Performance analysis of a volume shadowing model

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Performance analysis of a volume shadowing model

Citation for published version (APA):

Dekkers, A., & Wal, van der, J. (1990). Performance analysis of a volume shadowing model. (Memorandum COSOR; Vol. 9022). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1990

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

Memorandum COSOR 90-22 Perfonnance analysis of a volume shadowing model

A.Dekkers J. van derWal

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box513

5600 MB Eindhoven The Netherlands

Eindhoven, July 1990 The Netherlands

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*

A.Dekkers and J. van der Wal Eindhoven University of Technology

Deparunent of Mathematics and Computing Science P.O. Box 513

5600MB Eindhoven, The Netherlands e-mail: wscoade@win.tue.nl

Abstract

The term Volume Shadowing refers to the fact that there are two (or more) disks containing exactly the same data. Ifpart of the data is changed or extended the data are changed on all disks.

The disks are visited by two types of jobs: "reads", which can be send to either one of the disks, and "writes", which have to visitallthe disks.

This paper considers a system with two shadow disks. The analysis is based on the

M[XliM12 model. Reads and writes together are treated as a Poisson arrival stream of jobs with reads requiring the attention of only one disk, but writes requiring both disks. Writes are therefore considered as two simultaneously arriving jobs. Thus the arrival stream consists of batches of size one or two. Reads and writes have the same exponential service time. The two disks are the two servers.

This is only an approximate model for the Volume Shadowing problem, since the M[XliM12model assumes that a write canbeprocessed twice on one disk instead of once on both disks. Therefore refinements of the model are derived. Numerical results show that this approach leads to good approximations.

·The investigations were supported (in part)bythe Foundation for Computer Science in the Netherlands (SION) with financial aid from the Netherlands Organization for the Advancement of Scientific Research (NWO).

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-2-1. INTRODUCTION

1.1.What is Volume Shadowing?

Volume Shadowing is a technique in the storage of data used to increase the reliability and/or the perfonnance. Nonnally data are stored on just one disk. With Volume Shadowing the same infonnation is stored several times on different disks.

The disk system is visited by two types of jobs, "reads" and "writes". A read can be served by either one of the disks, but a write has to be perfonned on all disks so that the different disks remain identical copies of one another. Ifall jobs are reads then the system behaves better than a system where the infonnation is divided over all disks. If most jobs are writes then dividing the infonnation over the disks has to be preferred.

When the load on a heavily used disk, such as the system disk, becomes too high, Volume Sha-dowing might become interesting from a perfonnance point of view. Banking and insurance companies often use Volume Shadowing for reliability reasons.

Here we concentrate on the perfonnance aspects. Other problems with Volume Shadow-ing such as architecture, design and the prevention from, or updatShadow-ing after, a breakdown are beyond the scope of this paper.

We will be looking at a system consisting of two disks, visited by reads and writes. We are interested in the usual perfonnance parameters such as sojourn times, waiting times and queue lengths. We will first describe the implementation of a Volume Shadowing system and then approximate it by a fairly standard queueing model, theM[X) 1M 12 queue. This model is only a rough approximation and we will give several refinements later on. Numerical results show that these refinements yield a good approximation for the perfonnance of the Volume Shadowing system.

1.2.Description

Inthis section we will describe in more detail the implementation of a Volume Shadowing sys-tem.

Inthe Volume Shadowing system, as implemented on VAx\VMS[1,2],the queueing dis-cipline is Shortest Seek Time First (SSTF). SSTF gives an extra perfonnance gain when using Volume Shadowing. Here we only consider the First Come First Served (PCFS) discipline. We feel that it should be possible to obtain approximations for the SSTF discipline from the FCFS discipline results, by adapting the service times.

