A scheme for adaptive biasing in importance sampling - CiteSeerX

AEÜ, Specual issue on Rare Event Simulation

1

A scheme for adaptive biasing in importance sampling Poul E. Heegaard Norwegian University of Science and Technology, Department of Telematics, N-7034 Trondheim, Norway, e-mail: [email protected], url: http://www.item.ntnu.no/~poulh Abstract This paper considers simulation of large networks. The quantities of interest, such as system failure, blocking or cell loss probability, are dependent on observations of rare events. For evaluation of such systems, previous work have shown that simulation with a speed-up technique called importance sampling is an efficient means, provided that a good biasing of the simulation parameters exists. This paper addresses the unsolved problem of parameter biasing in large networks with well balanced resources. A new algorithm for adaptive biasing of the simulation parameters is introduced. In addition, a flexible framework for modelling of both traffic and dependability aspects of the network is described. As a feasibility demonstration, the applicability of the proposed simulation framework is demonstrated by evaluation of time blocking probabilities in a network example. This network has traffic classes with different quality of service requirements, different capacity requirements, alternative routing strategies and preemptive priorities. Rerouting occurs on overloads and after link and node failures.

1

INTRODUCTION

Simulation is a flexible means for system evaluation of complex data and telecommunication networks. The network users may, in addition to different traffic parameters, have different quality of service requirements. This requires existence of alternative routing strategies and preemptive priority mechanisms. However, under extreme quality of service requirements, the performance measure depends on the occurrence of rare events, such as trunk line blocking, buffer overflow or system failure. With direct simulation this requires very long experiments. Hence, a simulation speed-up technique should be applied, see [Hee97b] for an overview over various speed-up techniques. Importance sampling is one such technique known to be very efficient for rare event simulation. The efficiency of importance sampling is critical to the change of measure, i.e. the change of the simulation parameters, denoted as parameter biasing in the following. It is essential for the success of importance sampling that methods exist for optimal parameter biasing, or at least heuristics for obtaining good parameters. On traffic models, much work has been done on (asymptotically) optimal biasing in single server networks, or networks where the system performance is dominated by the performance of one single server. In dependability simulations, robust biasing of the failure rates are defined. An excellent survey of existing biasing methods is given in [Hei95], both for traffic and dependability models. In a well-engineered network, the system performance depend on the performance of several interacting queues. This implies that to obtain good estimates of the system performance it is not sufficient to bias the parameters to observe, for instance, the overflow in only one specific

AEÜ, Specual issue on Rare Event Simulation

2

queue. This paper proposes an adaptive parameter biasing where the objective is to obtain a significant sampling of, for instance, the overflow from each queue with significant contributions to the system performance. The proposed adaptive biasing scheme combines optimal solution from single server networks [PW89, CFM83] with ideas from a technique called failure distance biasing [Car91]. The main objective of this work has been to develop a biasing scheme that is robust and efficient for large networks, rather than being optimal for all kinds of general arrival and service processes. The adaptive biasing has been improved in several steps since it was first introduced and applied to a small model in [Hee95]. Application to a large Erlang loss network was presented in [Hee96], and simulations of both a traffic and a dependability model in [Hee97a]. 4 i j

Node i Link j

- service classes

6 3

- quality of service requirements - preemptive priority level

4 2 1 1

9

8 7 5

2 3 6

5 10

- rerouting on overload and failure - link and node failure

Figure 1 System example of a typical complexity

In section 2, the modelling framework is generalised to handle systems like the network topology example in figure 1. The principles of importance sampling, and the adaptive strategy for parameter biasing, are given in section 3. Section 4 describes the use of a regeneration box, which is a state sub-space, to conduct efficient regenerative simulations on a large network model. Finally, some simulation results are included in section 5, and closing comments in section 6. 2

FLEXIBLE SIMULATION MODEL FRAMEWORK

The network topology given as an example in figure 1, is offered traffic from a variety of users. Thy have different traffic parameters, and quality of service requirements. This implies that preemptive priorities must be allowed between users of different service classes. Furthermore, a specific user is connected to an origin node with a fixed routing through the network to a destination node. Alternative routes exist for some users, and this is chosen when a link is unavailable due to traffic congestion and failure. Performance analysis of a system with these characteristics requires a flexible modelling framework. Hence, to provide accurate models, this paper proposes a modelling framework including: - Network aspects like topology, resource capacities and service rates.

AEÜ, Specual issue on Rare Event Simulation

3

- Traffic aspects of the users, including parameters like arrival rates, capacity constraints, priority, etc. This section describes the flexible modelling framework. How the model is mapped to a state space diagram is also included. Figure 2 shows an example of a mapping between a sub-system of the example in figure 1, and its state space description. Discrete event semi-Markov simulations are conducted on state space models like the one in figure 2. 4

6 3 4

9

blocking pool 10 => TARGET 10

8 7 5 2

3

5 10 6

5 2 2

1

3

1

generator 1

2 1 1

10 6

generator 2

Figure 2 Mapping of system example to state space model

2.1

Basic concepts and notations

The models in this paper consists of K generators of entities requesting resources from J resource pools, defined as follows: Resource pool: a finite set of identical and interchangeable resources (e.g. a link in a network is considered as a pool of channels): - N j is the number of resources in pool j , i.e. the resource pool capacity. - Γ j is the set of generators with pool j as a constraint. Entity, e k , k = 1, …, K : an item that holds c kj resources from pool j , for all j in a fixed set of pools, Φ k (e.g. a connection k between A and B parties holds c kj channels from each link along the route between A and B). Generator (of entities): a component which explicitly models a process that generates the events operating on the entities: - e k is an entity from generator k , - Φ k is the routing set, a fixed set of resource pools, - λ k(ω) is the state dependent arrival rate, ˜ - M k is the population size, - µ k(ω) is the state dependent departure rate, ˜ - S k is the number of servers for generator k entities,

AEÜ, Specual issue on Rare Event Simulation

4

- c kj is the capacity that entities from generator k requires from pool j , The relations between traffic and network models and the basic concepts are given in table 1. Table 1 Relations between traffic and network models and the basic building blocks. Queueing network model

Generator, k

Resource pool, j

Network model

Routing set, Φ k

Resource pool capacity, N j Generator constraint set, Γ j

Traffic model

Population size, M k Arrival rate, λ k Number of servers, S k Departure rate, µ k Capacity requirement, c kj

Observe that the topology of the network model is implicitly defined through the sets Φ k and Γ j . Note also that the departure rates µ k are associated with the generators as part of the traffic –1 model. This is because µ k is considered to be the mean duration of a call or the mean repair time. 2.2

