Tài liệu Hiệu suất của hệ thống thông tin máy tính P17 docx

17.4.1 From iSPN to the underlying QBD be noted, however, that iSPNs have an unbounded number of states so that a ‘standard” place that may contain an unbounded number of tokens is know

Chapter 17

methods, it also avoids the state-space explosion problem that is so common in traditional

examples in Section 17.5

approach is the rapid growth of the state space Various solutions have been proposed for

Performance of Computer Communication Systems: A Model-Based Approach.

Boudewijn R Haverkort Copyright © 1998 John Wiley & Sons Ltd ISBNs: 0-471-97228-2 (Hardback); 0-470-84192-3 (Electronic)

efficient mean-value analysis style of solution becomes feasible [134, 751 In all these cases,

that the analysis of models with an unbounded state space is often simpler than analysing

models Of course, not all SPNs can be used for this purpose; we require them to exhibit

many SPNs do fulfill the extra requirements

iSPNs

The sections that follow are devoted to the specific parts indicated in this figure Defini-

17.2 Definitions 385

is then discussed in Section 17.4

17.2.1 Preliminaries

17.2.2 Requirements: formal definition

sufficient, rather than necessary

R’(Z) We denote L = lR’(n)I

hold for the so-called repeating portion of the state space:

2 inter-level one-step increases only:

1 no boundary jumping:

3w2 E T: (K - Lm,) tl\ (&ml,), (I‘$?& -3 (K - l,?&);

17.2.3 Requirements: discussion

tokens in place Pa It is for this reason that the levels Ic 2 K are called the repeating portion

We have tried to visualise the fact that starting from level K upwards, the levels repeat one

form:

(17.5)

17.3 Matrix-geometric solution 389

&c+1 =z,R, z,+~ =zn+rR=gKR2; , (17.6)

or, equivalently,

polynomial:

fi+1

c ZjBj,K = ~tc 1Bt+c + x,B,,, + z,+JL+l,ra, j=n-1

= z~-~% I,K + s,(B,,, + R&) = 0 (17.9)

as many unknown vectors However, as these vectors are still dependent, the normalisation

trices Ai have been computed

17.4 iSPN specification and measure computation

processes In Section 17.4.2 we then present a number of techniques to evaluate reward- based measures for iSPNs in an efficient way

17.4.1 From iSPN to the underlying QBD

be noted, however, that iSPNs have an unbounded number of states so that a ‘(standard”

place that may contain an unbounded number of tokens is known in advance Indeed, when

17.4 iSPN specification and measure computation 391

levels are the same, all levels beyond these will also be the same The 3-page proof, based

17.4.2 Efficient computation of reward-based measures

i=o E@?(i)

expected reward

We discuss a number of these special cases below

Pa, when computing

For these cases, we can write:

i=o

so that we have:

can be derived:

K-1

i=O j=o Pi-1

i=O j=O j=O

K-l

= xi(Zi.1) +Kz.i, i=O

(g-) I+& (gi-‘) 1 n-l

cm E[X] = LT + x r(m) xzi,,

E[X] = LT + c r(m)z,(I - R)-kE (17.17)

present a queueing model with delayed service in Section 17.5.1; this model is so simple

17.5.1 A queueing model with delayed service

Considers a single server queueing system (see Figure 17.3) at which customers arrive as

server awake and start its duties This is enforced by an enabling function associated with

until the buffer becomes empty, after which it resumes sleeping

start

buffer2T

Figure 17.3: iSPN of a single server queueing system with delayed service

where C is chosen such that the row sums equal 0 From this matrix we observe that the

From these matrices we derive pR2 - (X + p)R + X = 0 for which the only valid solution

Denoting zi = zi, for i = 0 or i = 3,4, ., and zi = (zi,~, z~,A), for i = 2,3, the boundary

17.5 Application studies 395

Figure 17.4: QBD of the single server queueing system with delayed service in case T = 3

&A - (A + &2A +pz3 = 0,

&,5 + &A - (A + /.+3 + I.Lz4 = 0,

20 + 21s + XlA + z2,5 + Z2A + x3(1 - p)-’ = 1

(17.18)

probabilities:

{

of customers in the system E[N] = 3.00

17.5.2 Connection management in communication systems

Figure 17.5: iSPN model of the OCDR mechanism

An iSPN model for such a system is given in Figure 17.5 Packets arrive via transition

the source remains in the off and on state The service rate is ,Q Mbps and the average packet length is denoted 1

Measures of interest we could address are: (i) the average node delay E[D] (in seconds);

Wwl

0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12

set 1 - set 2

Let us now turn to some numerical results We assume that a = 1.0, p = 0.04, c = 10.0,

is less sensitive to changes in X, especially towards higher values For smaller values of X,

and no packets present decreases with larger X

17.5.3 A queueing system with checkpointing and recovery

In this section we present an analysis of the time to complete tasks on systems that change

as different ways in which failed services might be resumed [46] The above studies aim at

of the server, jobs experience a delay in the queueing system Related models occur when

always arr buffer serve start-chk

reserve

ing

recovery

that are subject to failures and repairs [40, 221, 2371 When increasing the checkpointing frequency, the amount of overhead increases; however, the amount of work to be done after

situations

arrivals form a Poisson process The corresponding iSPN is depicted in Figure 17.10 In Table 17.1 we summarise the meaning of the places and transitions

17.5 Application studies 401

centage of time that the processor is available for normal processing and the probability

states in the repeating levels equals 2K + 1 This can be explained as follows As long as the system is up, i.e., there is a token in place up, there can be 0 through K - 1 tokens

already Then, when the server is down, i.e., when there is a token in place down, there are

up to K - 1 possible tokens in place done, thus making up another K different situations

In total, this yields 2K + 1 states per level As a consequence of this, the matrices Ai and

“by hand” would have been very impractical

be computed recursively

17.5 Application studies 403

(scenario 1) or K = 5 (scenario 2); a choice of K too small causes the system to make

Then, in Figure 17.12, we present the percentage of time the server is making check-

exactly alike lies in the fact that the server can only fail when it is serving regular jobs

most 10m3 The other two curves show the decreasing percentage of time the server is busy

percentage of time needed for normal processing equals X/p = 8/10 = 80% On top of

extra work required when failures occur, it becomes clear that the server cannot do all the work when K = 2

Let us now address the cases where the system is stable The upper curve shows the

This extra work is not earned back by the shorter recovery time that results On the other hand, for K > 6, the recovery times become larger so that the gain of less checkpointing

1.00

0.90 w+

0.85 0.80

0.70

ii

in performance

17.6 Further reading 405

using the model of Section 17.5.3 in [119] In that paper, models with block sizes up to

[226]

setup- and release-times?

1 Find the 2 x 2 matrices Ao, Al and A2 in symbolic form

17.3 Checkpointing models

Extend the model of Figure 17.10 such that:

first Model the timer as an Erlang-Z distribution

17.4 Two queues in series

system using iSPNs