Jobs arriving in the Volume Shadowing system are queued until one of the disks is ready to treat their request As stated before, we consider a system consisting of two disks. Apart from the word "job" indicating a read or a complete write we will also use the word "part" to indicate a read or one of the halves of a write; the two halves aretobe perfonned on different disks. The two parts of a write may be served simultaneously, but this is not necessary. Ifa

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disk becomes available the first part in the queue can start service, provided that it is a read or the suited part of a write. Ifit is the wrong part of a write, the second part in the queue can start service under the same condition, etc. The service time of a part is assumed to be exponentially distributed with mean 1I~for reads as well as for writes at both disks. The service times of the two parts of a write are assumed tobeindependent. By the sojourn time of a write we mean the elapsed time between its arrival and the completion of the last of the two parts. The sojourn time for a read is simply given by the completion time minus the arrival time.

In our discussion of the Volume Shadowing system we will consider Poisson arrival streams for reads and writes with arrival ratesA.,. and

A.w

respectively.

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4

-2. MODEL

2.1. Introduction

In this section we will give a Markov queueing model for the Volume Shadowing system and we fonnulate a set of equilibrium equations which canbederived for this model. The definition of a state in such a system is complicated, because the two parts of a write have to be served at different disks.

We will consider a queueing model with two servers and three queues. The two servers are the two disks, they both work with an exponentially distributed service time with mean 111.1.. Each disk has its "own" queue with parts that are to be served at that specific disk. In the third queue complete jobs are waiting. Jobs arrive in this system according to a Poisson process at queue 3 with arrival rate(A,.+Aw).

(A,.+Aw) ---..-H+_<

Fig.2.1: Volume Shadowing system

The disks serve the parts in their queue according to the FCFS discipline. If the queue in front of a disk becomes empty the disk grabs the first job waiting in queue 3, if this queue is not empty. Ifit is a write the other half of the write is put at the end of the queue at the other disk. Note that this implies that the queue of the disk which took the last job from queue 3 contains at most one pan. Funhennore this implies that the waiting parts in queues 1 or 2, are always writes, and if there is a read in one of these queues, then it has to be in service.

Without loss of generality we assume that queue 1 is the shonest one. Then we can define a state in this system as the triple(i,j,k) withi denoting the length of the queue 1(i=O or 1),j denoting the length of the queue 2 U~i) and k denoting the length of queue 3 (k~ with k>Oonly ifi>0). The events which can alter the state of the system are the arrival of a job and the completion of a part. If a disk grabs a job from queue 3 it gets a read with probability A,./(A,.+Aw)and a write with probabilityAw/(A,.+Aw).

This leadstoa set of equilibrium equations, which does not have a product fonn solution (Baskett, Chandy, Muntz & Palacios [3]) and we see no way to solve it in tenns of the input parameters. Of course it is possible to truncate the system and to solve it numerically. For

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higher utilizations of the disks the truncated sets will have to be in the order of a few hundred states. We prefer totryto derive simple and straightforward fonnulae based on an approxima-tion with a standard queueing model.

2.2. Basic model:MfXl\ M12.

The system described in the previous section resembles a standard queueing model, the MlXllM12-queue. The work arrives at an exponential two server station according to a Pois-son process in batches. A batch of size I corresponds to the arrival of a read, a batch of size 2 to the arrival of a write (see e.g.• Kleinrock [5] or Gross& Harris [4]).

So in this model we have random batch-arrivals: there is an arrival stream of singular parts with parameter

A,.

and an arrival stream of pairs of parts with parameter

Aw.

The parts are indis-tinguishable, there is no stamp on a part indicating whether it is a read or the first or second part of a write. All parts are queued in one queue and served on the first available disk. So in this simple model it is possible that the two parts of a write are processed on the same disk.

The probability that there arei parts in the system is denoted byPi. Then the equilibrium equations of thisM[XIIM12 system are easily obtained from the Markov chain in fig. 2.2.

~ 211 I I (1) 211 2).1 I I (2)

A,.+Aw

Aw

Defining a. and~by a.

