
A Xi
P
: factor reflecting the autocovariance of
*
A Xi
P
11
Importance Function for Jackson Networks (I)
Importance function for three-queue Jackson tandem network:
 if 1 > 2 > 3
ln 1
ln 2
  Q1
 Q2
 Q3
ln 3
ln 3
If 1  2  3 ,or if 2  1  3 ,   Q1  Q2  Q3
If 2  3  1 ,   Q1
ln 1
 Q2  Q3
ln 3
If 1  3  2 ,or if 3  1  2 ,  

ln 2
 Q1  Q2   Q3
ln 3
Villén-Altamirano, J. 2010. Importance function for RESTART simulation of
general Jackson networks. European Journal of Operation Research 203
(1), 156 – 165.
12
Importance Function for
General Jackson Networks (II)
P  Pr Qtg  L
H
*
ln  tg tgi

i 1
ln tg
  1i
tg 
j 1
tg
tgj 
K
1i  1 
  p
l i
1l
j 1
lj

ln  tg tgj

j 1
ln tg
Q1i   2 j
Q2 j  Qtg
K
K
 tg    j p jtg
K
tg

tg
*
tgi

 tg   Min 2 j   1i  1i  pij , 2 j  p jtg  tg ptgtg
j 1
tg
 tg  2 j p jtg   2l pltg  tg ptgtg
l j
tg
H
K
p jtg    2 j p jtg   tg
j 1
K
1i  pij p jtg
j 1
;
2 j  1
  p
i 1
1i
l j
il
pltg    2l pltg   tg
l j
 j p jtg
13
Importance Functions for Non-Jackson Networks (I)

Would be fit for other networks the importance function derived
for Jackson networks? Or at least, would be easy to modify it?

The importance function is a linear combination of the queue
length of the nodes. The coefficients are function of the load of
the nodes. In general: the lower the load of a node, the higher
the value of the coefficient.
14
Importance Functions for Non-Jackson Networks (II)

For Jackson networks the value of the load (), can be
calculated from the formulas:
P(X>=n) = P(X>=2n / X>=n) = ^n

(1)
For non-Jackson networks the value of  that will be used in
previous formulas of the importance function (derived for
Jackson networks) is calculated with Equation (1).
We will call it “effective load” and it does not match the
actual load. The probability P(X>=n) is evaluated by crude
simulation for a low value of n.
CONTENTS

Description and previous results

Effective Importance Function

Simulation results

Conclusions
16
Models Simulated (I)

Example 1: 2-node network with strong feedback
0,2
0,2
 =2
1
2
0,8
 =2
0,8

1  2 / 3; 2  1/ 3
Example 2: Three-queue tandem network .
1
Three sets of loads:
2
3
1  2/ 3; 2  1/ 2; 3  1/ 3
1  1/ 3; 2  1/ 2; 3  1/ 3
1  1/ 5; 2  1/ 4; 3  1/ 3
Models Simulated (II)
Example 3: Network with 7 nodes . Arrival rate γi = 1; i = 1, …,7

Transition Probability Matrix
1
2
3
4
5
6
t
Ext.
1
0.1
0.1
0.1
0.1
0.2
0.2
0
0.2
2
0.1
0.1
0.1
0.1
0.2
0.2
0
0.2
3
0.1
0.1
0.1
0.1
0.2
0.2
0
0.2
4
0.1
0.1
0.1
0.1
0.2
0.2
0
0.2
5
0.1
0.1
0.1
0.1
0
0.1
0.3
0.2
6
0.1
0.1
0.1
0.1
0.1
0
0.3
0.2
t
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
Three sets of loads: 1i  0.5; 2 j  0.41; tg  0.33
1i  0.32;  2 j  0.41; tg  0.33
1i  0.28;  2 j  0.30; tg  0.33
18
Models Simulated (III)
Example 4: A large network with 15 nodes:

4 of the nodes are at “distance” 3, and so their queue lengths are
not included in the importance function.

A customer leaving a node can go to 8 nodes with probability 0.1 (to
each one) or can leave the network with probability 0.2.

The load of the target node is similar to the loads of other 2 nodes,
and lower than the loads of the other 12 nodes.

