Tài liệu Queueing mạng lưới và chuỗi Markov P3 pdf

It is worthwhile pointing out that for any given ergodic CTMC, a DTMC can be constructed that yields an identical steady-state probability vector as the CTMC, and vice versa.. If, on the

A comprehensive source on algorithms for steady-state solution techniques is the book by Stewart [Stew94]

From Eq (2.15) and Eq (2.58), we have v = VP and 0 = nQ, respective-

ly, as points of departure for the study of steady-state solution techniques

Eq (2.15) can be transformed so that:

Therefore, both for CTMC and DTMC, a linear system of the form:

needs to be solved Due to its type of entries representing the parameters

of a Markov chain, matrix A is singular and it can be shown that A is of rank n - 1 for any Markov chain of size IS’] = n It follows immediately that the resulting set of equations is not linearly independent and that one of the equations is redundant To yield a unique, positive solution, we must impose

a normalization condition on the solution x of equation 0 = xA One way to approach the solution of Eq (3.2) is to directly incorporate the normalization condition

Queueing Networks and Markov Chains

Gunter Bolch, Stefan Greiner, Hermann de Meer, Kishor S Trivedi

Copyright  1998 John Wiley & Sons, Inc Print ISBN 0-471-19366-6 Online ISBN 0-471-20058-1

into the Eq (3.2) This can be regarded as substituting one of the columns (say, the last column) of matrix A by the unit vector 1 = [I, 1, , llT With

a slight abuse of notation, we denote the new matrix also by A The resulting linear system of non-homogeneous equations is:

An alternative to solving Eq (3.2) is to separately consider normalization

Eq (3.3) as an additional step in numerical computations We demonstrate both ways when example studies are presented It is worthwhile pointing out that for any given ergodic CTMC, a DTMC can be constructed that yields an identical steady-state probability vector as the CTMC, and vice versa Given the generator matrix Q = [qij] of a CTMC, we can define:

where Q is chosen such that Q > max;,jcs )qij] Setting q = maxi,jeS lqijJ should be avoided in order to assure uperiodicity of the resulting DTMC [GLT87] Th e resulting matrix P can be used to determine the steady-state probability vector x = u, by solving u = VP and vl = 1 This method, used

to reduce a CTMC to a DTMC, is called randomization or sometimes uni- formixation in the literature If, on the other hand, a transition probability matrix P of an ergodic DTMC were given, a generator matrix Q of a CTMC can be defined by:

2Modulo round-off errors resulting from finite precision arithmetic

SYMBOLIC SOLUTION: BIRTH-DEATH PROCESS 105

Fig 3.1 Birth-death process

Problem 3.1 Show that P - I has the properties of a CTMC generator matrix

Problem 3.2 Show that Q/q+1 has the properties of a stochastic matrix

Problem 3.3 Define a CTMC and its generator matrix Q so that the corresponding DTMC would be periodic if randomization were applied with

q = rnaxi,jeS 1qijl in Eq (3.5)

3.1 SYMBOLIC SOLUTION: BIRTH-DEATH PROCESS

Birth-death processes are Markov chains where transitions are allowed only between neighboring states We treat the continuous time case here, but analogous results for the discrete-time case are easily obtained

A one-dimensional birth-death process is shown in Fig 3.1 and its generator matrix is shown as Eq (3.7):

The transition rates Xk, Ic 2 0 are state dependent birth rates and ~11, L 2 1, are referred to as state dependent death rates Assuming ergodicity, the steady- state probabilities of CTMCs of the form depicted in Fig 3.1 can be uniquely determined from the solution of Eq (2.58):

Solving Eq (3.8) f or ~1, and then using this result for substitution with

lc = 1 in Eq (3.9), and solving it for 7r2 yields:

7Tl = -7r0, IT2 = -

plP2ro*

(3.10)

Eq (3.10) together with Eq (3.9) suggest a general solution of the following form:

(3.11)

Indeed, Eq (3.11) provides the unique solution of a one-dimensional birth- death process Since it is not difficult to prove this hypothesis by induction,

we leave this as an exercise to the reader From the law of total probability,

xi ri = 1, we get for the probability ~0 of the CTMC being in State 0:

1

co k-l l+ ,F;, go ik

Problem 3.4 Consider a discrete-time birth-death process with birth prob- ability ‘bi, the death probability di, and no state change probability 1 - bi - di

in state i Derive expressions for the steady-state probabilities and conditions for convergence [Triv82]

Section 3.1 shows that an infinite state CTMC (or DTMC) with a tridiag- onal matrix structure can be solved to obtain a closed-form result In this section we consider two other infinite state DTMCs However, the structure

is more complex so as to preclude the solution by “inspection” that we adopt-

ed in the previous section Here we use the method of generating functions (or z-transform) to obtain a solution The problems we tackle originate from non-Markovian queueing systems where the underlying stochastic process is Markov regenerative [Kulk96] One popular method for the steady-state anal- ysis of Markov regenerative processes is to apply embedding technique so as to produce an embedded DTMC from the given Markov regenerative process An alternative approach is to use the method of supplementary variables [Hend72]

We follow the embedded DTMC approach

HESSENBERG MATRIX: NON-MARKOVIAN QUEUES 107

We are interested in the analysis of a queueing system, where customers arrive according to a Poisson process, so that successive interarrival times are independent, exponentially distributed random variables with parameter X The customers experience service at a single server with the only restriction

on the service time distribution being that its first two moments are finite Order of service is first-come-first-served and there is no restriction on the size

of the waiting room We shall see later in Chapter 6 that this is the M/G/l queueing system Characterizing the system in such a way that the whole history is summarized in a state, we need to specify the number of customers

in the system plus the elapsed service time received by the current customer

in service This description results in a continuous state stochastic process that is difficult to analyze

But it is possible to identify time instants where the elapsed time is always known so they need not be explicitly represented A prominent set of these time instants is given by the departure instants, i.e., when a customer has just completed receiving service and before the turn of the next customer has come

In this case, elapsed time is always zero As a result, a state description given

by the number of customers is sufficient Furthermore, because the service time distribution is known and arrivals are Poissonian, the state transition probabilities can be easily computed It is not difficult to prove that the stochastic process defined in the indicated way constitutes a DTMC This DTMC is referred to as embedded into the more general continuous state stochastic process

Conditions can be identified under which the embedded DTMC is ergodic, i.e., a unique steady-state pmf does exist In Section 3.2.1 we show how the steady-state probability vector of this DTMC can be computed under given constraints Fortunately, it can be proven that the steady-state pmf of the embedded DTMC is the same as the limiting probability vector of the original non-Markovian stochastic process we started with The proof of this fact relies on the so-called PASTA theorem, stating that “Poisson arrivals see time averages” [Wolf82] The more difficult analysis of a stochastic process of non-Markovian type can thus be reduced to the analysis of a related DTMC, yielding the same steady-state probability vector as of the original process

The service times are given by independent, identically distributed (i.i.d.) random variables and they are independent of the arrival process Also, the first moment E[S] = 9 and the second moment E [S2] = s2 of the service time S must be finite Upon completion of service, the customers leave the system

To define the embedded DTMC X = {X,; n = 0,l .}, the state space is chosen as the number of customers in the system (0, 1, } As time instants where the DTMC is defined, we select the departure epochs, that is, the points

in time when service is just completed and the corresponding customer leaves the system

Consider the number of customers X = {X,; n = 0, 1, } left behind by

a departing customer, labeled n Then the state evolution until the next epoch n + 1, where the next customer, labeled n + 1, completes service, is probabilistically governed by the Poisson arrivals

Let the random variable Y describe the number of arrivals during a service epoch The following one-step state transitions are then possible:

we uncondition using the service time distribution:

cm

ak = I

The transition probability matrix P of the embedded stochastic process, which

is a Hessenberg matrix, is then given by:

HESSENBERG MATRIX: NON-MARKOVIAN QUEUES 109

We know from Section 2.1.2.1 that the states of an irreducible DTMC are all of the same type Therefore, positive recurrence of state 0 implies positive recurrence of the DTMC constructed in the indicated way A formal proof of the embedded DTMC being positive recurrent if and only if Relation (3.18) holds has been given, for example, by Cinlar [Cin175] We also know from Section 2.1.2.1 that irreducible, aperiodic, and positive recurrent DTMCs are ergodic Therefore, the embedded DTMC is ergodic if Relation (3.18) holds Assuming the DTMC to be ergodic, the corresponding infinite set of global balance equations is given as:

by B(t) and its LST B”(s) be given by:

00 B'(s) =

s eestdB(t)

G(x) = euk,zk = u()Eakz' + ~A~Yiak-i+lZk,

k=O k=O k=Oi=l

= ‘OGA(X) + (G(z) - VO)GA(Z), or:

G(4 x - GA(X) = u bG&) - GA(d) 7

coo0

=

cs j=o 0 e~“t~&?(t)zi

= i-(x(1 - 2))

Thus, the generating function for the {aj} sequence is given by the LST of the service time distribution at X(1 - z) Hence, we get an expression for the generating function of the DTMC steady-state probabilities:

G(z) = u zB-(X(1 - ‘d> - B-(X(1 - ‘>>

2 - &(X(1 - 2)) B-(X(1 - z))(z - 1)

HESSENBERG MATRIX: NON-MARKOVIAN QUEUES 111

The generating function in Eq (3.25) allows the computation of the infinite steady-state probability vector u of the DTMC embeđed into a continuous- time stochastic process at the departure epochs These epochs are given by the time instants when customers have just completed their generally distributed service period The steady-state probabilities are obtained by repeated dif- ferentiations of the probability generating function G(x), evaluated at x = 1:

(3.26)

If we set z = 1 on the right hand side of Eq (3.25) and since GẴ) = ET=, aj = 1, we have O/Ọ Differentiating the numerator and the denomina- tor, as per L’Hospital’s rule, we obtain:

or:

Taking the derivative of G(z) and setting z = 1, we get the expected number

of customers in the system, E[X] = x, in steady state:

x=x9+ x2s2

Equation (3.28) is known as the Pollacxek-Khintchine formulạ Remember that this formula has been derived by restricting observation to departure epochs It is remarkable that this formula holds at random observation points

in steady state, if the arrival process is Poisson The proof is based on the PASTA theorem by Wolff, stating that “Poisson arrivals see time averages” [Wolf82]

We refrain from presenting numerical examples at this point, but refer the reader to Chapter 6 where many examples are given related to the stochastic process introduced in this section

Problem 3.5 Specialize Eq (3.28) above to:

(a) Exponential service time distribution with parameter 1-1

(b) Deterministic service time with value l/p

(c) Erlang service time distribution with mean service time l/p and Ic phases

In the notation of Chapter 6, these three cases correspond to M/M/l, M/D/l, and M/EI, / 1 queueing systems, respectively

Problem 3.6 Given a mean interarrival time of 1 second and a mean service time of 2 seconds, compute the steady-state probabilities for the three queueing systems M/M/l, M/D/l, and M/Ek/l to be idle Compare the results for the three systems and comment

Problem 3.7 Show that the embedded discrete-time stochastic process

X = {X,;n = O,l, .} defined at the departure time instants (with transition probability matrix P given by Eq (3.17)), forms a DTMC, i.e., it satisfies the Markov property Eq (2.2)

Problem 3.8 Give a graphical representation of the DTMC defined by transition probability matrix P in Eq (3.17)

Problem 3.9 Show that the embedded DTMC X = {X,;n = O,l, } defined in this section is aperiodic and irreducible

Problem 3.10 Consider a single-server queueing system with indepen- dent, exponentially distributed service times Furthermore, assume an arrival process with independent, identically distributed interarrival times; a general service time distribution is allowed Service times and interarrival times are also independent In the notation of Chapter 6, this is the GI/M/l queueing system

1 Select a suitable state space and identify appropriate time instants where

a DTMC X* should be embedded into this non-Markovian continuous state process

2 Define a DTMC X* = {X:; n = 0, 1, } to be embedded into the non-Markovian continuous state process In particular, define a DTMC X* by specifying state transitions by taking Eq (3.14) as a model Specify the transition probabilities of the DTMC X* and its transition probability matrix P*

