On two parallel queues with one-way overflow

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van Doremalen, J. B. M., & Wessels, J. (1982). On two parallel queues with one-way overflow. (Memorandum COSOR; Vol. 8215). Technische Hogeschool Eindhoven.

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Department of Mathematics and Computing Science

Memorandum COSOR 82 - 15

On two parallel queues with ona-way overflow by

J. van Doremalen and

J. Wessels

Eindhoven, the Netherlands

J. van Doremalen *) and

J. Wessels *)

Abstract

This paper deals with a system consisting of two parallel multiserver queues with waiting places, which receive demands from two types of customers res-pectively. Customers for the first queue are allowed to overflow to the second queue, if they are blocked at the first queue.

The first sections of the paper are devoted to the numerical solution of the linear system of equilibrium equations in order to be able to compute loss probabilities and average system times. The algebraic structure of the linear system leads naturally to the use of a block iterative method. An iteration procedure based on the Accelerated Overrelaxation (AOR) method is described. Then the behaviour of systems with two parallel queues is discussed. Remarks are made with respect to some general notions on how to find queueing systems that satisfy certain requirements.

*) Department of Mathematics and Computing Science, University of Technology, P.O. Box 513,

1. Introduction

This paper deals with a queueing system where overflow is allowed for. The

-system consists of two finite queues with s servers, n waiting places and

t t

size k

=

s + n , £ = 1 and 2. Each queue has its own arrival stream of

t ~ t

customers. These two types of customers ~rrive according to independent Poisson processes with rates At > 0, t = 1 and 2. The service times are in-dependent and negative expontentially distributed with service rate ~t for queue!

=

1 and 2.

A customer of type 2 is served Urnnediately if a server in queue 2 is not busy and is queued if all servers are busy and a waiting place is free. Else he is blocked and cleared from the system. For a customer of type 1 the situation is different. If he is blocked at queue 1, he is not necessarily cleared from the system, but flows over to queue 2. At queue 2 he is served if a server is not busy and queued if all servers are busy and a waiting place is free. Only if he is blocked both at queue 1 and queue 2, he is cleared from the system.

In both queues the service pattern is first-come first-served and indepen-dent of the type of customer.

Private communications with Diane Sheng of Bell Labs at Holmdel (N.J.) brought the importance of such queueing systems as models for business telephone

systems to our attention. With John Rath she discussed an approximation

method in Rathand Sheng [1979J, which is based on the work of Kuczura [1972]. He analysis the queue with the overflow stream as a queue with mixed renewal

andPoissoninput. Morrison [1981J describes an exact direct method to analyse similar systems. In Brandwajn [1979J more general two-dimensional queueing

systems are analyzed with an iterative method based on probabilistic consi-derations.

We give an alternative exact analysis based on the algebraic structure of the set of equilibrium equations. As has been pointed out by Wallace [1974J, Muller [1981J et a1., the exploitation of the algebraic structure of a set of linear equations can be very useful in finding efficient solution methods. It appears that in our case this approach is very fruitful indeed.

Goal of our analysis is to find an efficient method to evaluate interesting steady-state quantities for the queueing system with overflow, for example loss probabilities and average system times. In Section 2 we give these quantities as functions of the steady-state probabilities. Apparently we have to compute the probabilities.

Let p . . denote the steady-state probability that i customers are in queue 1,J

and j in queue 2. Then the p . . 's have to satisfy a set of two dimensional 1,J

birth-and-death equations. A natural partitioning of the p . . 's leaves the 1,J

corresponding matrix of transition rates in a block tridiagonal structure, as we show in Section 2.

Consequently a block iterative method is a natural way to evaluate the p . . 's.

1,J

In Section 3 a method is described based on the Accelerated Overrelaxation (AOR) method developed by Hadjidimos [1978J. An efficient iteration procedure is discussed, which is based on interesting properties of the iteration

matrix. It is interesting to note that the method of Brandwajn [1979J for handling two-dimensional queueing systems, for this specific system, is strongly related to our method, but for the use of acceleration and over-relaxation parameters.

In Section 4 short notes with respect to the behaviour and the design of queueing systems with one-way overflow are given. As we have a method to evaluate steady-state quantities at our disposal, we are able to compare these queueing systems. Numerical examples support the intuitive ideas on the system with one-way overflow. As a consequence of the observations the design of a conversational computer program for the analysis of systems with two parallel queues is recommended.

Concluding remarks are made in Section 5.

