Two parallel queues with one way overflow : a matrix structure approach

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Two parallel queues with one way overflow : a matrix structure

approach

Citation for published version (APA):

van Doremalen, J. B. M. (1984). Two parallel queues with one way overflow : a matrix structure approach. (Memorandum COSOR; Vol. 8408). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1984

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Department of Mathematics and Computing Science

Memorandum COSOR 84-08

Two parallel queues with one way overflow; a matrix structure approach

by

J.B.M. van Doremalen

Eindhoven, The Netherlands August 1984

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TWO PARALLEL QUEUES WITH ONE WAY OVERFLOW A MATRIX STRUCTURE APPROACH

J.B.M. van Doremalen Abstract

This paper deals with a numerical analysis of a queueing network model of a system consisting of two parallel queues in which overflow takes precedence. The main pur-pose is to show how to develop efficient algorithms for the evaluation of steady-state probabilities based on the algebraic structure of the transition matrix.

1. Introduction

This paper deals with the numerical analysis of a two dimensional queueing network model of a system in which overflow takes precedence. Private communications with Diane Sheng of Bell Labs at Holmdel (N.J.) brought the impo~tance of such systems as models for business telephone systems to our attention.

The system consists of two finite capacity single server queues with a in first-out service discipline. At the first queue customers arrive according to a Poisson pro-cess with parameter Q. The capacity of the queue is N and the work rate of the server

is state dependent, saye , n

=

1,2, •.• ,N. If a customer finds upon arrival at the first

n

queue N customers present, he is allowed for to overflow to the second queue. At the se-cond queue, besides the overflowing customers, customers are arriving according to a Poisson process with parameter

A.

The queue has capacity M and the work rate is state dependent, say ~ , m =1,2, •.• ,M. If an arriving customer finds the second queue

satura-m

ted, he is cleared from the system. All service times are stochastically independent and negative exponentially distributed with parameter 1, normalized with respect to the state dependent work rates.

The main purpose of this paper is to show how one may develop efficient algorithms for the evaluation of the steady-state probabilities p(n,m) that n customers are at queue one and m customers at queue two. One may verify that interesting steady-state quanti-ties, for example loss prDbabiliquanti-ties, average waiting times and mean queue lengths, are simple functions of these probabilities. In section 2 it is shown that the set of

equilibrium equations for the probabilities p(n,m) has a nice matrix structure. As has been pointed out for instance by Wallace [llJ and Muller [8J, the use of the algebraic structure has to be exploited in order to come up with efficient solution methods.This is particularly true for our system as we will show in sections 3 and 4.

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In section 3 a direct method will be described, based on a straightforward decomposi-tion technique for tridiagonal matrices, which for example lias been discussed in a somewhat different setting in Varga [10: 194-197J. For larger values of Nand M this

_..

method is numerically instable and one has to resort to iterative methods.

In section 4 we will discuss a generalization of the Accelerated Overrelaxation method, which by now is a standard technique ( cf. Gaitanos,Hadjidimos and Yeyios [4J and the references mentioned there) •.

It is worthwhile to mention some other material. A very nice direct method, based on a seperation of variables, has been studied in Morrison [7J. An alternative iterative method, based on a decomposition like argument has been discussed by Brandwajn [1J. A queueing system related to our model is the system with an infinite capacity queue servicing ~e overflowing customers and a Poisson stream. Kuczura [6J discusses an embedded Markov chain approach based on the observation that the instream process of the second queue is a mixed renewal and Poisson process. Another approach is based on the technique of matrix geometric solutions. In a somewhat different queueing sys-tem Neuts and Kumar [9J discuss this approach in detail.

It should be noted that all these techniques need some adjustments to fit in with o~r queueing network model. Sometimes this will imply stronger assumptions to be made.

2. The equilibrium equations

The state of the system is described by a two dimensional vector (n,m), where nand m denote respectively the number of customers at the first and the second queue. We are interested in the evaluation of the steady-state probabilities p(n,m) that the system is in state (n,m). For n = a,1, •.. ,N,we introduce the row vectors x as

n

x_n = (p(n,0),p(n,1), ••• ,p(n,M». The equilibrium equations then can be stated in the following interesting form,

(1) _{xODO =-a.x} ₊ S1 x 1 n = a 0 '(2 ) x D = a.x n_1 (a. +

a

)x + l3n+1xn+1 n = 1 ,2, ••• , N-1 n n n n (3) _XNDN

₌

a.x N_1 SNxN n = N

where the matrices D , n-= a,1, •.. ,N, are tridiagonal matrices corresponding with the

n

transition matrices of finite capacity single server queues with Poisson instream processes with parameter A and state dependent service rates ~ , m = 1,2, ••• ,M, i.e.

