1

Working Paper NEW CONCEPT OF IMPORTANCE IN RESTART SIMULATIONS

Manuel Villén-Altamirano Eduardo Casilari Arcadio Reyes-Lecuona manolo.villen@uma.es ecasilari@uma.es areyes@uma.es

Universidad de Málaga Dpto. Ingeniería Electrónica, ETSI Telecomunicación

Campus de Teatinos s/n, 29071, Málaga, Spain

Abstract. RESTART is a widely applicable accelerated simulation technique that allows the evaluation of extremely low probabilities. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event of interest is higher. In previous papers the importance of a state, which is a measure of the chance of occurrence of the rare event when the state occurs, was defined. The mentioned regions should be defined by means of the importance of the states but, due to the difficulty of knowing them, they are defined by means of another function of the states called the importance function. The choice by the user of an importance function which reflects the importance of the states is crucial for the effective application of RESTART. The problem of the previous definition of importance is that the importance of a state does not only depend on the state itself but also on the chosen importance function. It produces a loop between importance and importance function which can make more difficult to find an appropriate importance function. The paper presents a new concept of importance which does not depend on the chosen importance function. It avoids the above mentioned loop and can make easier to find an appropriate importance function. The paper also presents the application of this new concept of importance to an M/M/1 queue and the work progress of its application to a two-queue Jackson tandem network.

1. INTRODUCTION

The study of critical events that occur very infrequently is of interest in many areas. As crude

simulations require prohibitively long execution times for the accurate estimation of very low

probabilities, acceleration methods are necessary.

Importance sampling is a well-known technique in rare event simulation, see e.g., [1]. Another

known method is RESTART. In this method a more frequent occurrence of a formerly rare event

is achieved by performing a number of simulation retrials when the process enters regions of the

state space where the chance of occurrence of the rare event is higher. These regions, called

importance regions, are defined by comparing the value taken by a function Φ of the system

state, the importance function, with certain thresholds Ti.

As indicated in [2], a suitable choice of the importance function is crucial for achieving a

high efficiency in the application of RESTART. An importance function is appropriate and

leads to an effective application of RESTART when the chance of occurrence of the rare event

when a certain system state occurs is similar for all the system states xi for which ( ) ii Tx =Φ .

2

For this purpose, [2] defines the importance of a state xi for which ( ) ii Tx =Φ , */ ixAP , which is

a measure of the chance of occurrence of the rare event when xi occurs. An importance function

is appropriate if all the states xi for which ( ) ii Tx =Φ have similar importance. The values of

( )ixΦ should reflect the values of * / ixAP in the sense that if */ ixAP is greater than *

/ jxAP for two

given states xi and xj , Φ(xi) should also be greater than Φ(xj).

The problem of the definition of */ ixAP is that the importance of a state xi does not only

depend on the state itself but also on the chosen importance function. When we apply

RESTART we have to find an importance function ( )ixΦ such that the values of ( )ixΦ reflect

the values of */ ixAP but, in its turn, the values of */ ixAP depend on the values of ( )ixΦ . This loop

can make more difficult to find an appropriate importance function.

The paper will present a new concept of importance of a state xi, i.e., a new measure of the

chance of occurrence of the rare event when xi occurs. This new importance, called ixI , does

not depend on the chosen importance function. Thus it avoids the above described loop and

makes easier to find an appropriate importance function. As with the previous concept of

importance, an importance function ( )ixΦ is appropriate if all the states xi for which ( ) ii Tx =Φ

have similar importance ixI .

Section 2 of the paper makes a review of RESTART and of the presently defined importance

*/ ixAP , showing the problem of the above mentioned loop. Section 3 presents the new concept of

importance ixI and shows that it is also a good measure of the chance of occurrence of the rare

event. Section 4 applies this new concept of importance to an M/M/1 queue and Section 5

shows the work progress of its application to a two-queue tandem Jackson network. Finally

Section 6 states the conclusions and provides guidelines for future work.

2. REVIEW OF RESTART AND OF THE PREVIOUS CONCEPT OF IMPORTANCE

RESTART has been described in several papers, e.g., [2]. Nevertheless, it is briefly described

here to provide a self-contained paper.