2. The equilibrium equations, the loss probabilities and the average system times

The system consisting of two parallel-queues with one-way overflow can be described as a continuous-time Markov process on a two-dimensional

state-space. Thus a set of two-dimensional birth-and-death equations and a nor-malization equation determine uniquely the steady-state probabilities p, .

:I., J

that i customers are in the first queue and j customers in the second. For i

=

O,l, ••• ,k

l and j

=

O,1, ••• ,k2, the equilibrium equations are given by

(1)

At(i-l)p'_l . + Jl₁(i+l)p'+1 . + _A2(i,j-l)p. '-I + Jl₂(j+l)p. '+1

1 ,J 3. ,J 1,] 1,]

where the state-dependent.instreamrates Al (i) and A

2(i,j) are defined by

(2) Al (i) .. { A01

i .. O,l, ••• ,k}-}

1

"'2

i

=

0,1, ••• ,k1 - 1 and j == 0, 1 , • , • , k2 - I

(3) _"'2(i,j) = _{Al + A2} i == k] and j := 0,1, ••• ,k₂ - 1

°

i == k _] and j ... k2

and where the state-dependent service rates ~l(i) and ~2(j) are defined by

(4) t := 1,2 and i == O,l, ••• ,k

i .

If kl and k2 are large, the equilibrium equations and the normalization equation form a large linear system, that can be solved efficiently only, if the underlying structure of the system is exploited. We show that a natural partitioning of the vector of steady-state probabilities to the first variable i leaves the set of equations in a block tridiagonal struc-ture.

Let the vector of steady-probabilities p . . be denoted as a row-vector

J.,J n p €

:m. ,

where n := (k 1+l)(k2+1), that is partitioned as k 2+1

with the row-vectors p. €

:m.

1.

(6) _{p. == (p. O,P. l'···'P. k )} J. 1, 1, 1., 2

i ::::: 0',1, ••• ,kI' defined as

Then the vector p is the solution of the linear system

nxn .

where e denotes a column-vector of ones and where Q E]E 1S the matrix

of transition rates that has the following block tridiagonal structure, corresponding to the partitioning of p in (5),

DO -F ₀ 0 0 -E _Dl -F

.

1

.

1

. .

.

.

(8) Q = 0

.

_.

.

.

0

.

.

-F_{k -)}

. .

_.

_.

_.

1

.

.

0 0 -E _Dk kl ₁

The diagonal blocks D

i, i ,.. O,l, ••• ,k}, are non-singular tridiagonal matri-ces given by aiO -ciO 0 -b il ail -Cit

. .

(9) D. =

.

. .

.

1

.

.

_.

.

.

_.

.

_.

_.

_.

.

-c ik -1

.

_.

.

• 2

.

_{. .}

0 -b. 1k2 a' k 1 2

where the transition rates a .. , b .• and c .. are defined by

1J 1J 1J (10) (11 ) (12) b.. ,.. J.I 2 (j) , 1J j = 0, 1 , ••• ,k2 ' j

=

_{1,2, ••• ,k2 ' and} j

=

O. I , ••• ,k2 - 1 •

The lower blocks E_i, i = 1,2,. •• ,kp.are diagonal matrices given by

The upper blocks Fi' i

=

O,l, ••• ,k} - 1, are diagonal matrices given by

In Section 3 the block tridiagonal structure of the matrix Q is exploited in order to evaluate the vector of steady-state probabilities p. Now we give the loss probabilities and average system times as functions of the

p. . IS. 1.,]

In order to do so it is convenient to define the marginal distributions

( 1) kl+l (2) k 2+1

P € lR and p ~ lR of queue land 2 respectively, that is

(15) and (I6) p~l) = 1. (2) p. J p. . 1.,] i O,I, ••• ,kl , j = 0, 1 , ••• , k2 •

Furthermore we define the auxiliary quantities w~R,) as

1.

(17) R, = 1,2 and i = 0,1, ••• , kR, - 1 ,

which denote the average system time for a customer accepted at queue'R. and seeing i customers before him.

Then the loss probabilities for customers of type 1 and 2, L} and L2 respec-tively, are

(I8)

and (19)

The average system times for accepted customers-of type 1 and 2, WI and W 2 respectively, are k -1 k -1

=[ \:

will

(1) 2 (2) (20) _W1 p. +

_L

w.

Pk~

I (1 - L 1) l. _{J I ,J} i=O j=O and k -1 2 (2) (2) (21) W₂

₌

_L

w. p. I (I - L₂) • J J j=O

As a marginal note we observe that the expressions (18) - (21) give a hint at how to find simple approximations for the loss probabilities and average system times.