This paper deals with networks with:
Interarrival times: Exponential, Erlang or Hyperexponential
Service times: Exponential or Erlang
CV Pearson: Erlang (3, β) = 0.58 ; Hyperexponential: 1.42
Simulation Results (I)

Example 1: 2-node network with strong feedback.
Rare event probability: P  Qtg  L   1015
Relative error = 0.1
 tg  0.33
Interarrival times: Exponential, Service times: Exponential
e
Interarrival times: Hyper-Exponential, Service times: Erlang  tg  0.22;
Importance function:   aQ  Q
1
tg

Events
Time
Gain
millions
minutes
(events)
0.36
3.3
2.1
8.2x1010
6.1
1,2x1010
6.9
0.19
1.8
1.6
6.1x1010
6.1
4.3x109
14
Model
L
P
1
a
Exp-Exp
31
1.4x10-15
0.67
Hyp-Erl
23
2.2x10-15
0.75
fV
Gain
(time)
f0
Robustness: Acceptable results for coefficients a between 0 and 0.21.
20
Simulation Results (II)
Example 2: Three-queue tandem network
15
Rare event probability: P  Qtg  L   10
Actual loads of the target network: t  0.33; Effective (H-E):  t  0.08;0.15;0.30


Importance function:   aQ1  bQ2  Qtg
Events
Time
millions
minutes
0.63
11.9
3.3
11
9.7x109
4.6
0.63
0.63
3.3
1.0
6.5
1.9x1010
4.3
1.6x10-15
1
1
1.0
0.25
1.0
1.0x1011
1.9
14
7.0x10-15
0.13
0.50
21.9
16.6
77
1.2x108
14
Hyp-Erl
19
1.3x10-15
0.32
0.32
9.1
6.4
19
2.5x109
13
Hyp-Erl
33
1.9x10-15
0.90
0.90
2.0
0.9
2.8
5,7x109
7.0
Model
L
P
a
b
Exp-Exp
31
1.7x10-15
0.37
Exp-Exp
31
1.6x10-15
Exp-Exp
31
Hyp-Erl

fV
Gain
(time)
f0
Robustness: Acceptable results for coefficients a and b up to 10%
lower or greater than optimal
21
Simulation Results (III)
Example 3: Network with 7 nodos:
15
Rare set probability: P  Qtg  L   10 ;


Importance function:
4
6
i 1
j 5
  a  Qi  bQ j  Qt
Events
Time
millions
minutes
0.45
3.8
1.1
1.8
4.4x1010
4.2
0.36
0.45
1.9
0.55
1.3
7.7x1010
3.7
2.6x10-15
0.40
0.61
1.6
0.50
1.0
8.7x1011
3.7
31
2.9x10-15
0.23
0.49
3.0
1.7
2.1
2.3 1010
6.1
Hyp-Erl
32
3.0x10-15
0.35
0.43
2.4
1.2
1.4
3.7 1010
5.4
Hyp-Erl
33
5.3x10-15
0.41
0.66
2.1
1.2
1.1
2.5 1010
6.0
Model
L
P
a
b
Exp-Exp
30
2.5x10-15
0.23
Exp-Exp
30
2.5x10-15
Exp-Exp
30
Hyp-Erl

fV
Gain
(time)
F0
Robustness: Acceptable results are obtained for coefficients a and b
up to 20% lower or greater than optimal ones. Similar results are obtained
(for the Hyp-Erl case) with the coefficients derived for Jackson networks.
22
Simulation Results (IV)

Example 4: Large network with 15 nodes
Rare event probability: P  Qtg  L   1015
Relative error = 0.1
Coefficients of the importance function:
0,0,0,0,0.22,0.20,0.20,0.22,0.37,0.35,0.35,0.37,0.34,0.42,1 (Exp-Exp)
0,0,0,0,0.25,0.22,0.22,0.24,0.42,0.39,0.38,0.41,0.35,0.43,1 (Hyp-Erl)

Events
Time
Millions
Minutes
1.1x10-15
10.5
3.1
1.9
1.4x1011
2.7
2.0x10-15
20.3
15.9
4.1
1.5x1010
6.6
Model
L
P
Exp-Exp
32
Hyp-Erl
31
fV
Gain
(time)
f0
Robustness: Acceptable results for coefficients up to 50% lower than those
given by the formula. Similar results are obtained (for the Hyp-Erl case) with
the set of coefficients derived for Jackson networks.
23
Simulation Results (V)
Analogous results (better than the Hyp-Erl case but worse than the ExpExp case) have been obtained for these 4 topologies with the following
distributions:
Exponential-Erlang, Hyper-Exponential-Exponential
Erlang-Exponential and Erlang-Erlang
For simulating the networks of 7 and 15 nodes the importance function
derived for Markovian networks has been used for these four cases.

For the three-queue tandem network and for the tho-node network
with strong feedback the same formulas have been used, but with the
“effective loads” calculated as:
P(X>=2n / X>=n) = ^n

24
Conclusions

Formulas of the importance function derived for Jackson
networks lead to very good results in Non-Jackson networks
changing the actual loads of the nodes by the “effective
loads”.

Probabilities of the order of 10-15 have been estimated, within
short or moderate computational times, in 48 types of
networks with different topologies and loads, and different
interarrival and service times. Most of them are “difficult”
networks for estimating rare event probabilities.


Worst results are obtained when the dependence of the target
queue on the queue length of the other queues is very high. As
a consequence, the efficiency of RESTART often improves with
the complexity of the system.
These type of formulas could be applied to many other nonJackson networks for estimating rare event probabilities.
25