Let Ω denote the state space of a process X(t) and A the rare set whose probability is to be

estimated. A nested sequence of sets of states Ci, 1 2 , , MC C C⊃ ⊃ ⊃K is defined, which

determines a partition of the state space Ω into the regions 1i iC C+− ; the higher the value of i,

the higher the importance of the region 1i iC C+− , i.e., the higher the chance of reaching the rare

3

set. These sets are defined by means of a function, : Φ Ω → ℜ , called the importance function.

Thresholds of Φ, Ti (1 ≤ i ≤ M), are defined so that each set Ci is associated with iTΦ ≥ .

Φ(t)

t (time)

L

T 3

T2

T1

B3

B2

B1

B3

B1

1

D2D2

D33

D1D1

D2

D3

A

1DD

D

Fig. 1: Simulation with RESTART

RESTART works as follows: a simulation path, called main trial, is performed in the same

way as if it were a crude simulation. Each time the main trial enters set C1, the entry state is

saved and 11 −R simulation retrials of level 1 are performed. Each retrial starts with the saved

state and finishes when the retrial exits set C1. When set C2 is reached in the main trial or in a

retrial of level 1, 2 1R − retrials of level 2 are performed, each one finishing when they leave set

C2, and so on. Note that the oversampling made in the region 1i iC C +− (CM if i = M) is given

by the accumulative number of retrials: 1

i

i jj

r R=

= ∏ .

Figure 1 illustrates a RESTART simulation with M = 3, R1 = R2 = 4, R3 = 3, in which the

chosen importance function Φ also defines set A as L≥Φ . Bold, thin, dashed and dotted lines

are used to distinguish the main trial and the retrials of level 1, 2 and 3, respectively.

Following [2], let us define events Bi and Di:

• Bi )1( Mi ≤≤ : event at which iT≥Φ having been iT<Φ at the previous event;

• Di )1( Mi ≤≤ : event at which iT<Φ having been iT≥Φ at the previous event.

Up to now the importance )1(*/ MiP

ixA ≤≤ has been defined as the expected number of rare

events occurred in a trial [Bi , Di ) (without counting upper thresholds retrials) when the system

state at iB is xi.

Let us call Xi to the random variable describing the state of the system at an event Bi (of the

main trial) randomly taken, and )1( Mii ≤≤Ω to the set of possible system states at an event Bi.

Thus )1(*/ MiP

iXA ≤≤ is also a random variable which takes the value*/ ixAP when Xi = xi .

4

The expected importance of an event Bi , ( )MiP iA ≤≤∗ 1/ , is given by:

[ ] )(/*

/*

/ ixAXAiA xdFPPEPi

ii ∫Ω∗==

where F(xi) is the distribution function of Xi. And the variance of the importance of an event Bi ,

( ) ( )MiPViXA ≤≤∗ 1/ , is given by:

( ) ( )[ ] 2*/

2*/

*/ )( iAXAXA PPEPV

ii−=

Let us define the probability AP Pr= of the rare set A as the probability of the system being

in a state of the set A at the instant of occurrence of certain events denoted reference events (an

example of a reference event is a packet arrival). Let us call rare event or event A to a reference

event at which the system is in a state of the set A. The estimator of the rare set probability P

used in RESTART is:

M

A

rN

NP=ˆ

where NA is the total number of events A that occur in the simulation (in the main trial or in any

retrial) and N the number of reference events simulated in the main trial.

Define as the random variable indicating the number of events A in the main trial and

( ) [ ]00AAA NENVK = . As proved in [2] (see formulas (4) and (5)), can be written as:

[ ]( )[ ]

−+= ∑=

0

0

1

)1()ˆ(

A

iAM

i i

i

M

A

NE

NEV

r

R

r

K

N

PPV

χ (1)

where ( )( )0

,...,,, 210 iNiiiii XXXN=χ , 0

iN being the random variable indicating the number of events

Bi occurred in the main trial of a simulation randomly taken and ),...,,(021 iN

iii XXX being the vector

of random variables describing the system states at those events Bi. According to formula (15)

of [2], | can be written as:

| = ⁄∗ + ⁄∗ (2)

where γI, of which an expression is provided in Section 5.3.1 of [2], is a measure of the

dependence of the importance of the system states Xi of the different events Bi occurring in the

main trial. If the random variables Xi were independent γi would be equal to 1. In most practical

applications, there may be some dependence between system states of close events Bi but this

dependence is negligible for distant events Bi. Thus γi is usually close to 1 or at least of the same

order of magnitude as 1.