3. The solution of the equilibrium equations

The block tridiagonal structure of the matrix Q in (8) leads in a natural way to the use of a block iterative methode in order to evaluate the vector of state probabilities and, consequently, the interesting steady-state quantities. One can verify that the structure of the matrix Q, especi-ally the tridiagonality of the diagonal blocks, implies that the computation

time of an iteration step in a block iterative method is comparable to that in an ordinary iterative method. In both cases the computation time is linear in the number of equations. As block iterative methods in general are con-verging faster than ordinary methods, these have to be preferred. For a de-tailed discussion of (block)~iterative methods we confer to Varga [1962J and Young [l971J.

In this section we describe a block iterative methode based on the Accelarated Overrelaxation (AOR) method developed by Hadjidimos [1978J. The convergence of the method is discussed. An interesting result on the eigenvalue structure of the iteration matrix of the method leads to an efficient iteration proce-dure.

Let us consider the linear system (8), namely

(22) pQ

=

0 pe

=

n n x n .

where p E JR is a row-vector and where Q E JR is a block matrl.x that can be split as

(23) Q

=

D-E-F ,

where D is a block diagonal matrix with non-singular diagonal blocks, where

E is a strictly lower and where F is a strictly upper block triangular matrix. The system has a unique solution.

For r,w E JR, with W :/: 0, the linear system (22) is equivalent with

(24) p(D-rF)

=

p[(l-w)D + (w-r)F + wEJ, pe

=

1 •

-1

(25) p

=

p[(l-w)! + (w-r)U + wL](I-rU)-l , pe = 1 ,

where U

=

FD-1 and L

=

ED-I.

If we define the iteration matrix T as r,w

(26) T ,.. [(l-w)I + (w-r)U + wL](I-rU) -1 , r,w

then a meaningful iteration scheme for the evaluation of p is given by

(27) m <i: 0 •

This scheme will be called the Block Accelerated Overrelaxation (BAOR)-scheme henceforward. Following Hadjidimos [1978J we will call r the acceleration and w the overrelaxation parameter.

The convergence of the BAOR-scheme with iteration matrix T to the solution r,w

of the linear system (22) depends on the properties of the matrix Q, the parameters rand wand the starting-vector p(O). If T • is a diagonalizable

r,w

matrix with an eigenvalue structure given by 10'1

₁

> 10'2

₁

<i: 10'3 1 <i: ••• <i:

I

ani

and if p(O) is not an element of the orthogonal complement of the eigenspace of 0'1"then the BAOR-scheme converges to a normalized eigenvector of Tr,w corresponding to the eigenvalue 0'1 with convergence factor p _r,w given by

10'2 1

I

IO'}I

(confer: Stoer and Bulirsch [1973: 45-46J ).

The convergence factor p of the BAOR-scheme with iteration matrix T

r,w r,w is defined as (28) = 1 . (m) Pr,w 1m p _r,w ~ where (m) II p(m) _ p (m-l)1I (29) co Pr,w ,.. II p (m-l) _ p (m-2) II 00

Apparently the eigenvalue structure of the iteration matrix T ~s

impor-r,w

tant for the convergence of the BAOR-scheme. If the largest eigenvalue of T _r,w is 1 and unique, then the scheme converges to a solution of (22) with a convergence factor p given by the second largest eigenvalue. Thus an

r,w

efficient iteration procedure has to be constructed in a way that conver-gence is assured and the converconver-gence factor is as small as possible. This implies that the eigenvalue structure of T as function of rand W has to

r,w be studied.

In the sequel of this section we discuss an iteration procedure to solve the set of equilibrium equations (7) with the matrix of transition rates Q (8). It appears that Q has interesting matrix properties, which give infor-mation on the eigenvalue structure of the corresponding matrices T • This

r,w notion is used in the design of an efficient iteration procedure.

From the theorey of matrix iterative methods it follows that Q is a

so-called "consistently-orderedl l _{2-cyclic matrix (confer: Varga [1962: 97-103J).}

This implies that the eigenvalue set of a corresponding iteration matrix T can be derived from the eigenvalue set of the Block Jacobi matrix

r,w

TO,}

=

(E+F)D-1 and vice-versa as a result of the following theorem.