Server vacations can be modeled as an extension of the approach presented

in Section 3.2.1 In particular, the impact of vacations on the investigated

HESSENBERG MATRIX: NON-MARKOVIAN QUEUES 113

performance parameters is of additive nature and can be derived with decom- position techniques Non-Markovian queues with server vacations have proven

to be a very useful class of models They have been applied in a great variety

of contexts [DoshSO], some of which are:

l Analysis of server brealcdowns, which may occur randomly and preempt

a customer (if any) in service Since breakdown (vacation) has priority over customer service, it is interesting to find out how the overall service capacity is affected by such breakdowns Such insights can be provided through the analysis of queueing systems with vacations

0 Investigation of maintenance strategies of computer, communication, or manufacturing systems In contrast to breakdowns, which occur ran- domly, maintenance is usually scheduled at certain fixed intervals in order to optimize system dependability

l Application of polling systems or cyclic server queues Different types

of polling systems that have been used include systems with exhaustive service, limited service, gated service, or some combinations thereof

3.2.2.1 Polling Systems Because polling systems are often counted as one of the most important applications of queueing systems with server vacations, some remarks are in order here While closed-form expressions are derived later in this section, numerical methods for the analysis of polling systems on the basis of GSPNs is covered in Section 2.2.3

The term polling comes from the polling data link control scheme in which

a central computer interrogates each terminal on a multidrop communication line to find out whether it has data to transmit The addressed terminal transmits data, and the computer examines the next terminal Here, the server represents the computer, and a queue corresponds to a terminal Basic polling models have recently been applied to analyze the performance

of a variety of systems In the late 195Os, a polling model with a single buffer for each queue was first used in an analysis of a problem in the British cotton industry involving a patrolling machine repairman [MMW57] In the 196Os, polling models with two queues were investigated for the analysis of vehicle- actuated traffic signal control [Newe69, NeOs69] There were also some early studies from the viewpoint of queueing theory, apparently independent of traffic analysis [AMM65] In the 197Os, with the advent of computer commu- nication networks, extensive research was carried out on a polling scheme for data transfer from terminals on multidrop lines to a central computer Since the early 198Os, the same model has been revived by [Bux81] and others for token passing schemes (e.g., token ring and token bus) in local area networks (LANs) In the current investigation of asynchronous transfer mode (ATM) for broadband ISDN (integrated services data network), cyclic scheduling is often proposed Polling models have been applied for scheduling moving arms

in secondary storage devices [CoHo86] and for resource arbitration and load

sharing in multiprocessor computers A great number of applications exist

in manufacturing systems, in transportation including moving passengers on circular and on back-and-forth routes, internal mail delivery, and shipyard loading, to mention a few A major reason for the ubiquity of these applica- tions is that the cyclic allocation of the server (resource) is natural and fair (since no station has to wait arbitrarily long) in many fields of engineering The main aim of analyzing polling models is to find the message waiting time, defined as the time from the arrival of a randomly chosen message to the beginning of its service The mean waiting time plus the mean service time is the mean message response time, which is the single most important performance measure in most computer communication systems Another interesting characteristic is the polling cycle time, which is the time between the server’s visit to the same queue in successive cycles Many variants and related models exist and have been studied Due to their importance, the following list includes some polling systems of interest:

0 Single-service polling systems, in which the

sage and continues to poll the next queue

server serves only one mes-

l Exhaustive-service polling systems, in which the server serves all messages at a queue until it is empty before polling the next queue

the

l Gated-service polling systems, in which the server serves only those messages that are in the queue at the polling instant before moving to the next queue In particular, message requests arriving after the server starts serving the queue will wait at the queue until the next time the server visits this queue

l Mixed exhaustive- and single-service polling systems

l Symmetric and asymmetric limited-service polling systems, in which the server serves at most Z(i) customers in in each service cycle at station i, with Z(i) = I for all stations i in the symmetric case