n m A n (4 ) D

=

n

-A

n -). n

(5)

3

=

1

x e n

(5)

For our specific system we have A

O = A1 = ••• = AN_1 = A and AN = A + a. It will be

*

convenient to introduce 0 = 0

0 = 01 = ••• = 0N_l and 0 = ON •

•Based on the theory of continuous-time Markov-processes it follows that the solution of (1) through (3) is unique up to a normalization. The normalization is introduced as

N

l.

n=O

where e is an appropriate column-vector of ones.

It is easily verified that the transition matrix has a block tridiagonal structure which, as a transformation of (1) through (3), gives the equation

I, n = 1,2, ... ,N.

n n

*

=

0

In sections 3 and 4 we will discuss a direct and an - iterative method to evaluate the equilibrium distribution. This section is closed with an interesting alternative to the normalization equation (5).

Lemma 1

For n O,l, ••• ,N we have that xne = ~n' where ~n is given by

N •

Thus, the ~_n's are the eqUilibrium distribution of a finite capacity queue. This is

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3. A direct evaluation method.

The structure of the transition matrix in Relation (6) is block tridiagonal. An efficient way of evaluating the solution is by studying the so called L-U decompo-sition of this matrix. Let us introduce this decompodecompo-sition as follows,

(8)

=

l1 0 -B 1 l11 -B l1 N N I -f

o

I - f 1 -f N-1 I where l1

0, l11,

...

,

l1Nand fO' f 1,

...

,

fN-1 can be computed recursively from,

(9) l1 0 and fO -1

=

A O l10

Co

r

,

n

=

N

If the scheme is consistent then it is not difficult to verify that the equilibrium distribution can be computed as the solution of the following recursive scheme. First compute x

N as the solution of

and then compute x , n = O,l, .•• ,N-l, from

n

(13) x == x B l1-1

n n+l n+l n

The following theorem proves the scheme to be consistent. Theorem 1

The matrices l1

0, l11, ••• , l1N_1 are non-singular and the solution of xNl1N = 0, subject to ~e = ~N is unique.

Proof~ A matrix theoretic proof is given in the Appendix.

o

The numerical stability of the scheme is not guaranteed. As a general remark it should be stated that especially for larger values of M the method has to be dissu-aded. The more so as the computational complexity of the scheme heavily depends on this value of M. Larger values of Ngive problems in the recursion as an accumalation of rounding errors takes place. It will not be clear on forehand whether this is as bad as it seems. Apperently one has to be careful.

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5

4. A Block Accelerated Overrelaxation method

In this section we will discuss a two-parameter block iterative method to solve for the equilibrium probabilities. Let us, therefore, consider the following reformulation of the equations (6) and (5),

(14) xQ

=

0 and xe

=

1, where x

=

(x

O,x1, ••• ,xN) and where Q

=

A - B - C, with A a block diagonal matrix having

non-singular diagonal blocks AO, ••• ,A

N, with B a strictly lower and C a strictly upper

block triangular matrix having respectively the blocks B

1, ••• ,BN and the blocks CO' •• "

C

N_1• The matrix Qis a block tridiagonal matrix.

o

(0)

where p is a starting vector. This scheme with iteration matrix T will be called

r,w

the Block Accelerated Overrelaxation (BAOR) scheme, with parameters r and w. It can be viewed as an extrapolation of the successive overrelaxation method with extrapolation parameter w/r and overrelaxation parameter r ( cf. Hadjidimos and Yeyios [5J ).

The convergence of the BAOR scheme with iteration matrix T to the unique solution

r,w

of the linear system (14) depends on the properties of the matrix Q, the parameter

(0)

values r and w and the choice of the starting vector p • But even if the convergence can be assured, it still is to be questioned whether the block iterative method has to be preferred to a point iterative method. This last item, of course, heavily depends on the computational complexity of one iteration step. We will remark on these subjects, not with the intention to give an elaborate overview , but merely to show which compli-cations one might meet and which lines of argument one may follow in solving for these problems.

The convergence of the BAOR scheme is a consequence of the following well-known theorem Theorem 2

If T_r,w is a diagonalizable matrix with an eigenvalue structure given by

1011 > _{1°2 '}

~

1°31

~

...