The state space

The global state space, Ω , consists of a set of system states, ω ∈ Ω , defined as ˜ ˜ ˜ System state, a representation of the number of entities at any time, i.e. ω = { ω 1, …, ω K } = { #e 1, …, #e K } (e.g. #e k is the number of end-to-end connections of ˜ source type k ). There is a one-to-one mapping between a generator and a dimension in the state space. This means that in a model with K generators, the state space description has K dimensions. The boundaries of the state space are determined by the relations between the generators and the resource pools, given as rerouting sets, Φ , and the resource pool capacity, N j. ˜ 2.3 The model dynamics To explain the dynamics of the model, consider how a simulation experiment operates on the global state space, Ω . Each observation or sample from the ˜ is a path which consists of a sequence of experiment events, where: Event: an occurrence that trigger a request or release of c resources (e.g. a connection attempt is an event that requests c resources). A request event results in a new entity if sufficient number of resources are available for all j ∈ Γ j . A release event removes an entity. Path (trajectory): any sequence of events s = { ω 0, ω 1, …, ω n – 1, ω n } , where ω i is the sys˜ state ˜ after ˜ event ˜ i and˜ n is the total ˜ number of tem events in path s . A regenerative cycle is a path where ( ω 0 = ω l ) ∈ Ω˜ 0 , where Ω 0 is the regenerative state. ˜ ˜ ˜ ˜

path, sr

Release entity: departure of entity e2

ω={#e1, #e2,}

Request event and spare capacity: arrival of entity e2

Figure 3 The dynamics in the state space.

AEÜ, Specual issue on Rare Event Simulation

5

Observe that a request event not necessary results in a change in the system state. For instance, if a new call arrives to a loss system where all trunks are occupied, the call is rejected. The call arrival is a request event. If the system state is the number of calls of each type, then it will not be updated. However, the request is still an event, because it may be a part of the statistics. Generally, each event within the path causes the system state ω i to be updated according to: ˜ if next event is release of e k  –1k  ˜ ωi = ωi – 1 +  1k if next event is request of e k and capacity is available in Φ k (1) ˜ ˜  ˜ otherwise  0 where 1 k = { 0, 0, …, 1 k, … } is a vector of size K where all elements are 0 except position k ˜ 1. which is 2.4

The target

This paper only considers performance measures dependent on the network constrains given by the resource pool capacities, e.g. time blocking, rerouting probability, etc. Estimation of these properties requires observations of visits to a sub-space Ω j , denoted targets. ˜ Target, Ω j = { ω ∑k ∈ Γ ω k ⋅ c kj ≥ N j } : a subset of Ω where the capacity N j of resource j ˜ pool j is ˜reached,˜ (e.g. all channels on link j are occupied). Rare event: a visit to the target when P(Ω j) « 1 , i.e. a visit is a rare event. ˜ A single target model is a network with only one dominating resource pool, while a multiple target model has several pools with significant contributions to the quantity of interest. 2.5

Applicability and extensions

The generality of the framework is illustrated in table 2 where models for both traffic and dependability aspects of the network example in figure 1 is given. Observe that a link failure is represented by a generator with events requesting all resources of the link. Using the concepts defined for traffic models provide a flexible modelling also for dependability aspects. With state dependent arrival rates, failure propagation can be modelled by making the failure rate of a generator dependent on state changes in other generators. The concept of routing set makes it easy to model common mode failures, i.e. a fault may cause several failures to occur. Both links and nodes can experience failure, by defining the links as resource pools. A node failure occurs when all links connecting to it is defined in the same routing set and hence fails at the same time. By describing the failure process with more than two states (“ok” and “defect”), a graceful degradation where partly failure in of a link or node can be modelled. The framework does not handle events that are pre-emptive by nature. For instance, a failure event must be able to pre-empt a call if this holds resources of a pool that fails, otherwise the fail is rejected! Furthermore, different priority levels are required to be able to differentiate between quality of service requirements. Alternative routing should be provided to model redundancy, e.g. to let high priority traffic change its routing if an overload exists, or if a link or node failure has occurred along its primary route.

AEÜ, Specual issue on Rare Event Simulation

6

Table 2 Mapping of discrete event model concepts and simulation models, an example. concepts

traffic simulation model

dependability simulation model

entity

connection

failure

generator

source type

link or node failure type

request event

connection arrival

failure of a link or node

release event

connection completion

repair of a link or node

capacity requirement

1 ≤ c kj ≤ N Φ

c kj = N Φ

quantity of interest

blocking probability

k

k

unavailability

The following restrictions and extensions are made: State dependent transition rates redefines the arrival and departure rates to λ k(ω k) and µ k(ω k) , respectively. This is a restriction of the general definition in section 2.1 which implies that the transition rates are only dependent on the state within its own generator. The following functional relations apply for the arrival and departure rates:  ( M k – ω k )λ k λ k(ω) = λ k(ω k) =  ˜  λk

Mk < ∞

(2)

otherwise

µ k(ω) = µ k(ω k) = min(ω k, S k)µ k ˜ The restrictions are made to simplify the adaptive algorithm.

(3)

Pre-emptive priority implies that request events from a generator with a high priority level is allowed to preempt an entity with a lower priority level. -

p k is the priority level of generator k , p = 0 is the highest priority level,

- ζ p is the set of generators having p as its priority level, ( p)

resource pool 1 p2=1

1

2

p1=2

generator 1: low priority

- Γ j is the generator constraint set where the generators with priority level p have resource j as a constraint.   ( p) - Ω j = ω ∑ ω k c kj ≥ N j  is the target sub-set for priority level p . ( p) k ∈ Γj ˜ ˜  Ω1(2)

New transitions due to preemption

Ω1(1) generator 2: high priority

Figure 4 Influence on state space limitations by adding rerouting, an example.

AEÜ, Specual issue on Rare Event Simulation

7

In figure 4 it is shown how the preemption add new transitions to the state space. With no spare capacity when a high priority request event occurs, two simultaneous update takes place; - a sufficient number of lower priority entities is identified and disconnected, hence the corresponding #e is decreased, - the arriving high priority entity from generator k increases #e k by 1. Rerouting sets are defined to allow the generator k to allocate resources from an alternative set of resources. - R k is the number of rerouting alternatives for generator k . - Φ k = { Φ k0, …, Φ kRk } is the set of alternative routes for generator k . ˜ K - ω = { ω k } k = 1 is the new system state, where the system state of generator k is extended to ˜ R

R

k k ω k = { ω kr } r = 0 = { #e kr } r = 0 with #e kr representing the number of entities of generator

k at routing r , r = 0, …, R k . 