5

From formulas (1) and (2) can be deduced that the smaller the variances ( )*/ iXAPV , the smaller

the variance of the estimator of the rare set probability and thus the higher the efficiency of the

RESTART application. It is crucial for an efficient application of RESTART the choice of an

importance function ( )xΦ such that all the states xi for which ( ) ii Tx =Φ have similar importance.

In this way ( )*/ iXAPV is small and is also small.

Note that, roughly speaking, the importance */ ixAP of a state xi is equal to the expected

number of events A occurred in a trial (without counting upper thresholds retrials) from the

instant at which the system state is xi until the next instant at which the system is in a state x

such that Φ < Φ . Thus the importance of a state xi does not only depend on the state

itself but also depends on the chosen importance function Φ. As was said in the Introduction,

when we apply RESTART we have to find an importance function ( )ixΦ such that the values of

( )ixΦ reflect the values of */ ixAP but, in its turn, the values of *

/ ixAP depend on the values of

( )ixΦ . This loop can make more difficult to find an appropriate importance function.

3. NEW CONCEPT OF IMPORTANCE

To avoid the above mentioned loop we propose a new definition of importance:

The importance xI of a state x is defined as the limit when Nr approaches infinite of the

expected number of rare events occurring (without counting the retrials) in the Nr reference

events following the occurrence of state x, minus (i.e., minus the expected number of rare

events occurring in the Nr reference events following a random state).

Although the definition of Ix is independent from the importance function and thus from the

thresholds Ti, we can define I i and ! for each threshold Ti as:

[ ] )( iXXi xdFIIEIi

ii ∫Ω==

( ) ( )[ ] 22 )( iXX IIEIVii

−=

As proved in the appendix, a good approximation of | is given by:

[ ]( ) ( ) ( ) [ ] ( ) iXiiiAiA iIVNENVPNEV βχ 002*

/0 | += (3)

where " , whose expression is given in formula (23), is a factor lower than 1 and usually close

to 1.

6

From formulas (1) and (3) can be deduced that, as occurred with ( )*/ iXAPV , the smaller the

variances !, the smaller the variance of the estimator of the rare event probability and thus

the higher the efficiency of the RESTART application. Thus we can say that, also with this new

definition of importance, an importance function ( )xΦ is appropriate if all the states xi for which

( ) ii Tx =Φ have similar importance.

The advantage of this new definition is that, as the values of Ix do not depend on the values

of ( )xΦ , the loop existing with the previous definition of importance does not exist with this

new one and, consequently, the choice of an appropriate importance function can be easier.

4. APPLICATION OF THE NEW IMPORTANCE TO AN M/M/1 QUEUE

Consider an M/M/1 queue, in which customers with Poisson arrival enter the queue. The mean

arrival rate is λ and the service time is exponentially distributed with mean service rate µ. The

load is # = $ %⁄ . The buffer space is assumed to be infinite. The system state is defined by (j),

where j is the number of customers at the queue. The rare set is defined as & ≥ (, i.e. as the

number of customers in the queue exceeding a predefined threshold L. The rare set probability is

= #). The reference event is defined as the arrival of a customer. Thus a rare event is a

customer arrival occurred when & ≥ (. (j does not include the arriving customer). Let I j denotes

the importance of the state (j).

The importances I j may be derived from a system of equations in which each equation relates

the importance of a state (j) to the importance of the states to which the system can move from

it, plus an additional equation of normalization. From the definition of importance, it is easy to

see that the importance I j can be written as:

!* = ∑ ,*,./*,. − /*,.1 + !.∀. (4)

where:

• Pj,k is the transition probability from state (j) to state (k);

• /*,. is the number of rare events occurring in the transition from state (j) to state (k).

/*,. = 1 if he transition is due to the arrival of a customer when & ≥ (, and /*,. = 0 in

the remainig cases;

• /*,.1 is the number of reference events occurring in the transition from state (j) to state

(k). /*,.1 = 1 if he transition is due to a customer arrival and /*,.1 = 0 if the transition is

due to an end of service.