3.2.2.2 Analysis As in Section 3.2.1, we embed a DTMC into a more general continuous-time stochastic process Besides non-exponential service times S with its distribution given by B(t) and its LST given by B * (s), we consider vacations of duration V with distribution function C(t) and LST C”(s), and rest periods of the server of duration R, with distribution D(t) and LST D’(s) Rest periods and vacations are both given by i.i.d random variables Finally, the arrivals are again assumed to be Poisson Our treatment here is based on that given in [KingSO]

If the queue is inspected and a customer is found to be present in inspection

in epoch n, that is X, > 0, the customer is served according to its required service time, followed by a rest period of the server (see Fig 3.2) Thus, after each service completion the server takes a rest before returning for inspection

of the queue If the queue is found to be empty on inspection, the server takes a

HESSENBERG MATRIX: NON-MARKOVIAN QUEUES 115

Fig 3.2 Phases of an M/G/l queue with vacations

vacation before it returns for another inspection We embed a DTMC into this process at the inspection time instants With E denoting the arrivals during service period including following rest period and F denoting the arrivals during a vacation, Eq (3.14) d escribing possible state transitions from state

X, at embedding epoch n to state X n+r at embedding epoch n + 1 can be restated:

X n-t1 = X,+=-l, if X, > 0,

With ek = P[E = k] and fi = P[F = Z] d enoting the probabilities of Ic or I arrivals during the respective time periods, the transition probabilities of the embedded DTMC can be specified in analogy to Eq (3.16):

(3.31)

Note that this transition probability matrix is also in Hessenberg form like the M/G/l case

Let p denote the state probability vector of the embedded DTMC at inspec- tion instants Then its generating function is given by:

= ii-(x(1 - z>>o-(X(l- x)),

G&z) = C-(X(1 - z))

(3.35) (3.36)

With generating functions G(x), GE(Z), and GF(x) defined, and the transi- tion probability matrix as given in Eq (3.31), an expression for the probability generation function at inspection time instants can be derived for the server with vacation:

HESSENBERG MATRIX: NON-MARKOVIAN QUEUES 117

ak = P[Y = k] d enoting again the probability of Ic arrivals in a service period, the probabilities vn, pi, and ak can be related as follows:

xC”(X(1 - x)) - x

1

x - B-(X(1- x))D”(X(l - z)) ’ (3.40) and:

x=x9+ X2(S + R)2 - xv2

If there is no rest period, that is, R = 0, and if the server takes no vacation, that is, V = 0, then Eq (3.42) re d uces to the Pollaczek-Khintchine formula

of Eq (3.28) Furthermore,

given by the term:

the average number of arrivals during a vacation,

is simply added to the average number of customers that would be in the system without vacation With respect to accumulating arrivals, the rest period can simply be considered as an extension of the service time

Problem 3.11 Give an interpretation of servers with vacation in terms of polling systems as discussed in Section 3.2.2.1 Give this interpretation for all types of polling systems enumerated in Section 3.2.2.1

Problem 3.12 How can servers with breakdown and maintenance strate- gies be modeled by servers with vacation as sketched in Fig 3.2? What are the differences with polling systems?

Problem 3.13 Give the ergodicity condition for an M/G/l queue with vacation defined according to Fig 3.2

Problem 3.14 Derive the steady-state probabilities ve and ~1 of an M/G/l queue with vacation Assume that the following parameters are given: Rest period is constant at 1 second, vacation is a constant 2 seconds, arrival rate

X = 1, and service time distribution is Erlang Ic with k = 2 and mean l/p = 0.2 seconds Check for ergodicity first

Problem 3.15 Use the same assumptions as specified in Problem 3.14 and, if steady state exists, compute the mean number of customers in system for the following cases:

1 M/G/l queue with vacation according to Fig 3.2

2 Parameters same as in Problem 3.14 above but with rest period being constant at 0 seconds

3 Parameters same as in Problem 3.15 part 2 above with vacation being also constant at 0 seconds

3.3 NUMERICAL SOLUTION: DIRECT METHODS

The closed-form solution methods explored in Sections 3.1 and 3.2 exploit-

ed special structures of the Markov chain (or, equivalently, of its parameter matrix) For Markov chains with a more general structure, we need to resort

to numerical methods There are two broad classes of numerical methods to solve the linear systems of equations that we are interested in: direct methods and iterative methods Direct methods operate and modify the parameter