∑k = 1 ∑r = 0 I ( j ∈ Φkr)ωkr ckj ≥ N j  is the sub-set of target j . K

Rk



(a) Without rerouting

blocking pool 1

2

1

primary route of generator1

generator 1

te rou r 2 y r o t a ra primgene f o

blocking pool 2

 - Ω j = ω ˜ ˜

generator 2 (b) With rerouting of generator 2 generator 1

1 ol

primary route of generator1 secondary route of generator 2

po

2

1

blocking pool 1

ng ki oc bl

te rou r 2 y r a rato primgene of

blocking pool 2 => rerouting

generator 2

Figure 5 Influence on state space limitations by adding rerouting, an example.

A generator choose an alternative route only if the primary route r = 0 is not available. The alternative routes are checked in sequence from r = 1 up to R k until a route with sufficient capacity on all links is found. If no route is available, a blocking has occurred, i.e. a visit to Ω j ˜ is observed. Figure 5 illustrates how the state space is changed when the high priority generator 2 is allowed to reroute to the same link as the primary route for the low priority generator 1. In its most general form with both priority on rerouting, the target sub-space j is defined as: ( p)

Ωj ˜

 = ω ˜



∑k ∈ ζ ∑r = 0 I ( j ∈ Φkr)ωkr ckj ≥ N j  Rk

p



(4)

AEÜ, Specual issue on Rare Event Simulation

3 3.1

8

ADAPTIVE BIASING STRATEGY FOR IMPORTANCE SAMPLING Basics of importance sampling

Importance sampling changes the dynamics of the simulation model. The dynamics must be changed to increase the number of rare events of interests. Let f (s) be the original sampling distribution where s is, for instance, a path as defined as in section˜ 2.3. This sampling distribution is changed to a˜ new distribution f ∗(s). If g(s) is the property of interest, the original ˜ g(s)˜> 0 is very small when taking samples from problem is that the probability of observing ˜ f (s) . This probability should be significantly increased with samples from f ∗(s). The observa˜ tions made must be corrected because the samples s are from f ∗(s) and not f (˜s) . ˜ ˜ ˜ The quantity of interest γ = E(g(s)) can now be rewritten: ˜ f ( s) γ = E f (g(s)) = ∑ g(s) f (s) = ∑ g(s) ----------(5) ˜ - f ∗(s) = E f ∗(g(s) ⋅ L(s)) ∗ ˜ ˜ ˜ ˜ ˜ ˜ ˜ f ( s) ∀s ∀s ˜ ˜ ˜ where L(s) = f (s) ⁄ f ∗(s) is denote the likelihood ratio. Observe that the expected value of ˜ under ˜ f is equal ˜ observations to the expected value of the observations under f ∗ corrected for bias by the likelihood ratio, E f (g(s)) = E f ∗(g(s) ⋅ L(s)) . ˜ ˜ ˜ An unbiased estimator for γ taking samples s from f ∗(s) is: ˜ ˜ 1 R γˆ IS = --- ∑r = 1 g(s r)L(s r) (6) R ˜ ˜ with variance 1 1 Var(γˆ IS) = ---Var f ∗(g(s)L(s)) = ---E f ∗(( g(s)L(s) – γ ) 2) R R ˜ ˜ ˜ ˜ 3.2

(7)

Parameter biasing in semi-Markov simulation

The change of measure, i.e. the change of sampling distribution from f to f ∗ , is the critical factor with respect to importance sampling efficiency. This section illustrates one approach to change the measure in semi-Markov models. The is a scaling of the arrival and departure rates by a common BIAS-factor when the transition probabilities P x, y are determined. Let s be a sequence of events, and G = ˜

˜ ˜

∑k = 1 λk + ∑k = 1 µk , then K

K

f (s) is defined as: ˜

 λk ⁄ G =   µk ⁄ G

if y = x + 1 k ˜ ˜ (8) f (s) = ∏ P ωi, ωi + 1 with P x, y ˜ if y = x – 1k ˜ ˜ ˜ ˜ ∀( ω i ∈ s ) ˜ ˜ ˜ ˜ Changing the f (s) is done be changing the transition rates in determination of P x, y . The new ˜ ˜ ˜ sampling distribution is defined as: f ∗(s) = ˜ now G∗ =

∏

P∗ ω , ω

 λ∗ k ⁄ G∗ with P∗ x, y =  ˜ ˜  µ∗ k ⁄ G∗

˜i ˜i+1 ∀( ω i ∈ s ) ˜ ˜ K K λ∗ k + k = 1 µ∗ k . k=1

∑

∑

if y = x + 1 k ˜ ˜ if y = x – 1 k ˜ ˜

(9)

AEÜ, Specual issue on Rare Event Simulation

9

In section 3.2.2, heuristics is given for biasing of the sampling distribution with general arrival and departure rates in a semi-Markov model. For Erlang loss system an exact solution is proposed, see section 3.2.3. But, first the BIAS-factor is properly introduced. 3.2.1

The BIAS factor applied to arrival and departure rates

In this paper, a general biasing technique is applied where the transition rates in (9) are scaled according to a common BIAS -factor: λ∗ k = λ k ⋅ BIAS µ∗ k = µ k ⁄ BIAS

(10)

Hence, the optimization of the change of measure is reduced to a very convenient single BIASfactor problem. This is similar to the failure biasing techniques when only the transition probabilities are changed, see e.g. the survey in [Hei95]. With state dependent rates, the BIAS-factor is also state dependent, BIAS(ω) , i.e. the biasing ˜ targets, an indiwill change as the system state changes. Furthermore, if the system have several vidual BIAS factor for every target exists, BIAS k(ω) . Finally, as the special case of Erlang loss system in section 3.2.3 indicates, the BIAS factor is˜ generator dependent if the resource requirements c kj is different, BIAS jk(ω) . Hence, equation (10) is rewritten: ˜ λ∗ k(ω) = λ k(ω) ⋅ BIAS jk(ω) ˜ ˜ ˜ ∗ µ k(ω) = µ k(ω) ⁄ BIAS jk(ω) (11) ˜ ˜ ˜ The following section give heuristics for choosing this BIAS-factor. 3.2.2