Theorem 1. Let Q be a "consistently-ordered" 2,..cyclic matrix, which can be splitted as Q = D-E-F, where D is a block diagonal matrix with non-singular diagonal blocks, where E is a strictly lower and where F is a strictly upper block triangular matrix. If T is an eigenvalue of the corresponding Block Jacobi matrix TO,l' then cr is an eigenvalue of the corresponding matrix T ., when cr satisfies

(30) (u-l+w) 2

=

WT 2 (r(u-l) + w) and vice versa.

Proof. The theorem is a two-parameter generalization of Theorem 4.3 in

Varga [1962: 106-l07J. The proof proceeds analogously.

o

As a consequence it is possible to give a relationship between the

eigen-value sets of two arbitrary iteration matrices T and T , This

x

J ,WI r2,w2

relationship shall exploited in the development of the iterationprocedure.

Let us consider a converging w(O) = I and starting vector

iteration procedure with parameters r (0)

=

1 and

(0) • • . T

P • We note that the ~terat~on matr~x 1,1

of this scheme is the well known Block Gauss-Seidel matrix, It is easily verified that 0 and 1 are eigenvalues of TI,I' In the course of the proce-dure an approximation PI,l ~or the convergence factor PI,t of the scheme can

(\I)

be evaluated from the subsequent values PI l'

_,

\I

=

2,3, •••• With this in-formation we try to find values for the parameters rand w for which a fas-ter convergence might be obtained. Considering the specific structure of the matrix Q, it is not unreasonable to use the following corollary.

Corollary 1. Let Q be defined as in Theorem 1. If the eigenvalue structure of TI,1 is given by TI

=

1, T2

=

T3= •••

=

_{Tn- J}

=

p and Tn

=

0, where

o

< p < 1, then the minimal convergence factor for the BAOR-scheme is ob-tained for the pair of values for die parameters rand w r

*

and w

*

defined by

The corresponding convergence factor P"I<

*

~is given by r ,w (32) Proof. p .-= (1 -

It -

p)

/ 0

+

II -

p) •

* *

r ,w

From Theorem 1 it follows, that, if T € [0,1] is an eigenvalue of

TI,l' then 0₁ (r,w,T) and 02(r,w,T) are eigenvalues of Tr,w ' where

(33) with and (35) - 1 - wf.(r,T) , 1. i

=

1,2 , i == 1,2 •

Consequently the eigenvalues of T in this specific case are r,w 0 1 (r, w, 1)

=

1 - w (2 -r ) and (j 2 (r , w, I)

=

1 • If we define 0 as r,w (36) o _r,w =max{ll-wl, !t-w(2-r)l, 10t(r,w,p)L 102(r,w,p)l} , j.

then the BAOR-scheme with iteration matrix T converges if 0 < 1

r,w r,w

*

with convergence factor 0 • It is elementary but tedious to determine r r,w

*

and w such that

(37) _p

_*

_*

== min { p

I

r, w E JR, w :J: 0 and 0 < 1} •

r,w r,w

In the iteration procedure we set reO) and w(O) consequently tor(l) and w(l) defined by

r (1) __

,J

I ) /

(38) w

=

2 / (1 + il ,.. PI ,J) •

If we had started the iteration procedure with arbitrary rCO) and w(O) for which the scheme converges, we could have found new values reI) and wet) also. With an approximation

P

(0) (0) for the convergence factor p (0) (0) an

r ,w r ,w

approximation PI,I for the convergence factor of the TI , I-scheme can be ob-tained form Theorem 1, namely

(39) PI,t

=

(p (0) (0) - 1 + w)2 / (w(r(p (0) (0) - 1) + w»

r ,w r ,w

And we can reset reO) and w(O) to rei) and wei) as in (38).

We executed numerical experiments with the above described procedure. The results were good. It appeared that the BAOR-scheme with an adjustment of the acceleration and overrelaxation parameters performed 3 to 10 times as fast as the ordinary BGS-scheme with iteration matrix T_l,l'

Especially for the computation of loss probabilities and average system times the iteration procedure is efficient, as these quantities have to be computed in three or four decimals precision only. Furthermore, one can consider an extrapolation for these quantities in an early stage of the procedure.

It should be noted that the choice of good starting values reO) and w(O) for the acceleration and overrelaxation parameters is important. Less crucial is the choice of the starting vector p(O). This partly is a consequence of the use of a block-iterative method instead of a point-iterative method, as deviations are much faster spreaded in the block iterative method for this particular problem.

