Comparison of the throughput of the M|Er|1|N and the M|M|1|N queues

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Comparison of the throughput of the M|Er|1|N and the

M|M|1|N queues

Citation for published version (APA):

Wal, van der, J. (1987). Comparison of the throughput of the M|Er|1|N and the M|M|1|N queues. (Revised, 1988 ed.) (Memorandum COSOR; Vol. 8736). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1987

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

Memorandum COSOR 87-36

COMPARISON OF THE THROUGHPUT OF THE M I Er I 1 I N AND THE M I M I 1 I N QUEUES

Jan van der Wal

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513 5600 MB Eindhoven The Netherlands Eindhoven. December 1987 (Revised November 1988) The Netherlands

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COMPARISON OF THE THROUGHPUT OF THE

M

I

Er

11 I

N

AND THE

M

I

Mill

N QUEUES

ABSTRACT

It is shown that the throughput in the M I Er I 1 I N queue is at least equal to

the throughput in the M I Mil I N queue if the interarrival times and average

service times in the two queues are equal.

1. Introduction

It is well-known that in most queueing systems the performance decreases if the randomness increases. For instance, the average responsetime in the MIG I 1 I FCFS queue is equal to

02:

2

•

1~

with I..l and a the mean and variance of the service time and p the server utilization.

And thus the responsetime is monotonically increasing in the variance of the service time.

In this paper the queueing systems, M I Er I 11 N and M 1M 11 I N, further referred to as the

Erlang system and the exponential system, with equal arrival rate A. and average service time 1/1..l will be compared. It will be shown that the throughput of the Erlang system is at least equal to the throughput of the exponential system.

In order to answer monotonicity questions various techniques have been used. One of most powerful ones seems to be the coupling method, which for queueing networks very often works, cf. Adan and Van der Wal [1], Shanthikumar and Yao [4] and Tsoucas and Walrand [6]. For sin-gle queues Stoyan [5] provides a number of useful tools. A third technique is the one used here based on the comparison of rewards in Markov chains, see e.g. Van Dijk and Lamond [2], Van Dijk and Van der Wal [3] and Van der Wal [7].

Here we consider as rewards the expected busy period lengths of the two systems. Since the idle periods are equal it suffices to show that the mean busy period in the Erlang system is at least equal to the one in the exponential system.

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-

2-2. Some notations and preliminaries

In order to prove that the Erlang system has the higher throughput, first some properties of the

M I Mil I N queue are studied.

The mean busy period duration in the M I Mil I s queue is denoted by B (A, f.l. s) or shortly

B (s). Further. wj(N) or simply Wj denotes the expected remaining busy period in the

M I Mil I N queue when there are i jobs in the system.

Note that the remaining busy period with i jobs in the system is the time to reduce the number of jobs in the system from i to i - I plus the time to clear the system with i - I jobs. So the time to reduce the number of jobs from i to i - I in the M I Mil I s queue is the same as the time to reduce the number of jobs from 1 to 0 in the M I Mil I s - i

+

1 queue.

The Wj satisfy

WI =B(N)

(2.1) W2 =B(N-l)+w}

Writing p

=

1Jf.l the B (i) are given by the follOwing standard result.

Lemma!

For all A and f.l. and all i

(2.2) B (0

=

(p + p 2

+ .. , +

p') -. 1 A

Proof. The mean busy period is equal to the mean idle period 111.. multiplied by the ratio of the busy fraction (p + p2

+ ... +

pi)! (1

+

p

+ ' .. +

pi) and the idle fraction 1/(1

+

P

+ ,., +

pi).

which yields (2.2), []

In the sequel we need the following result which compares the busy periods of the M I Mil I s

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3 -Lemma 2

For all A, J1 and s

(2.3) 1 +AB(s)-J1B(s

+

1)=0 .

Proof. Straightforward from (2.2).

o

Further, let Uj denote the remaining busy period in the M I Er I 1 I N queue if there are i phases in the system. (For i = nr

+

k this means n

+

1 jobs in the system and the one in service has already completed r - k of its r phases.)

Our aim is to prove

(2.4) ur~ WI

i.e., the busy period in the Erlang system is larger than or equal to the one in the exponential sys-tem.

First note that the Ui, i = 1, ... ,Nr, satisfy the following set of linear equations, where we

assume (without loss of generality) that 1.+ rJ1 = 1.

{

unr+k

=

1

+

M(n+l)r+k

+

rJ1Unr+k-l

U(/,/-l)r+k = 1

+

M(/,/-l)r+k

+

rJ1UCN-l)r+k-I

, n

=

0, 1, ... ,N - 2. k

=

1, 2, ... ,

r

, k

=

1,2 •...• r

where Uo is defined 0 and 1 is the expected time until the next event, an accepted or rejected arrival with probability A and a phase completion with probability rJ1.

The solution of this scheme can be shown to be unique.

To prove that the busy period U_rsatisfies (2.4), we will use some properties of the method of

suc-cessive approximations.