Heuristics for biasing with general arrival and departure rates

The change of measure is optimal if the variance in (7) is minimised. This is generally very difficult task, see [Hei95] for a thorough survey of parameter biasing in importance sampling. However, for some specific systems, explicit (asymptotic) optimal solutions exists. For instance, with K = 1 and J = 1 (M/M/1/N-queue), an optimal solution exist, which is an interchange of the arrival and departure rates [PW89, CFM83], i.e. BIAS = µ ⁄ λ . Then, a positive drift is induced towards the target, Ω 1 = { N } . With state dependent transition rates, the factor is state dependent, i.e. BIAS(ω) = µ(ω) ⁄ λ(ω) , see [Buc90]. To extend these results, identify that the optimal solution interchange the transition probabilities. It can also be expressed as a reversion of the drift, where the drift of generator 1 is: Positive drift towards Ω 1

: δ 1+(ω) = λ 1(ω) ⋅ c 11 ˜ ˜ Negative drift towards Ω 1 : δ 1-(ω) = µ 1(ω) ⋅ c 11 (12) ˜ ˜ The optimal solutions to BIAS-factor for K = 1 and J = 1 imply a reversion of the drift of generator 1 towards target 1, i.e. : δ 1+∗(ω) = λ 1∗(ω) ⋅ c 11 = µ 1(ω) ⋅ c 11 = δ 1-(ω) ˜ ˜ ˜ ˜ Negative drift towards Ω 1 : δ 1-∗(ω) = µ 1∗(ω) ⋅ c 11 = λ 1(ω) ⋅ c 11 = δ 1+(ω) (13) ˜ ˜ ˜ ˜ Verify the optimal solutions by substitution of BIAS(ω) = µ(ω) ⁄ λ(ω) and (11) into (13). Positive drift towards Ω 1

AEÜ, Specual issue on Rare Event Simulation

10

Using this heuristics for K > 1 and J = 1 , then δ 1+∗(ω) = ˜ δ 1-∗(ω) = ˜

∑k = 1 ckj ⋅ λk∗(ω˜ ) = ∑k = 1 ckj ⋅ µk(ω˜ ) = δ1-(ω˜ ) K K ∑k = 1 ckj ⋅ µk∗(ω˜ ) = ∑k = 1 ckj ⋅ λk(ω˜ ) = δ1+(ω˜ ) K

K

=> BIAS(ω) = max(1, [ ∑k = 1 c kj µ k(ω) ] ⁄ [ ∑k = 1 c kj λ k(ω) ]) (14) ˜ ˜ ˜ is proposed as a rough but efficient approximation. It is obtained by making a superposition of the generators and ignoring the boundary conditions [Hee97a]. The maximum function is added to ensure that the drift towards the target is never reduced. K

K

In the remaining of this section, an adaptive parameter biasing is described where the case of K > 1 and J > 1 is treated. The optimisation of the bias parameter applies results from the large deviation theory and requires that one specific target exists [Buc90]. With J > 1 the model has multiple targets and each generator might have different constraint sets. The previous results no longer apply. Using the heuristics from (13) and study each target separately, then J individual “optimal” BIAS-factors can be provided: δ j+∗(ω) = ˜ ∗ δ j- (ω) = ˜

∑∀( k ∈ Γ ) ckj ⋅ λk∗(ω˜ ) = ∑∀( k ∈ Γ ) ckj ⋅ µk(ω˜ ) = δ j-(ω˜ ) ∑∀( k ∈ Γ ) ckj ⋅ µk∗(ω˜ ) = ∑∀( k ∈ Γ ) ckj ⋅ λk(ω˜ ) = δ j+(ω˜ ) j

j

j

j

=> BIAS j(ω) = max(1, [ ∑∀( k ∈ Γ ) c kj µ k(ω) ] ⁄ [ ∑∀( k ∈ Γ ) c kj λ k(ω) ]) j j ˜ ˜ ˜

(15)

Applying BIAS j(ω) to all generators in Γ j , see equation (11), gives an efficient change of ˜ a positive drift towards target j . This new distribution is denoted f ∗(s) . measure by inducing j ˜ Both (14) and (15) are less precise when the resource requirements, c kj , are significantly different for the generators in Γ j . Then, a generator dependent factor should be included, see the example in the following section. 3.2.3

Exact biasing for Erlang loss system

Let λ k(ω) = λ k ( M k infinite) and µ k(ω) = min(ω k, S k)µ k ( S k > 1 ), with K > 1 and J > 1 . ˜ Erlang loss system for which˜ an optimal scaling of the parameters with respect to a This is an dominating target exists. The infimum of following function must be obtained [Kel86]: –

∑

K λ k=1 k

⁄ µk ⋅ e

–

∑

c ∀( j ∈ Φ k ) kj

⋅ xj

+ ∑j = 1 x j ⋅ N j J

(16)

with respect to a single free scaling variable x j , letting all other scaling factors i ≠ j be x i = 0 . Then the optimal scaling, valid for large N j , with respect a single target j is obtained by: x j where

∑∀( k ∈ Γ ) ckj λk ⁄ µk ⋅ e

– c kj ⋅ x j

j

This solution is discussed in [Man96].

= Nj

(17)

AEÜ, Specual issue on Rare Event Simulation

11

The optimal scaling with respect to target j results in the following change of parameters: λ∗ k ⁄ µ∗ k = ( λ k ⁄ µ k )e

– c kj ⋅ x j

The proposed BIAS-factor is with λ k(ω) = λ k and µ k(ω) = min(ω k, S k)µ k : ˜ ˜ 2 2 λ∗ k(ω) ⁄ µ∗ k(ω) = λ k(ω) ⁄ µ k(ω) ⋅ BIAS j (ω) = λ k ⁄ ( min(ω k, S k)µ k ) ⋅ BIAS j (ω) ˜ ˜ ˜ ˜ ˜ ˜ To compare the two solutions, compare: – c kj ⋅ x j

(18)

(19)

2

with BIAS j (ω) ⁄ min(ω k, S k) (20) ˜ Observe that the scaling in (18) is state independent but dependent on the generator k , while the opposite applies for (19). e

The system example used for a feasibility demonstration in section 5, is an Erlang loss system, and hence (18) is applied in combination with the adaptive parameter biasing described in the following. 3.3