~

10nl, and if p(O) is not an element of the othogonal com-plement of the eigenspace of 01 then the BAOR scheme converges to a normalized eigen-vector of T corresponding with the largest eigen value 01. The convergence factor

r,W

Pr,w a measure for the rate of convergence is given by 1°21/1°11. Proof: Confer Stoer and Bulirsch [1973J

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Apparently, the eigenvalue structure of T is of importance. From the theory of r,w

matrix iterative methods it follows that the transition matrix Q of Relation (14)

is a so-called "consistently ordered" two-cyclic matrix ( cf. Varga [10:97-103] ).

The relationship between the eigenvalue sets of two iteration matrices T and r,w

T

*

*,

which are derived from the same consistently ordered two-cyclic matrix, is

r ,w .

given in the next theorem. Theorem 3

*

Let a be an eigen value of the matrix T • Then 0 is an eigen value of the matrix r,w T

* *

if a* satisfies r ,a (18) -.--;:.;;;...---.;;;....~-(0* - 1 + w*)2 =

* * *

*

w

(r (0 - 1) + W ) (0 - 1 + w)2 w(r(o - 1) + w)

Proof. A consequence of Theorem 1 in Van Doremalen and Wessels [2]

o

A consequence of the theorem is that it is theoretically possible to obtain an

"optimal" choice for r and

w,

if the eigenvalue structure of a given standard matrix, e.g.· the Gauss-Seidel matrix T

1,1' has been evaluated. In practice one will not know this eigenvalue structure in detail and assumptions for this structure will have to be made in order to be able to obtain approximate values for the optimal choice of r and

w.

In Van Doremalen and Wessels [2] we have studied an algorithm in which iteratively better approximations for these optimal rand ware constructed. The algorithm is based on the following lemma, which is a consequence of Theorem 3.

I

Lemma 2

If the eigen value structure of T

1,l is given by 01

=

1, O2

=

03

=.•.. =

an_1

=

(21) x(i+1)A

₌

(1 w)x(i}A _{+ rX}(i+1) C + (w - ) (i) C + (i)B

n_ 1 n-1 wx

n n n n r xn_1 n-1 n+1 n+1

(22) _~(i+1)~

₌

(1 _{- w)xN AN +}(i) rX(i+1) (w - ) (i) C_N_₁ N_1 CN_1 + r ~-1 and for the normalization

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The following lemma shows that the normalization can be omitted. Lemma 3

If x(O) is such that x(O)e

=

~

,

n

=

O,l, ••• ,N, then the evaluation of Relations (20)

. n n . . (i+1) (i+1)

through (22) yields 1n each new 1terat10n step x such that x e c ~ ,

n n

n = O,l, ••• ,N.

Proof. Use the relations (1), (2) and (3) and the technique of the proof of Lemma 1

0

As a consequence, in each iteration step N + 1 linear-systems of size M + 1 have to be solved. Observing that the matrices AD, A

1, ••• , AN are tridiagonal and that the matrices B₁, B₂, ••• , B_N and CO' C

1, ••. , CN_1 are diagonal, it is easily verified that the computation of x(i+1) n = O,l, ••• ,N has complexity O(M). Consequently the evaluation

n

of one iteration step is O(NM).

Thus the complexity of the computation of one step in the block iterative method is linear in the number of states, as it would have been in the corresponding point ite-rative method. So with relatively little extra effort a block iteite-rative method can be implemented. The implementation of a block iterative method seems to be a good choice, as the rate of convergence in general will be much faster than for the corresponding point iterative method (cf. Varga [10:87-93J). For more details on the method we refer to Van Doremalen and Wessels [2J

5. Conclusions.

The discussions in section 3 and 4 on the use of matrix structures in the evaluation of steady-state probabilities in two dimensional queueing network models have shown this approach to be very fruitful. Other methods could have been studied also, as we have indicated in the introduction. But we have chosen for a direct way, based on the ele-mentary analysis of matrices and numerical analysis, to show that it is not a priori necessarry to have a profound knowledge of deeper stochastic theories to come up with elegant solutions of non trivial problems in queueing theory.

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Refe~ences

/1/ A BRANDWAJN

AN ITERATIVE SOLUTION OF TWO-DIMENSIONAL BIRTH-AND-DEATH PROCESSES, OPERATIONS RESEARCH, 27 (1979): 595-605.

