Tài liệu Hiệu suất của hệ thống thông tin máy tính P13 pdf

Chapter 13 present the BCMP result in Section 13.1, and then we discuss a number of computational 13.1 Queueing network class and solution first present this class of queueing models in

Chapter 13

present the BCMP result in Section 13.1, and then we discuss a number of computational

13.1 Queueing network class and solution

first present this class of queueing models in Section 13.1.1, after which we discuss the

specified A class can either be open or closed and jobs are allowed to change classes when changing from queue to queue The queueing stations can be of 4 types:

ISBNs: 0-471-97228-2 (Hardback); 0-470-84192-3 (Electronic)

of all classes need to be the same and must be negative exponential, albeit possibly

FESCs

2 In PS nodes, jobs are served in a processor sharing fashion All jobs are processed

although only the first moment does play a role in the computations

(per class) may depend on the queue length at the node Service time distributions must be of Coxian type; only the first moment needs to be specified

type 4 nodes

When leaving node i, a job of class r will go to node j as a class s job with probability

ri,r;j,s * Jobs leave the network with probability T+;~ Depending on which routing possi-

fraction T();i,r of the arrivals goes as a class T job to station i; or (2) every routing chain has

(denoted X,(rCc) with c E C; see below) A fraction ro;i,c of these arrivals arrive at queue i

brO;i,T (arrivals per chain/class)) As a result of this, one obtains the throughputs & for

tomers present in queue i and let Ni = C,“=, Ni,r be the total number of jobs in queue i

customers in the QN is given by K = CE, Ni

form:

i=l

(13.2)

When node i is of type FCFS, we have in the load-independent case

(13.3)

(13.4)

(13.5)

(13.6)

(13.7)

(13.8)

number of customers in the queueing station (ni) In the BCMP paper, other

rate of a class r customer at queue i depends on the number of customers of that class at that station (ni,,); we do not address these cases here

Important to note is that although the service time distribution in PS, IS and LCFSPR

Pr{N = n> = fi p&i),

i=l

with

Pi(%) =

I

e-Pz &.Y

where pi is defined as

(13.9)

(13.10)

(13.11)

with Ri the set of classes asking service at station i Notice that the value A(n) = Xk is not

operate as if they are MIMI 1 queues studied in isolation (see also Chapter 6 where we found

1”

with

V

%! II,“=, 5 ( 2 >

nqr

Note that ni = C,“=, ni,r

(13.13)

0

13.2 Computational algorithms

complexity

QNs in Section 13.2.2

follows:

i=l

(13.14)

nEZ(M,K) i=l

Kj+M-1 M-l

computational steps (one for each state) where each step consists of multiple multipli-

recursive relation holds:

K is negative

R Kr

ising constants with one node less, and with one customer less in each of the classes Note that for a node i of FCFS type, the value pi,r = pi, for all classes r

x

tomer classes, even if the number of customers remains the same, increases the number of operations to be performed significantly

adhere to the same class of QNs as in the previous section If we define the average service

time (per passage) for a class T customer at node i is given as follows

E[fii,r(K)] = Di,T (C;“=i E[Ni,j(K - 4)] + 1) , FCFS, PS, LCFSPR nodes,

D

\

(13.19)

The throughput for class r customers (through the node with I,$r = 1) is given as:

(13.20)

as

to break the recursion

at node i, one can estimate this value as

same way An advantage of this approach is that the system of recursive MVA equations is

so-called operational analysis method which also addresses QNs with a variety of character-

[169] The books by Bruell and Balbo [30] and by Conway and Georganas [65] deal ex-

by BCMP nodes of types 1 through 4 (FCFS, PS, IS or LCFSPR)