Parameter biasing in networks with a dominating target

As a general guideline to minimise the variance of γˆ IS , the new distribution, f ∗(s) , should be chosen to be proportional to the original sampling distribution and the property of˜ interest at every sample s : ˜ (21) g(s) f (s) ∝ f ∗(s) ˜ ˜ ˜ This is the objective of the parameter biasing and it indicates that the change of measure should increase the frequency of the most important of the unlikely paths leading to a rare event of interest. The importance of a path s is given by the product g(s) f (s) . ˜ ˜ ˜ Let the property of interest depend on the performance of a more than one resource pool, i.e. J > 1 . This means that the model consists of multiple target sub-spaces, Ω j . This is, as pointed out in the previous section, a challenge because existing optimisation no ˜longer apply. According to [Buc90], the results from large deviation theory applies only when K = 1 and J = 1 . The approximation is applicable when the number of generators is more than one, K > 1 . The heuristics is that the same approximation can be applied to change the measure successively with respect to every target in the model. Each change of measure must satisfy: (22) g j(s) f (s) ∝ f j∗(s) ˜ ˜ ˜ which means that the optimal change of measure towards target j , f j∗(s) , must be proportional ˜ to the property of interest with respect to target j and the original sampling distribution of s . ˜ Under some restrictive conditions the relation in (22) directly applies to obtain efficient importance sampling simulation: 1. A single dominating target: If the overall property of interest is dominate by a single target, j max , i.e. g(s) ≈ g max(s) , then an efficient change of measure is [FA94, Man96, PW89]: ˜ ˜ ∗ ∗ (23) f (s) = f jmax (s) = max ∀ j( f j∗(s)) ˜ ˜ ˜ J 2. Disjoint target sub-spaces: If the targets constitutes of disjoint sub-spaces, ∩ j = 1 Ω j = ∅ , ˜

AEÜ, Specual issue on Rare Event Simulation

12

then independent simulation experiments can be conducted. Each experiment have a biasing towards one single target [Fra93]. Using stratified sampling, each target is a strata: Conduct R j experiments with samples from f j∗(s) , 1 ≤ j ≤ J , where R j = p j ⋅ R (24) ˜ The probability of strata j , p j , is the relative probability of visit to target j given that a visit to a target have occurred. In a more realistic system, the various target sub-spaces are overlapping and visits to each of them contributes significantly to the overall performance. Neither of the two approaches above will apply, nor will the use of an all generator biasing. The latter biasing assumes that all K generators meet only one and common constraint and uses equation (14). Experiments have shown that this results in samples of unlikely paths because no single target is in focus. Instead, a new parameter biasing strategy is proposed that will be described in more detail in the following section. 3.4

Adaptive parameter biasing in well engineered networks

In a realistic system, the overall property of interest has significant performance contributions from multiple resource pools. The target sub-spaces associated with these resource pools will J generally be overlapping, i.e. ∩ j = 1 Ω j ≠ ∅ . ˜ A biasing strategy is proposed that applies to models under these conditions. The idea was first presented in [Hee95], and later improved in [Hee96] and generalised in [Hee97a]. The idea is to assign importance to each target, including both the probability of a path s and the contribution to the overall performance from this target. In the following section the ˜target importance is discussed in more detail. The relative target importance is basis to decide from which distribution the next event in a path s should be sampled. Since the relative importance will change as the system state changes, the˜ change of measure must adapt the current state. Importance sampling indicates in general that it should be more likely to visit the most important target j than the others. The heuristics is then that this should be applied at every step in a simulation. The target j should not be chosen once for the entire path, giving a fixed change the measure during s . Instead, a new target must be sampled at every state according to the changes in the relative˜importance. The f ∗(s) will change, correspondingly. Hence, this can be considered as ˜ adapts to the current system state, ω , by assigning a sampling having a change of measure that distribution that is proportional to the current value of the property of˜ interest, and the probability of the remaining path s' given start at ω , i.e. ˜ ˜ (25) g(s' ω) f (s' ω) ∝ f ∗(s' ω) ˜ ˜ ˜ ˜ ˜ ˜ The parameter biasing will always induce a positive drift towards a single target only, but not necessary to the same target along the entire path. As a summary, the following algorithm is proposed to accomplish this: (i) Choose at random a target j from the current relative target importance at state ω , (ii) Bias the parameters to induce a positive drift towards this target, i.e. change to f ˜j∗(s) , ˜ (iii) Move to the next state with the new set of parameters. For every state ω in the path s , the algorithm is repeated until a target is visited or the experi˜ ˜ e.g. by completion ment is terminated, of a cycle.

AEÜ, Specual issue on Rare Event Simulation

13

In [Car91] a similar idea was proposed, denote failure distance biasing. The importance is then the number of failure transitions from current state to the nearest system failure. This is obtained from a failure tree established initially before the simulation is started. All system failures are equally severe. Correspondingly, in the adaptive biasing technique the importance is the probability of these sequences instead of the number of steps, multiplied by the contribution observed at this system failure (target) instead of assuming all failures equal. The adaptive biasing algorithm depends on the existence of a good estimate of the relative importance. In the following section it is described how this target distribution is obtained. 3.5

Target distribution

The importance of a target is denoted the target importance, g j(s) f (s) , which consists of the tar˜ ˜importance between targets get contribution, g j(s) , and the target likelihood, f (s) . The relative ˜ is denoted the target˜distribution, p j(ω) . ˜

Ω 10

ω3

ω2

1

Target 2

Target 3 10-1 10-2

1 23 g1=0.1

Target 1

ω1

ω0 (origin)

10-3 10-4 10-5

Figure 6 Changes in target distribution illustrated by iso-likelihood curves, an example.

As previously mentioned, the target likelihood is dependent on current state ω , and hence the ˜ relative importance that determines the target distribution will change as the simulation evolves along a path in Ω . Figure 6 shows a state space consisting of 3 target sub-spaces. The subspaces are made˜disjoint to simplify the illustration. The contour lines represent the “iso-likelihoods” of each target, i.e. every state at a specific contour line have the same conditional target likelihood. The contour intervals are logarithmic with base 10, i.e. the iso-likelihoods represent probabilities at every order of magnitude, i.e. 10-1, 10-2, 10-3, etc. The path in the figure, s = { ω 0, ω 1, ω 2, ω 3 } , demonstrates how the relative importance ˜ have ˜ the˜ same ˜ importance ˜ changes. At ω 0 all targets and the sampling distribution will be cho˜ sen by sampling a target from a uniform distribution, see histogram assign to the system state ω 0 in the figure. Stepping to ω 1 , the picture is changed. Now the process has approached ˜ ˜ target 1, and the target distribution reflects this by increasing the probability of target 1 relative to target 2 and 3, see histogram. This means that it is now most likely that the sampling distribution which is optimal with respect to target 1 is chosen, f 1∗(s) . Correspondingly, at state ω 2 ˜ ˜ and ω 3 , the most likely targets are 2 and 3, respectively. The numerical values of this example ˜ found in table 3, and the resulting target distribution in table 4. can be Generally, the conditional iso-likelihoods does not have the nice circular contour lines as the example in figure 6. Hence, it is not a trivial matter to determine the target importance and the