/2/ J. VAN DOREMALEN AND J. WESSELS

ON TWO PARALLEL QUEUES WITH ONE-WAY OVERFLOW

COSOR MEMORANDUM 82 -15, EINDHOVEN UNIVERSITY OF TECHNOLOGY, EINDHOVEN (1982) •

/3/ J. VAN DOREMALEN

TWO PARALLEL QUEUES WITH ONE-WAY OVERFLOW: A MATRIX STRUCTURE APPROACH COSOR MEMORANDUM 84-08, EINDHOVEN UNIVERSITY OF TECHNOLOGY, EINDHOVEN

(1984) •

/4/ N. GAITANOS, A. HADJIDIMOS AND A. YEYIOS

OPTIMUM ACCELERATED OVERRELAXATION METHOD FOR SYSTEMS WITH POSITIVE DEFINITE COEFFICIENT MATRIX,

S.I.A.M. J. OF NUM. ANALYSIS, 20(1983) : 774-783. /5/ A. HADJIDIMOS AND A. YEYIOS

THE PRINCIPLE OF EXTRAPOLATION IN CONNECTION WITH THE ACCELATED OVER-RELAXATION METHOD

LIN. ALG. AND ITS APPL. 30 (1980) : 115-128. /6/ A. KUCZURA

QUEUES WITH MIXED RENEWAL AND POISSON INPUTS THE BELL SYSTEM TECHN. J. 51 (1972) : 1305-1326. /7/ J.A. MORRISON

AN OVERFLOW SYSTEM IN WHICH QUEUEING TAKES PRECEDENCE THE BELL SYSTEM TECHN. J. 60 (1981) : 1-12.

/8/ B. MOLLER

DECOMPOSITION METHODS IN THE CONSTRUCTION AND NUMERICAL SOLUTION OF QUEUEING NETWORK MODELS

IN: PERFORMANCE 181, KIJLSTRA ED., 99-112, NORTH-HOLLAND PUBL. COMP., AMSTERDAM (1981).

/9/ M. NEUTS AND S. KUMAR

ALGORITHMIC SOLUTIONS OF SOME QUEUES WITH OVERFLOW MAN. SCI. 28 (1982) : 925-935.

/10/ R.S. VARGA

MATRIX ITERATIVE ANALYSIS

PRENTICE HALL, ENGLEWOOD CLIFFS (N.J.) (1962). /11/ V.L. WALLACE

ALGEBRAIC TECHNIQUES FOR THE NUMERICAL SOLUTION OF QUEUEING NETWORKS IN: PROC. CONF. OF MATH. METHODS IN QUEUEING THEORY, LECTURE NOTES IN ECONOMICS AND MATHEMATICAL SCIENCES 98 : 295-304 (1974).

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Appendix

A1. Some elements of matrix theory.

We will introduce some elementary theorems from matrix theory which we will need in the proof of Theorem 1. Let us consider a matrix A EJRn)tn with a set of eigen-values a(A) = {AE~IA-AI is singular}.

-1

Any such matrix A can be transformed into sAs where A is a special type of block diagonal matrix and where S is a regular matrix which defines a basis of the space JRn. We will introduce an adjusted description of the Jordan-normal form.

Theorem A1

Assume that AEJRnxn has an eigenvalue structure a(A) = {A

1, . . . ,Ap} and that the algebraic multiplicities of the eigenvalues are given by ~, k=1, . . . ,p. Then there is a regular matrix S ( with inverse S-1) and a matrix A = diag(1I.

1, .... ' Ap)

9

such that A sAs-1. The matrix A is a block diagonal matrix with diagonal blocks 1I.

k = Aklk + Hk EJR~xmk, where Ik is the unitary matrix and where Hk is a strictly upper diagonal matrix.

Proof: Confer Zurmuhl and Falk [1984J

This theorem can be used to prove some interesting lemmata on eigenvalue sets.

Lemma A1

o

Let p (A) .EOa.Am i be a polynomial in A, then p(A)Ea(p(A» if AEa(A).

~= ~

m i -1

Proof: Use of Theorem A1 yields p(A) = iEOaiSA S . Noting that det(p(A) - p(AI»

m i i

det (iEOai (A -A I » , we see that for AEa(A) det(p(A) - p(AI») = O.

Lemma A2

-1 -1

Let A be a non-singular matrix, then a(A ) = a(A) . -1 -1 -1 .

Proof: Use of Theorem A1 yields A = S1I. S . Apparently the diagonal blocks

o

A

(12)

-1 -1

*

A k

=

Ak Ik + Hk, where Hk -1 -1 we have that

A

k

E a(A ).

is a strictly upper diagonal matrix. As a consequence

*

The construction of H

k is discussed in A3.

o

Another important theorem gives bounds on the set of eigenvalues.

Theorem A2 nxn

Let A E ~ be a matrix with elements a. 0'

lJ

Gershgorin-circles C. are defined by

l

C. = {z E {[

I

Iz-a ..