=

and~

=-

we get

211 211 PI =2C1{Jo Pi=C1{Ji-l+~Pi-2 00 :!:Pi=l. i~

Fig. 2.2:Markov chain

fori~

(1) (2) (3)

In order to solve this set of equilibrium equations we define the generating function

00

P(z)

=

:!:

Pizi. Summing (2) overalli~yields: i~ P(z)-Po-pIZ

=

w(P(z)-Po)

+

~z2p(z). So. by inserting (1), 1

+w

P(z) =Po 2 . 1-w - ~z (4) (5)

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-6-WithP (1)=1 we get from (5)

_1-a-[3

Po - 1

+

a

"

After some algebraP(z)can also be written as

1 1 P(z)=C1 " +C 2" -l-z/z1 _{l-z/z 2} with (6) (7)

_ -a+

"a2+4~

zl - 2[3 , po(1+zla)

C

I

=

,

[3zl(ZI-Z2)

_ -a-

"a2+4~

z2 - 2[3 , Po(1+Z2 a ) C2= . [3z2(Z2-z

I>

One may verify that the ergodicity condition 211

>

A.,.

+ 2A.

w (or analogously

a+[3<I)

implies

z I

>

1 and z2

<

-1. (The ergodicity condition states that the total arrival intensity of jobs is less than the total capacity of the servers.)

So fori~we have

(8)

(9)

withXj

=

lIzj , j=I,2.

The average number of parts in the system L0, can be derived from formula's (5) and (6) (the superscript 0 denotes that this quantity is derived from theM[X11M 12model):

L

°

= d P(Z) =~

+ a+2fL

dz z=l

l+a

l-a-[3·

The distribution WO(t) of the waiting time is equal for reads and writes. The waiting time is equal to 0 with probabilityPO+PI and with probabilityPi the time until i-I parts have been completed, thus Erlang distributed withi-I phases of mean 1I(2Jl).

So after some algebra

o x21 211(l-xI) X22 211(I-X2) E_e-sW (t)_=P(0)_+P(1)

+

C_{1 - -} t'"

+

C_{2 - -} t'" _, I-XI s+2Jl(I-xI) l-X2 S+21l(1-X2) thus X2

x

2 WO(t)=CI-I_(1-e-~(l-XI)t)

+

C 2- 2-(1-e-~(I-x2)t). I-XI l-X2

Or alternatively the mean waiting timeWO follows from

WO=

:E

Pi" [ i-I]

i=2 211

1 00

=

-(:L

Pi"i - (l-po-p

I»

2Jl ;=2

(10)

(11)

(9)

(13)

The sojourn time of a read is the sum of the mean waiting time and its mean seIVice time which is the completion time of one part, 1I1!. For a write the computation of the sojourn time is more complicated. The mean generalized seIVice time of a writeB~ is detennined by the part of the write which completes second. Ifa write arrives in an empty system the completion time of the second part is the maximum of two exponentially distributed seIVice times for the individual parts. Otherwise a write has to wait until one of the disks becomes free. Then the first part of the write canbeseIVed directly and the second part will start afterwards on the first disk which becomes free. Thus in this case the mean generalized seIVice time is

°

3 [1 1 3 1 IJ

B

w=

Po-

+

(I-po)

+ +

-2J.l 21! 2 2J.l 2 I!

The sojourn time of a write now is the sum of the waiting time and this generalized seIVice time.

As said before, the M[Xl1M12 model has one great disadvantage in describing the Volume Shadowing system. Both parts of a write can be seIVed on the same disk. Inreality it is possible to have 2 or more parts in the system while one of the disks is idle. So the M[XlIM12 system is more efficient than the Volume Shadowing system as described in Section 2.1. Hereafter we subsequently consider two improvementstotheM[XllM 12 model in order to obtain a better approximation for the Volume Shadowing system. In these improvements we will make use of the definition of a state in Section 2.1. In particular the longest disk queue where parts of writes are waiting will be important.