AEÜ, Specual issue on Rare Event Simulation

14

Table 3 The target importance from figure 6. State

ω0 ˜ ω1 ˜ ω2 ˜ ω3 ˜

g 1(s) ⋅ f 1(s ω) ˜ ˜ ˜

g 2(s) ⋅ f 2(s ω) ˜ ˜ ˜

g 3(s) ⋅ f 3(s ω) ˜ ˜ ˜

Σ

0.1 * 1.0E-03 = 1.0E-04

1.0 * 1.0E-04 = 1.0E-04

10 * 1.0E-05 = 1.0E-04

3.00E-04

0.1 * 1.0E-02 = 1.0E-03

1.0 * 6.8E-04 = 6.8E-04

10 * 6.8E-08 = 6.8E-07

1.68E-03

0.1 * 3.5E-03 = 3.5E-04

1.0 * 1.0E-02 = 1.0E-02

10 * 1.4E-07 = 1.4E-06

1.04E-02

0.1 * 2.9E-03 = 2.9E-04

1.0 * 3.8E-04 = 3.8E-04

10 * 1.0E-04 = 1.0E-03

1.67E-03

Table 4 The target distribution from the relative importances in table 3. State

p 1(ω)

p 2(ω)

p 3(ω)

ω0 ˜ ω1 ˜ ω2 ˜ ω3 ˜

0.33

0.33

0.33

0.60

0.40

~0

0.03

0.97

~0

0.02

0.27

0.71

corresponding target distribution. Normally, an approximation for this distribution must be obtained, with the following objectives: - It must be close to the exact target distribution. Observe that this requires only determination of the relative, not the absolute, importance. - The solution must be robust to changes in parameters and state space structures. - It must be very computer efficient, because it will be invoked at every state along the path. Several algorithms and approximations have been proposed since the adaptive biasing was introduced in [Hee95]. Through relaxation of restrictions and extensions in the model, both the solutions in [Hee96] and [Hee97a] have been improved. The current solution are described in this section. Consider a single target j , 1 ≤ j ≤ J . The first step is to determine the most likely path from the current state ω up to any state in the target subspace Ω j . Denote this as the probe path of target ˜ j , s j(ω) . Obviously, only generators in the constraint˜ set of this target, Γ j , are allowed to ˜ change˜ state in the probe path. The idea is to apply the probe path to construct a state model of the resource allocations of pool j . This is considered to be a mapping, or “projection”, from the multiple dimensional model spanned by the generators in Γ j , onto a single dimensional model representing the resource allocations of pool j . In figure 7, a state space model is described where the generator constraint set Γ j contains two generators with resource requirements, c 1 j = 1 and c 2 j = 3 , respectively. The example in this figure resulted in a probe path s j(ω) , see [Hee98] for the ˜ ˜ 2, followed by 3 arrivals details. The probe path in the example contains one arrival of generator of generator 1. This gives a state model of resource pool j as in figure 7. First arrival has intensity λ 2 and allocates c 2 j = 3 resources, next arrival as intensity λ 1 and allocates c 1 j = 1 resource, and so on. From this resource state model, the target likelihood is assigned to the conditional steady state probability of blocking in resource pool, π j(N j ω) . ˜

AEÜ, Specual issue on Rare Event Simulation

λ2

15

Target subspace, Ωj

{#e1,#e2}

2µ2 λ

λ1 λ1 end state µ1 2µ1 3µ1 λ2 λ2 λ2 λ2 µ2 λ µ λ µ2 λ µ2 λ λ1 2 1 1 1 1 µ1 4µ1 5µ1 2µ1 3µ1

Probe path, sj(ω)

1

λ1 6µ1

c1j =1, c2j =3

start state, ω “Projection” on to resource pool

0

µ2

1

2

λ2

3µ

1

λ1

λ 4 2µ 1 5 1

λ1 6 3µ1

{#resource allocated} = {c1j*#e1 + c2j*#e2}

Figure 7 A projection onto a resource pool state space from the probe path, an example.

The i ’th step of defining the probe path is expressed as: ω i = ω i – 1 + 1 k (ω ) ˜ ˜ ˜ j ˜i–1

(26)

where k j(ω i) = { k max ∀( j ∈ Γ j )(π(ω i + 1 k)) } ˜ ˜ ˜ –1 π(ω i + 1 k) = π(ω i) ⋅ λ k(ω) ⁄ µ k(ω) ⋅ G i (27) ˜ ˜ ˜ ˜ ˜ i with normalisation constant G i = ∑x = 0 π(ω x) and initial condition π(ω 0 = ω) = 1 . Observe ˜ that the probe path only contains resource request events. This is because˜ it is ˜expected that the most likely path to a target only contains a combination of request events from the generators within Γ j . Any release event will reduce this likelihood. The target contribution, g(s j(ω)) , is the contribution of the probe path. For instance, if the time ˜ blocking is the property of˜interest and ω e is the end state of the probe path, –1 K K g(s j(ω)) = ( ∑k = 1 λ k(ω e) + ∑k = 1 µ k(˜ω e) ) , i.e. the expected state sojourn time at ω e . ˜ ˜ ˜ ˜ ˜ To summarize the target distribution approximation: 1. Determine the probe path, s j(ω) , from current state ω up to an end state ω e in Ω j . ˜ ˜ allocations ˜ 2. Let the sequence of resource in s j(ω) be a˜ projection from the ˜state space ˜ ˜ spanned by Γ j onto a single dimensional model of the resource pool. 3. Assign the target likelihood of j to the steady state probability of blocking in resource pool j , π j(N j ω) . 4. Obtain ˜the target contribution, g(s j(ω)) ,for the probe path s j(ω) . ˜ 5. Repeat step 1-4 for all pools 1 ≤ j˜ ≤ ˜J , and assign the target˜ distribution J p j(ω) = ( π j(N j ω) ⋅ g(s j(ω)) ) ⁄ ∑ j = 1 ( π j(N j ω) ⋅ g(s j(ω)) ) . ˜ ˜ ˜ ˜ ˜ ˜ ˜ Even though a target is chosen after every state change, feasibility experiments have shown that the resulting sequence of events is far more directed towards the targets than for experiments with biasing towards all targets simultaneously. It is also observed that the targets are visited nearly in accordance to their relative contributions to the system performance, whenever a good estimate of the target distribution exists. See section 5 for a feasibility demonstration.