4. Remarks on the behaviour and design of overflow systems

In real-life situations where the installation of an overflow mechanism is considered, it is necessary to obtain information on the behaviour of queueing systems with overflow. Namely, according to requirements with respect to

loss probabilities and average system times, a satisfying design for the system has to be found. Of course a comparison of systems with and without overflow has to be possible.

For a queueing system consisting of two parallel queues with one-way over-flow a procedure to evaluate the performance quantities has been deve~oped in Section 3. The analysis of a system consisting of two independent queues is elementary, as each queue can be regarded as a M / M / s / k-queue

(a queue with a Poisson arrival process, s servers, each with exponentially distributed service times, and a maximum of k customers in the system). The equilibrium distribution of such a queue is easy to obtain as the normalized solution of a set of one-dimensional birth-and-death equations.

Consequently we are able to analyze arbitrary queueing systems with and without overflow and a comparison of such systems is possible. In practice we suggest the development of a conversational computer program to find a satisfying system. There are three major reasons for this suggestion. In the first place a general optimization method seems inadequate due to the large number of both integer and real variables involved and due to the difficulty

in formulating a representative object function and plausible restrictions. Secondly, a conversational program has the advantage that the decision maker gets a good insight in the behaviour of the systems he considers. Thus, he will not only find a satisfying system quickly, but he also has the possibility

to reset his goals if new ideas might rise in course ,of his conversation with the program. The third reason is that a conversational program can be extended with extra options, for example procedures for rude and sophisti-cated approximations and for all kinds of output.

In a subsequent paper we will go into the matter of developing such a con-versational program more closely. In this section we give some general re-marks with respect to the design and behaviour of queueing systems with and without overf1owo Three numerical examples support these ideas.

An important observation for the study of overflow systems is that the over-flow stream is a more irregular process than the main Poisson instream. As

has been observed earlier by Kuczura [1972J in the analysis of

GI + M I Mis I k queues, a consequence of this phenomenon is that over-flowing ,customers are more often blocked and have, if they are accepted, longer system times, than the Poissonian customers. The phenomenon in par-ticular makes the study of systems with overflow difficult. But some general ideas can be given.

A problem in designing this type of queueing systems is how many servers are needed. Usually, the total capacity of the system and the total instream rate have to be of the same order of magnitude.

For a system without overflow this means that sl~l ~ Al and that s2~2 ~ _A20

In a system with one-way overflow a lowerbound for the rate of the overflow process can be given by max{O,A₁ - ~lsl}o Consequently, we have the require-ments that SIP} +s2P2 ~ Al + A2 and that s2~2 ~ A2 + max{O,A) - ~lsl}' These observations are rahter accurate if the numbers of waiting places n₁ and n₂ are large. Thus, if we allow for relatively large system times, the

design of the system is easy. However, not much can be gained by implemen-ting an overflow mechanism. This is shown in Example 1.

Much more interesting is the situation if the numbers. of waiting places are small. Now, the allowance of overflow has a very positive influence on the loss probability of type 1 customers. It appears that more efficient systems are obtained allowing overflow. This is shown in Example 2.

As a general result it appears that overflow has to be considered in those situations where short system times are necessary and where the losing of type 1 customers is much worse than the losing of customers of type 2. Then there is the question how many waiting places have to be installed.

1 n₁ 1 n₂

Important quantities to answer this problem are -- + ---- and -- + ---- ,

PI _{slP l P2} s2P2

which indicate the maximum average system times in queues 1 and 2 respec-tively. For the system without overflow these quantities along with the requirements for the system times give good information on how to choose n

l

and n

2, In case overflow is allowed for, the situation is more complex, as for customers of type 1 both quantities become important. An example of such a situation is given in Example 3.

We find as a general result that overflow has to be considered if the require-ments for the system times WI and W

z

do not differ too much.

The three following examples try to give a picture of the above notions with respect to the design and behaviour of queueing systems with one-way overflow.