2.3. Improvement 1: The two parts of a write to different disks.

In the first improvement both parts of a write are seIVed at different disks, but we assume that before this write is going into seIVice the system has always behaved as aM [Xl 1M12 queue. So the waiting time for arriving jobs is the same as in theM [Xli M 12 model

WI =Wo. (14)

Here and in forthcoming fonnulae the superscript indicates that the quantity corresponds to the improvement with that number.

After the waiting time, which is equal for reads and writes, the seIVicing starts. For a read the seIVice time is unchanged. For a write the generalized seIVice timeB~is different fromB~. Ifa write arrives in an empty system the last part will still finish after the maximum of two exponentially distributed seIVice times for the single parts, in this case the two parts are seIVed on different disks. Otherwise a write has to wait until there is one disk free. One part of the write is then seIVed directly and the other part will start afterwards; not on the first available disk, but on the other disk. Note that in this definition it is possible that during the generalized seIVice time of a write, neither one of the disks is actually working on that write! The general-ized seIVice time is the maximum of an exponentially distributed seIVice time for the first part

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(15)

(16)

8

-and an Erlang-2 distributed service time for the second part (this part has to wait on the part on the other disk to complete before it may start itself).

This yields 1

3 [1 12 13J

B =Po- +(I-po) + + -w 2Jl 2Jl 2 Jl 2 211 3 9 =Po-

+

(l-po)-· 2Jl 4Jl

Because the sojourn time of writes increases, the average number of parts in the systemL1is

larger than LO. We can use Little's formula (Little [6]), to approximate L1. Itis the arrival

intensity of reads multiplied by the sojourn time of reads plus the arrival intensity of writes mul-tiplied by the sojourn times of the first and second part of writes:

L1=

A,r(Wl+~)

+

A.w(Wl+~

+

Wl+pol..+(l-po)~)

Jl Jl Jl Jl

= (A,r+2A.w)(Wl+1..)

+

A.w(1-po)l...

Jl 11

Indeed, we ignored the fact that the delay of second parts of writes might be advantageous to the reads.

2.4. Improvement 2: The situation at the start of service.

In theM[XIIM 12 model and in Improvement 1 we assumed that a job has to wait until there is

at most one part in front of it before its service can start. Inthe Volume Shadowing system this is not true in general. Ifa part of a write, which has to be served on disk 2, is first in the waiting queue and disk 1 becomes available, this part can not be served there and (if present) a part behind it in the waiting queue may start at disk 1 even though there is at least 1 part in front of it. But again this can be a part of a write that has to be served on disk 2 and then the next part may start at disk 1, and so on. Interms of the Volume Shadowing model of Section 2.1 this corresponds to the situation that a part of a write in queue 2, which arrived before the jobs in queue 3, will not go into service when disk 1 becomes free. Instead disk 1 grabs a job from queue 3. Thus a part can start its service at a disk if it is a read or the suited part of a write and if there are no parts waiting in front of it, which may be served at that disk. So the number of parts, which arrived earlier but are still waiting when the job starts, can be 0, 1,2, etc.

This has two consequences:

1. The waiting time foralljobs might be less than estimated by WI, since notalljobs have to wait until there is at most one part in front of it.

2. The waiting time for the second part of a write however will be larger, because it has to wait for a specific disk. At the moment the first part starts its service there may be several parts waiting on the other disk that have to be completed first. So the generalized service time of a write, defined as the interval from the moment upon which the first part goes into service until the moment both parts have finished, willbe larger than

B

l.

(11)

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(19) In Improvement2we will use the Volume Shadowing model of Section 2.1 to get an estimate for the length of queue 2 at the moment a job is grabbed from queue 3. Recall that in Section

2.1 we assumed that queue1was the shortest one. With the estimation for the length of queue2

the two consequences stated above appear in the performance formulae for Improvement 2. We denote the probability to start service in a state when there arej parts in queue2byqj

and call this an imbalance ofj. The only way to obtain the

q/s

exactly would be to solve the set of equilibrium equations of Section2.1 and sum the probabilities for all states withj jobs in queue 2. As mentioned before we want to avoid this and prefer another method. We will approximate the probabilitiesqjby a geometrical distribution.