AEÜ, Specual issue on Rare Event Simulation

16

Introducing rerouting will not affect the determination of the target distribution directly. The current routing must be identified, and the target distribution is assigned according to these constraints. Priorities will change the target distribution because some generators can be pre-empted. Instead of obtaining the probe path from the current state, the start state of the probe path is changed to a state where all lower priority generators were pre-empted. i.e.  #e k K ω = { ω k } k = 1 where ω k =  ˜ 0

if k ∈ ζ 0

.

(28)

otherwise

From this state, the probe path is obtained as describe in this section, irrespective of the different priorities of the generators. See [Hee98] for further details. 4

SIMULATION SETUP FOR REGENERATIVE CYCLES IN LARGE MODELS

When conducting a simulation experiment with importance sampling it is recommended to divide the experiment into independent regenerative cycles for the stability of results. A regenerative state, or regenerative subspace, must be defined. For dependability models and very small traffic models, this is normally a single state like the state “system intact” or “empty system”. However, for real-sized traffic models, choosing a single regenerative state, will make the expected regenerative cycle period too long. This paper introduces the concept of a regenerative box of K dimensions, denoted Ω 0 . Within ˜ this box, the regenerative cycles starts and ends. The box should contain the equilibrium states of the system, i.e. the sub-state space in which the simulation process spend most of its time. Furthermore, the regenerative box must be disjoint with all target subspaces, Ω 0 ∩ Ω j , for all j . ˜ ˜ process This is because no observations with respect to a target is recorded when the simulation is within Ω 0 . A similar concept denoted load cycles was applied in [HH94]. ˜ The identification of the regenerative box and the expected cycle time related to it, are not known in advance. Hence, a pre-simulation is required, consisting of the following phases: 1. Regenerative box identification is made by a block simulation experiment where the equilibrium states are identified. 2. Relative distribution of the states within the regenerative box is obtained by a second block simulation. 3. Regenerative cycle estimation. The cycle length is defined as the length between two departures from the regenerative box (identified after phase 1). Each new cycle starts at an arbitrary state within Ω 0 , sampled from the box distribution estimated after phase 2. Otherwise, the cycle lengths˜will not be independent. The pre-simulation is much less computer demanding than the main simulation. In the main simulation, importance sampling and the adaptive parameter biasing are applied. The target observations are made. The complete algorithm of the main simulation consist of the following steps (see figure 8 for an example of a cycle): - Sample an initial state (state A) according to the relative distribution of the states within the regenerative box. - Start a regenerative cycle with importance sampling at the first departure from the box (state B).

AEÜ, Specual issue on Rare Event Simulation

17

1. DEPARTURE: IMPORTANCE SAMPLING ACTIVE

1

OBSERVATIONS STARTS

C

3

TRANSIENT PERIOD D

B STARTING STATE

IMPORTANCE SAMPLING TURNED OFF

TRAGET OBSERVATION CYCLE ENDS A

E

REGENERATIVE BOX

2. DEPARTURE: REGENERATIVE CYCLE ENDS

2

Figure 8 A target observation cycle, an example.

- Turn off importance sampling biasing if a visit to a target is observed (state C). - End the cycle at first return to the regenerative box (state D). This will save simulation time without loss of target observations, because no targets can be observed within the regenerative box, i.e. in the remaining of the regenerative cycle (from state D to state E in the figure). 5

FEASIBILITY DEMONSTRATION

As a feasibility demonstration of the flexibility of the proposed simulation framework, this section includes a study of service interactions in a system example. Two traffic sources with the same priority and routing strategy have different resource requirements. The relative blocking is studied. In section 5.1, the system is described. The results in section 5.2 show the blocking for the two probe sources, also influenced by high priority pre-emptive traffic with rerouting, and by a link failure. The disconnection and the rerouting probabilities are also estimated. 5.1

System description

The objective of this study is to demonstrate the feasibility of the proposed simulation framework. The time blocking, the disconnection probability, and rerouting probability, are estimated. The network carries traffic between all nodes. Each pair of nodes has two traffic classes with different priority levels. The high priority class has alternative routing on overload and link failure, while the low priority class has only one route. High priority traffic may pre-empt low priority traffic. Furthermore, the traffic classes with the same origin and destination node have the same mean traffic, but the high priority traffic has larger capacity requirements than low priority traffic. A link failure is considered to be a “call” pre-empting all other calls and allocating all resources. Table 5 contains an overview of the 4 different generator types applied in this example, see figure 9 for an illustration of the network. Type 1 is the high priority traffic generators, type 2 and 3 are the low priority generators with different resource requirements, and type 4 is the generator causing link failures. The complete model consists of 32 generators, 15 of type 1, 15 of type 2, 1 of type 3, and 1 of type 4. The capacity of the 10 resource pools are N={33, 33, 37, 27, 25, 30, 28, 30, 21, 37}. For further details, see [Hee98].

AEÜ, Specual issue on Rare Event Simulation

18

OBJECTIVE: study service interaction, i.e. - relative blocking, or - relative disconnection between different generators affected by high priority traffic or failures in links/nodes.

4

type 1

6

pe ty 2

3

9

3 4

type 3

2

8 7 5

1

1

type 4

5

2 3

10

6

Figure 9 Description of generator types in the evaluation example Table 5 Generator types Type

Priority

Res.req.

Comments

1

1

4

rerouting on overload and failure

2

2

1

3

2

5

4

0

33 (all resources)

5.2

link failure

Results

Examples of results that can be obtained by the simulation experiments are given in table 6, 7, and 8. Each experiment contains 10,000 regenerative cycles, and the CPU time consumption varied from 50 minutes to nearly 4 hours on a 120 Mhz Sun sparc station. 5.3

Observations

Exact values have not yet been provided for comparisons with the simulation results. However, the blocking results are in the correct order of magnitude relative to what the network was dimensioned for. The blocking results in table 6 have a standard error to mean ratio in the range of 0.22-0.36 for generator 23 (with resource requirement of 1) and 0.14-0.24 for generator 31 (with resource requirement of 3). The importance sampling parameters were chosen to provoke blocking in generator 23 and 31. This mean that is should be expected less precision in the disconnection and rerouting estimates, see table 7 and 8, respectively, for results that confirms this. The mean likelihood ratio is close to the expected value 1 for all 3 experiments. This, combined with the precision of the estimates, implies that it is feasible to use the proposed simulation framework for simulations of complex networks as the example in this example.