NUMERICAL SOLUTION: DIRECT METHODS 119

matrix They use a fixed amount of computation time independent of the parameter values and there is no issue of convergence But they are subject

to fill-in of matrix entries, that is, original zero entries can become non-zeros This makes the use of sparse storage difficult Direct methods are also subject the accumulation of round-off errors

There are many direct methods for the solution of a system of linear equa- tions Some of these are restricted to certain regular structures of the parame- ter matrix that are of less importance for Markov chains, since these structures generally cannot be assumed in the case of a Markov chain Among the tech- niques most commonly applied are the well known Gaussian elimination (GE) algorithm and, a variant thereof, Grassmann’s algorithm The original version

of the algorithm, which was published by Grassmann, Taksar, and Heyman, is usually referred to as the GTH algorithm [GTH85] and is based on a renewal argument We introduce a newer variant by Kumar, Grassmann, and Billing- ton [KGB871 w ere interpretation h gives rise to a simple relation to the GE algorithm The GE algorithm suffers sometimes from numerical difficulties created by subtractions of nearly equal numbers It is precisely this property that is avoided by the GTH algorithm and its variant through reformulations relying on regenerative properties of Markov chains Cancellation errors are nicely circumvented in this way

ao,oxo + alp1 + + an-l,oxn-1 = bo,

If the system of linear equations has been transformed into a triangular structure, as indicated in Eq (3.44), the final results can be obtained by means of a straightforward substitution process Solving the first equation for

20, substituting the result in the second equation and solving it for ~1, and

so on, finally leads to the calculation of CC,-~ Hence, the xi are recursively computed according to Eq (3.45):

‘-More formally, for the lath elimination step, i.e, the elimination of x,-k from equations j, j = n - k, n - lc - 1, , 1, the (n - k + 1)th equation is to

be multiplied on both sides by:

(k-1)

an-k,j-l

- (k-l) ’ an-k,n-k

(3.46)

and the result is added to both sides of the jth equation The computation of the coefficients shown in the system of Eq (3.47) and Eq (3.48) for the lath elimination step is:

j = n - k - 1, n - k, ,O (3.48)

n-k,n-k

3Note that i n t he system of Eq (3.44) relation k = n - j >_ 0 holds

NUMERICAL SOLUTION: DIRECT METHODS

In matrix notation we begin with the system of equations:

121

an-i,0 an-l,1 - - - an-l,n-1

After the elimination procedure, a modified system of equations results, which

is equivalent to the original one The resulting parameter matrix is in upper triangular form, where the parameters of the matrix are defined according to

Eq (3.47) and the vector representing the right-hand side of the equations according to Eq (3.48):

In matrix terms, the essential part of Gaussian elimination is provided by the factorization of the parameter matrix A into the components of an upper triangular matrix U and a lower triangular matrix L The elements of matrix

U resulted from the elimination procedure while the entries of L are the terms from Eq (3.46) by which the columns of the original matrix A were multiplied

during the elimination process:

an-2,n-2 an-l,0 an l,n l

(3.50)

As a result of the factorization of the parameter matrix A, the computation

of the result vector x can split into two simpler steps:

for the vector of unknowns x is required

Note that the intermediate result y from Eq (3.51), being necessary to compute the final results in Eq (3.52), is identical to Eq (3.49) and can readily be calculated with the formulae presented in Eq (3.48) Since only the coefficients of matrix U are used in this computation, it is not necessary

to compute and to represent explicitly the lower triangular matrix L It is finally worth mentioning that pivoting is not necessary due to the structure

of the underlying generator matrix, which is weakly diagonal dominant, since )q+ 1 > qi,j, V’i, j This property is inherited by the parameter matrices Now the Gaussian elimination algorithm can be summarized as follows:

Construct the parameter matrix A and the right-side vector b

according to Eq (3.4) as discussed in Chapter 3

NUMERICAL SOLUTION: DIRECT METHODS 1.23

Carry out elimination steps or, equivalently, apply the standard algorithm to split the parameter matrix A into upper triangular matrix U and lower triangular matrix L such that Eq (3.50) holds Note that the parameters of U can be computed with the recursive formulae in Eq (3.47) and the computation of L can be deliberately avoided

Compute the intermediate results y according to Eq (3.51) or, equivalently, compute the intermediate results with the result from Eq (3.53) according to Eq (3.48)

Perform the substitution to yield the final result x according to

Eq (3.52) by recursively applying the formulae shown in Eq (3.45)

Example 3.1 Consider the CTMC depicted in Fig 3.3 This simple finite birth-death process is ergodic for any finite X and ,Q so that their unique steady-state probabilities can be computed Since closed-form formulae have

Fig 3.3 A simple finite birth-death process

been derived for this case, we can easily compute the steady-state probabilities

ri as summarized in Table 3.1 with X = 1 and ,Q = 2 and with:

1 7ro =

First, the generator matrix Q is derived from Fig 3.3:

NUMERICAL SOLUTION: DIRECT METHODS 125

is the intermediate solution vector y of Eq (3.51) with given lower triangular matrix L

The last step is the substitution according to the recursive Eq (3.45)

to yield the final results x as a solution of Eqs (3.52) and (3.53):

b(“) yo x0= &= -&=

>

>

8 17 8

4

a3,3 a3,3 a3,3 a3,3

=A!?- - uo,3 -x0 - u1,3 -X1 - -22 u2,3

u3,3 u3,3 u3,3 u3,3

15 = 15’

The computational complexity of the Gaussian elimination algorithm can be characterized by O(n3/3) multiplications or divisions and a storage require- ment of O(n2), where n is the number of states, and hence the number of equations

Note that cancellation and rounding errors possibly induced by Gaussian elimination can adversely affect the results This difficulty is specially relevant true if small parameters have to be dealt with, as is often the case with the analysis of large Markov chains, where relatively small state probabilities may result For some analyses though, such as in dependability evaluation studies, we may be particularly interested in the probabilities of states that are relatively rarely entered, e.g., system down states or unsafe states In this case, the accuracy of the smaller numbers can be of predominant interest

Grassmann’s algorithm constitutes a numerically stable variant of the Gaus- sian elimination procedure The algorithm completely avoids subtractions

and it is therefore less sensitive to rounding and cancellation errors caused

by the subtraction of nearly equal numbers Grassmann’s algorithm was originally introduced for the analysis of ergodic, discrete-time Markov chains

X = {X,; n = 0, 1, } and was based on arguments from the theory of regen- erative processes [GTH85] A modification of the GTH algorithm has been suggested by Marsan, Meo, and de Souza e Silva [MSA96] We follow the vari- ant presented by Kumar et al., which allows a straightforward interpretation

in terms of continuous-time Markov chains [KGB87]

We know from Eq (2.41) that the following relation holds:

-qi,i = c Qi,j- (3.54) j,i#i

Furthermore, Eq (2.57) can be suitably rearranged:

n-l

;TTiQi,i = c rjqj,i (3.55)

j=O,j#i Letting i = n - 1 and dividing Eq (3.55) on both sides by qn-l,+l yields:

n-2 -7rn-1 =

c

~ %,n 1 j=o ’ 4n-l,n-1’

This result can be used to eliminate rn-1 on the right side of Eq (3.55):

7riqi,i =

c rjqj,i - c iTT, qj,n-lqn-1,i

j=O,j#i j=o 3 Qn-l,n-1 n-2

(3.56)

Adding the last term of Eq (3.56) on both sides of that equation results

in an equation that can be interpreted similarly as Eq (3.55):

, O<i<n-2

(3.57) With:

qj,i = qj,i _ qj7n-lqn-lTi = qj i + qf+;lqn l,i,

qn-l,n-1 ’

C h-l,1 I=0

the transition rates of a new Markov chain, having one state less than the original one, are defined Note that this elimination step, i.e., the computation