Now one server may be idle, while there

are

two or more parts available in the system. Therefore the probabilty on an empty system will be less thanPo as given in theM[XllM12

model. However, we will still assumePoto be the probability on an empty system.

So what we need is the ratioqj+l Iqj. But this is just the probability of going from imbalancej

to j+l before going from j toj-l. There are two conditions to fulfil in order to find a higher imbalance.

1. There should be a job available in queue 3before the moment the job at disk2 finishes. The probability ~ on such an event is the probabilty of at least one arrival in the interval from the arrival of the current job at disk 1until the completion of the job in service at disk 2.

2. The job in service at disk 1 should be a write, and disk 1 should be ready before disk 2. The ratio" of growing imbalance, given that there are jobs in queue 3 can be determined by a birth-death process.

The ratio~

=

qj+l Iqjis then approximated by

(17) How do we obtain (approximations for)" and~?

For~ we first of all need the probability that a job arrives during the waiting time of the current job in queue1.This probability denoted byris obtained from(11),

xI

Ar+Aw x~ A.,.+Aw

r=CI - - +C2- - .

I-Xl A.,.+Aw+21l(l-XI) l-X2 A.,.+Aw+21l(l-X2)

But for the imbalance to increase, an arrival before the completion of the job in service at disk 2 may suffice if condition 2 is met. Given that there was no arrival during the waiting time of the current jobinqueue 1, the probability

s

on an arrival before its completion is

A.,.+A.w s =

-A.,.+Aw+JJ. .

The probability~now is

~=r+(I-r)s. (20)

Given that there are jobs in queue 3 it is easy to derive 11 from the simple birth-death pro-cess depicted below

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(21)

-

10-Fig. 2.3:Birth-death process

with the probability on a writePw=Aw/(Ar+Aw). Inequilibrium Ihpw

l1=~=Pw.

Now the ratio ~ is known, the distribution for the number of parts in queue 2 can be approxi-mated. Note thatqjis obtained from

and

qj

=

~qj-l forj

>

1, (22)

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(recall that we assumedPostill to be the probability on an empty system). So

qj

=(l-Po)(l--l;)~j-l

forj ~1. (24)

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Simulation experiments have shown that a geometrical distribution with the value for~derived above yields an estimation for the probabilities qj,that differs less than 10% from the simula-tion results.

With this geometrical distribution we can describe the two effects above: a shorter wait-ing time for all jobs and a longer generalized service time for writes. The waitwait-ing time W2 is the waiting time from theMlX1IM12modelWOminus the gained time, because a job may not have to wait for all parts in queue 2

2 00

[i

-lJ

00

[i

-lJ

W ="LPi - "Lqi

-i=1 21J. i=1 2Jl

o

1 ):

=

W - 2J.l(l-po)~.

The generalized service time for writes,B~, is now larger thanB~: the second part of a write can start at the same moment as the first part with probabilityPo and it now has to wait with probabilityqj forj partsinqueue 2 to complete before it can start its service. The service time of a write is the maximum of the completion times of the first and second part minus the arrival time. Thus:

2 3 1 00 1 [.

~

B =po--+ ~ q.- z+l+ .

w 2 IJ.

i:t

1Jl 21+1

(13)

(27)

The average number of parts in the system L2, is approximated similarly as L1 (see (16)).

Again it is the arrival intensity of reads multiplied by the updated average sojourn time of reads plus the arrival intensity of writes multiplied by the updated sojourn times of the first and second part of writes:

2 2

1 [1

1" i+lJ

L

=

A.,.(W

+-) +

"-w W

2

_+-

₊

_W

2

_+Po-

₊

_L

_{q t}

-I.L I.L I.L i =1 I.L

=

(A.,.+2"-w)W

2_{+ (A.,.+"-w).l +}

"-w[ Po.l+(1-PO).l2j .