Thus the following system of equations can be written:

7

PII −= 10 (5a)

( ) 1111 −≤≤+

+−+

= −+ LjIPII jjj µλµ

µλλ (5b)

( ) LjIPII jjj ≥+

++−+

= −+ 11 1µλ

µµλ

λ (5c)

∑∞

=

=0

0j

jj Iρ (5d)

Equation (5d) is the normalization equation. It takes into account that the weighted mean

value of the expected number of rare events from each of the system states, weighting with the

corresponding state probability 1 − ##*, is equal to the expected number of rare events from a

random state.

From the above system of equations the following expression of the importance I j is

obtained:

!5 =67879− ):;<=>?: if& = 0! + :>?:B 1 − #*1 + &1 − ##)?*if& ≤ (!) + :>?:;>?: & − (if& ≥ (

(6)

As could be expected, I j is a monotonically increasing function, being negative for low

values of j. Thus an appropriate importance function is Φ& = &. As the monotonicity and

increasing properties of I j were already known, we could have chosen the importance function

Φ& = & without need of deriving formula (6).

Nevertheless to derive formula (6) has been useful as a first step to gain insight on how is the

importance in queuing networks. For example, we can learn from formula (6) that, except for

low values of j, the growth of !* is approximately exponential when & ≤ ( (the ratio

!* !*?> ≈ 1 #⁄⁄ ), and linear when & > (, given that !* − !*?> = #1 − #) 1 − # ≈ # 1 − #⁄⁄ .

5. APPLICATION OF THE NEW IMPORTANCE TO A TWO-QUEUE TANDEM JACKSON NETWORK

Consider the two-queue tandem Jackson network shown in Figure 2. Customers with Poisson

arrival enter the first queue and, after being served, enter the second one. The mean arrival rate

is λ and the service time is exponentially distributed in each queue with mean service rates 1µ

and 2µ , respectively. The load at each queue is )2,1( == iii µλρ . The buffer space at each

8

queue is assumed to be infinite. The system state is defined by ( )ji , , where i and j are the

number of customers at the first and at the second queue respectively.

λ

Figure 2: Two-queue tandem Jackson network

The rare set is defined as & ≥ (, i.e. as the number of customers in the second queue

exceeding a predefined threshold L. The rare set probability is = #) . The reference event is

defined as the end of service of a customer in the first queue and its arrival at the second queue.

Thus a rare event is a customer arrival at the second queue occurred when & ≥ (. (j does not

include the arriving customer). Let ! ,* denotes the importance of the state (i, j). Following the

same reasoning as in Section 4, the following system of equations can be stated:

0,10,0 II = (7a)

( ) 11,11

10,1

10, ≥−

++

+= −+ iPIII iii µλ

µµλ

λ (7b)

11,02

2,1

2,0 ≥

++

+= − jIII jjj µλ

µµλ

λ (7c)

( )11

11,

21

21,1

21

1,1

21, −≤≤

≥++

+−++

+++

= −+−+ Lj

iIPIII jijijiji µµλ

µµµλ

µµµλ

λ

(7d)

( )Lj

iIPIII jijijiji ≥

≥++

++−++

+++

= −+−+

11 1,

21

21,1

21

1,1

21, µµλ

µµµλ

µµµλ

λ (7e)

∑∑∞

=

∞

==

0 0,21 0

i jji

ji Iρρ (7f)

Up to now, we have not been able to solve this system of equations. Any suggestion of the

audience on how to solve it will be very welcome.

However if we define !F,* as the mean weighted value of the importance of the states ( )ji ,

for a given value of j and any value of i, that is:

( ) jii

ijm II ,

011. 1 ∑

∞

=

−= ρρ

9

we have proved that !F,* satisfies the equations (5a) to (5d) of the M/M/1 queue (substituting I j

by Im,j , µ by µ2 and ρ by ρ2). Therefore Im,j may be obtained from formula (6) (making the same

substitutions).

6. CONCLUSIONS AND FUTURE WORK

We have presented a new concept of importance which avoids the loop between importance and

importance function existing with the previous concept, and thus can make easier to choose an

appropriate importance function.