Example 1. We consider a telephone exchange where two groups of servers are handling two types of customers. The respective instream rates and service rates are Al

=

A2

=

10 and ].11

=

\.1

2

=

1. For both types of customers- a

maxi-,

mum system time of twice the service time is acceptable and consequently we install n₁

=

51 and n

2

=

$2 waiting places in queue 1 and queue 2. For the loss probabilities L1 and L2 we would like to have Ll < 0.01 and L2 < 0.10. In Table 1 performance quantities for some reasonable designs are pictured.

configuration system with overflow system without overflow

n 1 n2 51 52 L} L2 WI W2 Ll L2 W} W2 10 10 10 10 .015 .103 1.41 1.47 .068 .068 1.40 1.40 11 9 11 9 .008 .148 1.28 1.56 .028 .133 1.27 1.53 9 11 9 11 .022 .081 1.53 1.42 .133 .028 1.53 1.27 11 10 11 10 .005 .082 1.28 1.43 .028 .068 1.27 1.40 10 11 10 11 .009 .053 1.41 1.35 .068 .028 1.40 1.27 12 9 12 9 .002 .137 1.16 1.54 .009 .133 1.16 1.53 9 12 9 12 .013 .042 1.52 1.31 .133 .009 1.53 1.16

Table 1. Steady-state quantities for a system with instream rates

For the system with overflow sl = 11 ans s2 = 10 gives a satisfying system; for the system without overflow 51 =]2 and 52

=

10 is satisfying. We gain 4.5% in the number of servers by the implementation of an overflow mechanism.

Example 2. We consider the same system as in Example 1. No~there are no ~aiting times allowed and~e set n

1

=

nZ

=

O. In Table 2 performance quanti-ties for a number of reasonable designs are pictured. For the system ~ith overflo~ s1

=

s2

=

14 is satisfying, and for the 5ystem~ithout overf1o~ 51

=

18, 52

=

13. Now 9.7% in the number of servers is gained by the imple-mentation of an overflow mechanism.

-configuration system with overf1o~ system without overflow

!

I

n 1 n2 s} s2 Ll L2 WI W2 L} L2 WI W2

a

0 14 14 .008 .073 1.00 1.00 .057 .057 1.00 1.00 0 0 15 13 .007 .097 1.00 1.00 .036 .084 1.00 1.00 0 0 16 12 .005 .128 1.00 1.00 .022 .120 1.00 1.00 0 0 12 16 .011 .046 1.00 1.00 .120 .022 1.00 1.00 0 0 17 11 .004 .168 1.00 1.00 .013 .163 1.00 1.00 0 0 18 11 .002 .166 1.00 1.00 .007 .163 1.00 1.00

2. Steady-state quantities for a system with instream rates

...---Al

=

A2

=

10 and service rates ~1

=

~2

=

1.

Example 3. The same system as in Example 1 is considered. No~ the require-ments for WI and W

_z

are WI < _{1.10 and W2} < 5.00. In Table 3 the performance

quantities for a number of reasonable designs are pictured. It appears that the requirements lead to an overf1o~ system with two nearly independent queues. Consequently, we find for the 5y~tem s with and without overflo~ a satisfying configuration ~ith sl = 13, n}

=

12, s2

=

_{9 and n2}

=

36.

configuration system with overflow system without overflow -n 1 n2 51 s2 LI LZ WI W2 Ll LZ WI W

z

.

0 40 12 10 .020 .103 1.33 4.12 .120 .022 1.00 2.88 0 40 13 10 .011 .078 1.22 3.89 .084 .022 1.00 2.88 1 40 13 10 .007 .061 1.16 3.68 .061 .022 1.01 2.88 1 25 13 10 .007 .069 1.09 2.48. .061 .034 1.01 2.13 2 36 13 9 .0lD .136 1.13 4.29 .045 .101 1.02 4.07 36 13 9 .001 .104 _1.10

I

4.09 .003 .101 1.08 4.07

Table 3. Steady-state quantities for a system with instream rates Al

=

A2

=

10 and service rates ~l

=

~2

=

1.

5. Concluding remarks

In this paper a system consiting of two parallel queues with one-way overflow has been analyzed.

An

iterative method to evaluate performance quantities has been developed by exploiting the algebraic structure of the equilibrium equa-tions. We note that the method can be extended easily to solve a more general class of two-dimensional birth-and-death processes. For example the interesting class of systems with two-way overflow might be considered.

As the computation time for large systems, with kl ~ k2 ~ 50, is in the order of tens of seconds, the use of approximations for the performance quantities has to be considered. We thereby think of the method described in Rath and Sheng [1979J , but also of more straightforward aggregation and decomposition

methods. Very promising is the use

of

data evaluated in previously analyzed systems. The latter approach is possible only, if for a great many systems performance quantities have to be evaluated.

In the second part of the paper we have discussed some general notions with respect to the behaviour and the design of queueing systems with one-way overflow. The development of a conversational computer program to find satis-fying queueing systems in real-life situations has been recommended.

In a subsequent paper we will study such a program. The use of approximations will be discussed also.

6. References

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