(14)

-

12-3. RESULTS

The Volume Shadowing model described in Section 2.1 has been simulated and we compared these results with our analysis. We considered a group of twelve examples, which are in our opinion representative for such a system. The ratio's of reads and writes vary from 1to 9 and the utilization varies from 0.25 to 0.9. The input parameters are given in table 1. The speed of the disks, J,1, is normalized to 1. The arrival intensities are given by A., for the reads and Aw for the writes. The utilization is given by (A.,+2Aw)/(2J,1).

J,1 A., _Aw uti!. A.,/Aw

Example 1 1.0 0.40909 0.04545 .25 9 Example 2 1.0 0.3 0.1 .25 3 Example 3 1.0 0.16667 0.16667 .25 1 Example 4 1.0 0.81818 0.09091 .50 9 Example 5 1.0 0.6 0.2 .50 3 Example 6 1.0 0.33333 0.33333 .50 1 Example 7 1.0 1.22727 0.13636 .75 9 Example 8 1.0 0.9 0.3 .75 3 Example 9 1.0 0.5 0.5 .75 1 Example 10 1.0 1.47273 0.16364 .90 9 Example 11 1.0 1.08 0.36 .90 3 Example 12 1.0 0.6 0.6 .90 1

table3.1:Inputfor the12examples

7.453 7.922 8.233 8.421 1.ge-02

table 3.2: Performance results

Table 2 shows that the improvements indeed are improvements. For the M [X] 1M 12 model the difference between approximation and simulation is up to 18.3%, for Improvement 1 this differ-ence is up to 8.5% and for Improvement 2 this differdiffer-ence is at most 3.8%.

Iftable 2 is studied more precisely, it appears that there is a dependency between the input for a model and the difference between approximation and simulation. Intable 3 the maximum difference in sojourn time for reads and writes between approximation and simulation is given for the ratio of reads and writes as well as for the average utilization of each disk.

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-

14-ArIAw

M[X1 IMI2 I. 1 1.2 9 15.05 1.42 0.83 3 15.78 3.40 1.36 1 18.31 8.46 3.80 util. 0.25 11.39 2.08 0.94 0.50 17.79 5.74 2.69 0.75 18.31 8.46 3.78 0.90 11.50 5.93 2.34

table 3.3: Maximum differences between approximation and simulation(%)

As can be seen from table 3, the approximation is less accurate if the fraction of jobs being a write increases and if the utilization is moderate (50% -75%). The explanation can be found in the probability on an empty system. As mentioned before we assume this probability to be equal to this probability in theM[X1IM12 model. But in the Volume Shadowing system there might be 2 jobs in the system while one disk is not working. This inefficiency will cause the system to be less frequently empty. The overestimation of the probability on an empty system will be larger if there are more writes; then the system will work inefficiently more often. Furthermore the approximations will be influenced mostly if the utilization is moderate. For low utilizations the sojourn time is mainly dependent on the service time of a job, the waiting time is less important. For high utilizations the system will be empty less frequently and the overestimation of the probability on an empty system will therefore be smaller.

We have have tried to derive a simple formula for a better approximation forthe probabil-ity on an empty system, but we did not find one based on straightforward or intuitive arguments. With a cumbersome hybrid method in which we combined the numerical solution of a set of equilibrium equations for situations with a small number of jobs in the system with the approxi-mation of section 2 for situations with a large number of jobs we obtained a better approxima-tion. This yielded differences between approximation and simulation of less than 1% for all the examples given above.

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4. CONCLUSION

In this paper we have discussed a Volume Shadowing system with two identical disks. In such a system there are two kinds of jobs, reads, which visit either one of the disks, and writes, which visit both of them. For such a system we have made a perfonnance analysis.