With this new concept we have obtained formulas of the importance of each state in an

M/M/1 and we have stated the system of equations from which the formulas of the importance

of each state in a two-queue tandem network could be derived. However we have not been able

up to now of solving this system of equations.

First future work is obviously to solve the above mentioned system of equations for

obtaining formulas of the importance of the states of the two-queue tandem network. We ask the

audience any suggestion or cooperation for solving this system of equations. Once we have

solved this problem, we will try to obtain formulas of the importance of other simple Jackson

queuing networks such as the following ones:

- A network as that of Figure 2 where a fraction of the customers leaves the system after

being served in the first queue without entering the second queue;

- A network as that of Figure 2 where a fraction of the customers coming from outside the

network directly enter the second queue;

- A network with both features mentioned in the two previous cases;

- A network with two queues in the first stage and one queue in the second stage;

- A three-queue tandem network.

If we are able of solving all the above simple Jackson networks, we think that we will have

gained insight which allows making approximations for more complex Jackson queuing

networks and choosing appropriate importance functions, hopefully in a more accurate way than

using the existing formulas [3].

Once we have new formulas of the importance function in Jackson queuing networks, we

will apply the concept of effective load introduced in [4] to obtain formulas of the importance

function in non-Markovian queuing networks.

10

APPENDIX: DEVELOPMENT OF GHIJK|LM AS A FUNCTION OF GNOM Let us define l

iZ as the random variable indicating the number of events A in the trial [Bi, Di)

starting in the lth event Bi of the main trial, and P Q as ∑=

=0iN

lk

ki

li ZU

As 10iA UN = , [ ]( )iANEV χ|0

can be written as:

[ ]( ) [ ]( ) [ ]( )[ ] ∑∑∞

=

∞

=

+ ∆=−=11

10 |||l

ll

ilii

liiA UEVUEVNEV χχχ (8)

being:

[ ]( ) [ ]( )ilii

lil UEVUEV χχ || 1+−=∆

As P Q = R Q + P QS>, we can write l∆ as:

[ ]( )[ ] [ ] [ ][ ] [ ]( ) [ ]( )21212||2| ++ +−⋅+=∆ l

ilii

lii

lii

lil UEUEUEZEEZEE χχχ (9)

The first term of this expression of l∆ can be written as:

[ ]( )[ ] [ ]( )[ ] [ ]( )[ ]200202|Pr||Pr| l

iliiii

liii

li XZEElNlNZEElNZEE ≥=≥≥= χχ (10)

The condition lNi ≥0 also exit in the last member of (10), but it has not been indicated

because it is implicit in the condition liX . If liX exists, then lNi ≥0 .

Manipulating in the same manner the second term of l∆ , we have:

[ ] [ ][ ] [ ] [ ][ ]li

li

li

liii

lii

li XUEXZEElNUEZEE ||Pr|| 101 ++ ⋅≥=⋅ χχ (11)

Substituting (10) and (11) in (9) and after some mathematical manipulations, we obtain:

[ ]( ) [ ]( )( ) [ ]( ) [ ]( )( ) [ ]( ) [ ]( )212201200

10

||Pr

||Pr

++

+

+−≥−≥≥+

+−≥=∆

li

lii

lii

lii

li

li

li

liil

UEUElNUElNUElN

XUEVXUEVlN

(12)

Let us call:

[ ]( ) [ ]( ) ∑∞

=

+−≥=Ω1

101 ||Pr

l

li

li

li

lii XUEVXUEVlN (13)

[ ]( ) [ ]( ) [ ]( )2120120

1

02 ||Pr ii

lii

li

li UElNUElNUElN −≥−≥≥=Ω +

∞

=∑ (14)

From (12), (13) and (14) we obtain 211

Ω+Ω=∆∑∞

=ll and, taking into account (8), we have:

11

[ ]( ) 210 | Ω+Ω=iANEV χ (15)

As [ ] 0|00 1 =+ ii N

iNi XUE and thus [ ] 0|

00 1 =

+ ii N

iNi XUEV , we can write Ω1 as:

[ ]( ) [ ]( )

[ ]( ) [ ]( ) [ ]( ) ∑

∑∑∞

=

−

+∞

=

∞

=

−≥+≥=

=+≥−≥=Ω

2

10110

1

1

0

1

01

||Pr|1Pr

|1Pr|Pr

l

li

li

li

liiiii

li

li

li

li

li

li

XUEVXUEVlNXUEVN

XUEVlNXUEVlN

(16)