I

~.1:.

Ia ..

I }

l I I J~l lJ

and furthermore, a(A)nc. is non-empty for i

l -

-Proof: Confer Stoer and Bulrisch [1973J

A2. Proof of Theorem 1.

n then a(A) C i~l C

i , where the

1,2, .••,n.

o

Theorem 1 is a consequence of the following three lemmata.

Lemma A3

a(D) C {Z E {[ I Re(z)~O}

Proof: A consequence of Theorem A2.

Lemma A4

o

6 is non-singular for n

n O,l, . . . ,N-l

Proof: The proof of the lemma is by induction. Crucial is the observation that a matrix A is singular i f and onlv if 0 E a(A). I t will be nroven that

a(D. )n{o}_n _- = ~, n'" O,l, •.. ,N-l.

First we will introduce some notations. The Jordan-normal form

ot

D is given as -1

D

=

sAs

.

Furthermore the functions f are defined by the following recursion. n Z E {[ and n = 0 f (z) n z + Sn f n_1(z) ,z E {[ and n 1,2, . . . ,N-l

(13)

f_n(a(o))_. + a 11 2. a(~ ) C {z E ~

I

R (z) ~ a} n e -1 -1 1 3. S ~ S = diag( (f (A ) + a)- r + H k) n k n k k n,

Statement 2 provep that~nis non-singular. The matrix Hn,k is a strictly upper diagonal block for k = 1,2, . . . ,p. Observe that by definition a(D) =+~-,~-;-;-,kp};

Induction step: n = 0

-1 -1

D + aI and D = sAs we have that ~o = S(A + aIlS . Thi5 implies

a(~o) fO(a(D)) + a (statement 1). From Lemma A3 i t follows that

a(~O) C {z E ~

I

Re(Z) ~ a} (statement 2) and consequently ~O is non-singular.

As a consequence of Lemma A2 we can construct H

O,1, . . . ,HO,p 3 holds.

Induction step: n > O.

such that statement

-1

Observing that ~ = D + (a + S ) r - as ~ 1 we find with the induction assumption

n n n

n-S-1~ns

= A + (a + Sn)I - aS

n

di~g(

(fn_1( \ ) + a)-1 rk + Hn-1,k) For A

k E a(D) this yields

a+ S ) - as (f 1(A

k) + a)-l E

a(~)

n n n- n

or fn(A

k) + a E a(~n) , and so a(~n)

=

fn(a(D)) + a (statement 1) From the induction assumption i t follows that ~(fn_1(A

k)) ~ 0, k and so elementary complex arithmetic yields

1, ...

,p,

Re

which settles for statement 2 and the fact that ~ is non-singular. n

-1

Furthermore, we can write S ~ S as n -1

S ~ S n

(14)

and as a consequence we can construct H 1, ...,H such that

n, n,p

which settles for statement 3. For the construction of H see A3. n,k

Lemma AS

o

The linear system X ~

mm 0, subject to Xme ~m' has a unique solution.

Proof: We have to prove two things. First that there exists a solution and second that the solution is unique.

To prove that there exists a solution i t suffices to show that 0 E a(~ ). n

From Lemma A4 we have that f (A) + a E a(~ ), n = O,l, ..• ,N-l. It is not ~iffioult

n n

to verify that a E a(~ ), as we have that 0 E a(D) and consequently, with n

f (0) = 0, that a E a(~ ). But we can even give a right eigenvector. With De = 0

n n

(where e is a vector of ones) we have ~Oe = (D + aI), e = ae. It is not difficult to verify that e is a right eigenvector of each ~ with eigenvalue a~

0, subject to X e

=

~ .

Then by virtue of Lemma A4

m m m m m

these two solutions lead in a unique way to two different solutions of the original linear system. This is in contradiction with the uniqueness of the

solution of the set of equilibrium equations ·of a continuous-time Markov-chain.

0

A3. The construction of an inverse matrix.

Assume that A E ~nxn is an upper diagonal matrix with non-zero diagonal elements.

*

,

JJ 13

For a matrix A AI + H, where A is non-zero and where H is a strictly upper

-1

*

diagonal matrix, this scheme yields an inverse matrix

A

I + H with H strictly upper diagonal.

11.4 References

J. Stoer and R. Bulrisch [1973J, Einfuhrung in die Numerische Mathematik II,

Springer Verlag, Berlin.

R. Zurmuhl and S. Falk [1984J, Matrizen und ihre Anwendungen, Springer Verlag, Berlin.