AEÜ, Specual issue on Rare Event Simulation

19

Table 6 Time blocking for the probe sources. Generator 23

Generator 31

Influenced by Mean

Std.err.

Mean

Std.err.

None

3.10 E-09

1.12 E-09

3.05 E-08

0.45 E-08

High priority traffic

3.86 E-06

0.88 E-06

1.47 E-05

0.24 E-05

High priority traffic and link failure

4.54 E-06

1.53 E-06

2.22 E-05

0.53 E-05

Table 7 Disconnection probabilities Generator 23

Generator 31

Disconnected by Mean

Std.err.

Mean

Std.err.

High priority traffic

6.07 E-07

4.35 E-07

3.02 E-05

1.13 E-05

High priority traffic and link failure

1.88 E-06

1.19 E-06

1.56 E-05

0.73 E-05

Table 8 Rerouting of high priority traffic [mean number per regenerative cycle] Rerouting due to overload only

Rerouting due to overload and failure on link 1

No of connections

Mean

Std.err.

Mean

Std.err.

Mean

Std.err.

1

1.00 E-07

1.00 E-07

1.43 E-06

0.78 E-06

0.684

0.014

2

2.22 E-07

1.43 E-07

8.28 E-07

4.42 E-07

0.803

0.016

3

1.18 E-07

0.79 E-07

1.02 E-06

0.47 E-06

0.586

0.013

9

2.06 E-08

1.53 E-08

8.46 E-07

4.78 E-07

1.282

0.023

High priority generator in Γ 1

6

CLOSING COMMENTS

In a well engineered network, previous biasing schemes for optimisation of importance sampling parameters are no longer efficient. This paper addresses this problem and proposes an adaptive strategy for parameter biasing which seems robust and flexible with respect to the network size and different viewpoints. The paper includes the following contributions: 1. A flexible framework for both traffic and dependability simulation models. 2. An adaptive parameter biasing for importance sampling: i. parameter biasing in semi-Markov models with multiple dimensions, ii. heuristics for reducing a multiple target model to single target model according to the target importance, adapting to the changes in system state during simulation, iii. estimation of the relative target importance which is essential for the success of the adaptive scheme. 3. The use of regenerative simulation in large network models: i. pre-simulations to identify a regeneration box of equilibrium states ii. regenerative simulation with importance sampling and adaptive parameter biasing generating cycles completed by states within the regeneration box.

AEÜ, Specual issue on Rare Event Simulation

20

The paper includes a demonstration of the feasibility of the proposed simulation framework with importance sampling and the adaptive parameter biasing. The results of this example show that it is feasible to use this framework for network simulations with different service requirements, pre-emptive priorities, and different rerouting strategies. Further investigations of the robustness and efficiency of all aspects of the adaptive biasing should be carried out, and possible improvements should be identified. The RESTART technique [VAVA91] should be considered for simulation of networks with the characteristics given in this paper. The problem with identifications of RESTART states in multiple dimensions must be solve [Hee97b]. A possible solution is to define “meta-event” like disconnection, rerouting, failures, etc. and use these as RESTART state definitions. It is interesting to study the efficiency, either a stand-alone RESTART experiment, or combined with the adaptive importance sampling scheme described in this paper. References [Buc90] James A. Bucklew. Large Deviation Techniques in Decision, Simulation, and Estimation. Wiley, 1990. [Car91] Juan A. Carrasco. Failure distance based simulation of repairable fault-tolerant computer systems. In G. Balbo and G. Serazzi, editors, Proceedings of the Fifth International Conference on Computer Performance Evaluation. Modelling Techniques and Tools, pages 351 – 365. North-Holland, Feb. 1517 1991. [CFM83] Marie Cottrell, Jean-Claude Fort, and Germard Malgouyres. Large deviation and rare events in the study of stochastic algorithms. IEEE Transaction of Automatic Control, AC-28(9):13–18, September 1983. [FA94] Michael R. Frater and Brian D.O. Anderson. Fast simulation of buffer overflows in tandem networks of GI/GI/1 queues. Annals of Operation Research, 49:207–220, 1994. [Fra93] Michael R. Frater. Fast simulation of buffer overflows in equally loaded networks. Australian Telecom Research, 27(1):13–18, 1993. [Hee95] Poul E. Heegaard. Rare event provoking simulation techniques. In Proceeding of the International Teletraffic Seminar (ITS), pages 17.0–17.12, Bangkok, Thailand, 28 Nov-1 Dec 1995. Session III: Performance Analysis I, Regional ITC-Seminar. [Hee96] Poul E. Heegaard. Adaptive optimisation of importance sampling for multi-dimensional state space models with irregular resource boundaries. In Peder J. Emstad, Bjarne E. Helvik, and Arne H. Myskja, editors, The 13th Nordic Teletraffic Seminar (NTS-13), pages 176–189, Trondheim, Norway, 20 - 22 August 1996. Tapir Trykk. [Hee97a] Poul E. Heegaard. Efficient simulation of network performance by importance sampling. In 15th International Teletraffic Congress (ITC15), Washington D.C., USA, June 23-27 1997. [Hee97b] Poul E. Heegaard. Speed-up techniques for discrete event simulations. AEÜ, 1997. Submitted. [Hee98] Poul E. Heegaard. Efficient simulation of network performance by importance sampling. PhD thesis, Norwegian University of Science and Technology, 1998. In preperation. [Hei95] Philip Heidelberger. Fast simulation of rare events in queueing and reliability models. ACM transaction on modeling and computer simulation, 5(1):43–85, January 1995. [HH94] Bjarne E. Helvik and Poul E. Heegaard. A technique for measuring rare cell losses in ATM systems. In J. Labatoulle and J.W. Roberts, editors, The Fundamental Role of Teletraffic in the Evolution of Telecommunications Networks, pages 917–930, Antibes Juan-les-Pins, France, June 6-10 1994. Elsevier. [Kel86] F. Kelly. Blocking probabilities in large circuit switched networks. Advances in applied probability, 18:473–505, 1986. [Man96] Michel Mandjes. Rare event analysis of communication networks. PhD thesis, Vrije Universiteit, December 1996. [PW89] Shyam Parekh and Jean Walrand. Quick simulation of excessive backlogs in networks of queues. IEEE Transaction of Automatic Control, 34(1):54–66, 1989. [VAVA91] Manuel Villen-Altamirano and Jose Villen-Altamirano. RESTART: A method for accelerating rare event simulation. In C. D. Pack, editor, Queueing Performance and Control in ATM, pages 71 – 76. Elsevier Science Publishers B. V., June 1991.