Taking into account that 1++= li

li

li UZU , [ ] *

/0| iAi

li PlNZE =≥ , [ ] [ ]0*

/1

iiAi NEPUE = and that

[ ] ∑∞

+=

+ ≥=≥≥1

0*/

010 Pr|Prlj

iiAilii jNPlNUElN , we can write Ωas:

( ) [ ]( ) ( ) ( )02*/

20

1 1

002*/2 Pr2Pr iiAi

l ljiiiA NVPNEjNlNP =

−

≥+≥=Ω ∑ ∑

∞

=

∞

+=

(17)

Substituting (16) and (17) in (15), we obtain:

[ ]( ) ( ) ( ) [ ]( )

[ ]( ) [ ]( )( )∑∞

=

−−≥+

+≥+=

2

10

11002*/

0

||Pr

|1Pr|

l

li

li

li

lii

iiiiiAiA

XUEVXUEVlN

XUEVNNVPNEV χ (18)

As an approximation, let us evaluate [ ]( )li

li XUEV | and [ ]( )1| −l

ili XUEV assuming that the

simulation has infinite length and thus: ∑∞

=

+=∞=1

0 ,j

jli

lii ZUN . Although R QS* depends on U Q

(or on U Q?>), this dependence is very small for high values of j. Thus to include in PQQ terms

R QS* with a high value of j has a small impact on [ ]( )li

li XUEV | and on [ ]( )1| −l

ili XUEV . As

the correlation between ...,,...,, 1 jli

li

li ZZZ ++ is positive in most applications, the

approximation will usually lead to a slight overestimate of [ ]( )li

li XUEV | and

[ ]( )1| −li

li XUEV and thus of [ ]( )iANEV χ|0 .

With this approximation, [ ]( )li

li XUEV | and [ ]( )1| −l

ili XUEV do not depend on the value

of l because they really depend on the value of − V, and if = ∞ the value of − V is

the same for any value of l. As [ ]∑∞

==≥

1

00Prl

ii NElN , formula (18) becomes:

[ ]( ) ( ) ( ) [ ]( )[ ] ( ) [ ]( ) [ ]( )( )100

002*/

0

||1Pr

|1Pr|

−−≥−+

+≥+=

li

li

li

liii

li

liiiiAiA

XUEVXUEVNNE

XUEVNNVPNEV χ (19)

12

If = ∞, the importance of the state Q, !XY, is equal to P Q| Q minus a term which does

not depend on Q . Thus:

[ ]( ) ( ) ( )il

iXX

li

li IVIVXUEV ==| (20)

In the same manner:

[ ]( ) [ ][ ]( ) [ ]

== −−− 111 |||| l

iX

li

li

li

li

li XIEVXXUEEVXUEV l

i (21)

Substituting (20) and (21) in (19) and taking into account that Z!Y[ − Z \!Y|U Q?>][ =

= \ Z!Y|U Q?>[] , we have:

[ ]( ) ( ) ( ) [ ] ( ) iXiiiAiA iIVNENVPNEV βχ 002*

/0 | +=

(22)

being:

" = Pr` ≥ 1a + b1 − Pr` ≥ 1a c Z!Y|U Q?>[! 23 As Z!Y|U Q?>[ is smaller than ! but usually close to !, factor " is smaller than 1

but usually close to 1.

REFERENCES

[1] Rubino, G., and Tuffin, B. (editors). 2009. Rare even simulation using Monte Carlo

methods. Chichester: Wiley.

[2] Villén-Altamirano, M., and Villén-Altamirano, J. 2002. Analysis of RESTART Simulation:

Theoretical Basis and Sensitivity Study. European Transaction on Telecommunications 13, 4,

373-385.

[3] Villén-Altamirano, J. 2010. Importance function for RESTART simulation of general Jackson

networks. Eur. J. Oper. Res. 203(1), 156-165.

[4] Villén-Altamirano, J., and Villén-Altamirano, M. 2013. Rare event simulation of non-

Markovian queuing networks using RESTART method. Simulation Modelling Practice and

Theory 37, 70–78.