Our analysis was based on three steps. In the first step we made an analysis of the M[X1IM12system, which looks like the Volume Shadowing system. Inthe second step the two parts of a write were sent to different disks, but before its arrival the system was assumed to behave as theM[X1IM 12system. Inthe third step the specific situation at the start of a service was taken into account, caused by the two parts of a write which have to go to different disks.

We have showed with some representative numerical examples that our approximation algorithm is a good one; only in cases with

a

large number of writes the difference between our analysis andasimulation exceeded the1.5%.

References

1. VAXIVMS Volume Shadowing Manual, Digital Equipment Corporation, U.S.A., April 1986.

2. Digital Equipment Corporation, private communication, 1989.

3. BASKETI, F., CHANDY, K.M., MUNTZ, R.R., AND PALACIOS, F.G., "Open, Gosed and Mixed Networks of Queues with Different Classes of Customers", J. ACM, vol. 22, pp. 248-260, 1975.

4. GROSS, D. AND HARRIS, C.M., Fundamentals of Queueing Theory, John Wiley & Sons, New York, 1985 (second edition).

5. KLEINROCK, L., Queueing Systems, Volume I: Theory, John Wiley & Sons, New York, 1975.

6. LITILE, J.D.C., "A Proof for the Queueing fonnula: L

=

'AW", Oper. Res., vol.9,

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=:INDHOVEN UNIVERSITY OF TECHNOLOGY )epartment of Mathematics and Computing Science

?ROBABILITY THEORY, STATISTICS, OPERATIONS RESEARCH AND SYSTEMS

rHEORY ).0. Box 513

,600MB Eindhoven - The Netherlands ;ecretariate: Dommelbuilding 0.03 relephone: 040 - 473130

.ist of COSOR-memoranda - 1990

~umber Month Author Title

.190-01 January I.J.B.F. Adan Analysis of the asymmetric shortest queue problem

1.Wessels Part 1: Theoretical analysis W.H.M.Zijm

.190-02 January D.A. Overdijk Meetkundige aspecten van de productie van kroonwielen

.190-03 February I.J.B.F. Adan Analysis of the assymmetric shortest queue problem 1. Wessels Part II: Numerical analysis

W.H.M.Zijm

190-04 March P. van der Laan Statistical selection procedures for selecting the best variety L.R. Verdooren

190-05 March W.H.M.Zijm Scheduling a flexible machining centre E.H.L.B. Nelissen

,f90-06 March O.Schuller The design of mechanizations: reliability, efficiency and flexibility W.H.M.Zijm

,f90-07 March W.H.M.Zijm Capacity analysis of automatic transport systems in an assembly fac-tory

190-08 March OJ.v. Houtum Computational procedures for stochastic multi-echelon production W.H.M.Zijm systems

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Laarhoven W.H.M.Zijm

M90-10 March F.A.W. Wester A hierarchical planning system versus a schedule oriented planning

1. Wijngaard system

W.H.M.Zijm

M90-11 April A.Dekkers Local Area Networks

M 90-12 April P. v.d. Laan On subset selection from Logistic populations

M 90-13 April P. v.d. Laan De Van Dantzig Prijs

M 90-14 June P. v.d. Laan Beslissen met statistische selectiemethoden

M 90-15 June F.W. Steutel Some recent characterizations of the exponential and geometric distributions

M 90-16 June 1. van Geldrop Existence of general equilibria in infinite horizon economies with C. Withagen exhaustible resources. (the continuous time case)

M 90-17 June P.C. Schuur Simulated annealing as a tool to obtain new results in plane geometry M 90-18 July F.W. Steutel Applications of probability in analysis

M 90-19 July U.B.F. Adan Analysis of the symmetric shortest queue problem 1. Wessels

W.H.M.Zijm

M90-20 July U.B.F. Adan Analysis of the asymmetric shortest queue problem with threshold

J. Wessels jockeying

W.H.M.Zijm

M 90-21 July K.vanHarn On a characterization of the exponential distribution F.W. Steute1

M90-22 July A.Dekkers Performance analysis of a volume shadowing model