Define the operator L on vectors (Xl> Xl • .••• XNr) in JRNr by

{

(Lx)nr+k

=

1

+

AxCn+l)r+k

+

rJ1Xnr+k-1 • n = 0, 1, ... ,N - 2, k

=

1,2, .. '. r

(Lx)(N-l)r+k = 1

+

Ax(/'/-l)r+k

+

rJ1X(/'/-I)r+k-I ,k

=

1,2, ... , r

wherexo := O.

So U can be seen as the unique solution of Lx = X, and U can be approximated by the successive approximation scheme:

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-4-{

Choose

uo

and compute for

n

=

0,1, ...

Ull+l :=Lull

•

We have

Lemma 3

(i) For all

x

E JRNr

(ii) The operator L is monotone, i.e.

x;?! y implies Lx ;?! Ly

Corollary

If Lx;?! x then u ;?! x.

Proof:

From Lemma 3(ii) it follows that Lx;?! x implies L 2x;?! Lx;?! x, and in general L IIX;?! X. SO with

Lemma 3(i) also u 2: x.

0

This corollary will be exploited as follows.

We construct a vector

z

such that Z/cr

=

Wk, k

=

1, 2, ... ,N which has, as will be shown, the

pro-perty that Lz;?! z.

Then by the corollary u;?! z, so in particular

ur;?! Zr =Wl

which is the result (2.4) we want to prove.

We define z as linear interpolation of W

(2.5) Znr+k=wlI+-(WII+I-WII) k ,n=O, I, "',N-I,k=I,2, " ' , r

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5

-3. The

proof of

Lz ~ z

In order to establish (Lz)j:2 Zj we distinguish two cases

A : i =

nr

+

k with

n

= 0, 1, ... , N - 2 and k = 1, 2, ... ,

r

B: i = (N - 1) r

+

k with k = 1, 2, ... ,r .

So A is the case that an arriving job can be admitted and B the case that a new arrival has to be rejected.

Case A:

(LZ)nr+k

=

1

+

Az(n+I)r+k

+

rJ..lZnr+k-l ,

Thus, using (2.5) and A

+

rll

=

1,

With (2.1) this can be rewritten as

k

=

1 +AB(N -n)+A- [B(N -n -l)-B(N -n)]-IlB(N -n) r

k

=

1

+

AB(N

-n

-l)-IlB(N - n)

+

(A-A -) [B(N -n)-B(N -n -1)]:2

° ,

r since by Lemma 2 1 +AB(N -n -1) -IlB(N -n) =0 and by Lemma 1 N-n B(N -n)-B(N -n -1)=

T

>

0 .

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·6-Case B: (LZ)(N-l)r+k - Z (N-I)r+k

=

1

+

A.z (N-l)r+k

+

rJ.l.Z (N-J)r+k-l - Z (N-l)r+k

=

1

+

rJ.l.(z (N-l)r+k-l - Z(N-I)r+k] k-l k = 1

+

rJ.l.[WN-l

+

--(WN - WN-l) - WN-l - -(WN - WN - 1)]

r

=

1 - J.I.(WN - WN-l)

=

1 - J.LB (1)

=

0 ,

where the last equality follows from B (1) being just the service time of one job, so

..!..

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7 -4. Conclusion

From the proof in the preceding section it follows that

u

r ~ WI, which implies that the throughput in the M I Er I 1 I N queue is larger than the throughput of the M I Mil I N queue with the same average interarrival and service times.

It is unclear how this line of proof can be extended to prove monotonicity in r (the number of Erlang phases) or to compare the M I Hr I 11 Nand M I M I 11 N queues (Hr denoting hyperex-ponential service times.

In a companion caper we will address the comparison of MIG I 1 I N systems using techniques from Stoyan [5].

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-8-References

[1] U.B.F. Adan and 1. van der Wal, Monotonicity of the throughput of a closed queueing net-work in the number of jobs, to appear in Operations Research.

[2] N.M. van Dijk and B.F. Lamond, Simple bounds for finite single server exponential tandem queues, Operations Research 36 (1988),470-477.

[3] N.M. van Dijk and 1. van der Wal, Simple bounds and monotonicity results for finite multi-server exponential tandem queues, to appear in Queueing Systems.

[4] J.G. Shanthikuman and D.D. Yao, General queueing networks: Representation and stochas-tic monotonicity, Proceedings of the 26th IEEE Conference on Decision and Control (1987), 1084-1987.

[5] D. Stoyan. Comparison methods for queues and other stochastic models, Wiley, New York (1983).

[6] P. Tsoucas and 1. Walrand, Monotonicity of throughput in non-Markovian networks. to appear in Journal of Applied Probability.

[7] J. van der Walt Monotonicity of the throughput of a closed exponential network in the number of jobs, Memorandum COSOR 85-21, Eindhoven University of Technology. Department of Mathematics and Computing Science.