Mathematical Methods of Operations Research (1996) 44: 49-74

Individual Blocking Probabilities in the Loss System GI + MIMINIO

ANDREAS BRANDT

Wirtschaftswissenschaftliche Fakult/it, Humboldt-Universit/it zu Berlin, Spandauer Str. 1, 10178 Berlin, Germany

Abstract: In this paper we investigate an N server loss system, where the input is a superposition of two types of traffics, namely of a renewal process and a Poisson process. The holding times of the two custome~ypes are exponentially distributed with different parameters. For this model, denoted by GI + MIMIN[O, we derive a numerical algorithm for computing the individual blocking (loss) probabilities. The analysis is given by constructing a two-dimensional embedded Markov chain and by using the intensity conservation principle as well as point process arguments. The results general- ize those of Kuczura I-8] and Willie 1-11]. Finally, for the G1 + GI]MII[O loss system we give a system of partial differential equations for the densities of the steady state distribution and discuss a special case.

Key Words: Individual blocking probabilities, loss system, superposition, embedded Markov chain, time stationary probabilities, intensity conservation principle, supplemented variables.

I Introduction

The traffic offered to service facilities in communica t ion systems is, in general, a superposit ion of different traffic streams, where the holding times may depend on the type of traffic. If one thinks of modern I S D N networks, where different types of traffic occur, the question of determining the individual blocking (loss) probabilities is of practical importance, since the desired individual blocking probabilities of the different traffic types may vary considerably.

Individual blocking probabilities were investigated in the last years for systems where the holding time distributions are identical for the different types of traffics. Kuczura [8] treated the GI + MIMIN]O system, i.e. where the input is a superposit ion of a renewal process and a Poisson process, cf. also [9]. In Willie [11] the superposit ion of a Poissonian stream with a semi-Markov stream and a general s tat ionary stream, respectively (SM + MIMINIO and G + MIMINIO), is considered. Simple explicit formulas for the individual blocking probabilities in the system G 1 + . . . + GNIM1110 are given in Willie [12] by using point process arguments.

0340- 9422/96/44:1/49 - 74 $2.50 O 1996 Physica-Verlag, Heidelberg

50 A. Brandt

In this paper we will analyze the GI + MI]~rINI0 system, where the input is a superposition of two types of traffics, namely of a renewal process and a Poisson process and the holding times are exponentially distributed where the parameter depends on the type of traffic. For this model we derive a numerical algorithm for computing the individual blocking probabilities. The approach is an extension of the ideas used in [8]. If both of the arrival processes are renewal processes, then, in general, the analysis becomes very complex and one has to look for approximations. For the case of N = 1, i.e. for the GI + GIIMIll0 system, we will illustrate these difficulties.

The paper is organized as follows. In Section 2 we describe the model and give the embedded two-dimensional Markov chain. Further a numerical algorithm for computing the stationary probabilities of the Markov chain is given. A numerical algorithm for computing the time stationary probabilities, which requires to solve N linear systems of equations recursively, is presented in Section 3. Section 4 deals with the individual blocking probabilities. The results of the previous sections provide a numerical algorithm for computing them. In case of N = 1 and equal parameters for ..the exponential service times, the formulas for the individual blocking probabilities arise as a special case of the formulas given in [12] for the G~ + "" + GNIMJll0 system. In Section 5 we present numerical results for two examples of GI streams: in the first one it is assumed that the interarrival time distribution of the GI stream is constant (i.e. D + MIMINIO) and in the second example that the GI stream corresponds to an overflow stream approximated by an interrupted Poisson process (i.e. IPP + MIJ~INI0). The numerical results show that the individual blocking probabilities m a y b e considerably different. In Section 6 we consider the loss system G! + GIIMI 110, i.e. the case of both arrival streams being renewal pro- cesses. Using the technique of supplemented variables we develop, for the den- sity of the steady state distribution, a system of partial differential equations with corresponding boundary conditions. The individual blocking probabilities can be expressed in terms of these densities. For equal holding time parameters the individual blocking probabilities can be obtained by using an argument given in [12].

2 The Model (71 + MIh~INI0 and the Embedded Markov Chain

By G1 + M[IvI[N[0 we denote an N server loss system where the input is a superposition of a renewal process with interarrival distribution A(t) and finite mean 271 = ~o (1 - A(t))dt and a Poisson process with intensity 22. The cus- tomers of the renewal process (Poisson process) are called type 1 (type 2) cus- tomers and require an exponentially distributed service time with parameter #1 (/~z). For technical reasons we assume that the renewal process (which may be

Individual Blocking Probabilities in the Loss System GI + M[ ~[N[0 51

delayed) and the Poisson process are given on the whole real line). Let Xi( t ) = X~(t + 0) be the number of type i customers in the system at t. Then _X(t) = (Xl( t ) , X2(t)) is the vector of the number of customers of different types in the system at t. The state process X = (X(t), t > 0) is not a Markov chain in general. It has the state space X = {(i,j): i + j < N, i , j e Z+}, where Z+ denotes the set of non-negative integers. Denoting by T, 1, n e Z (.-. < To ~ < 0 < T1 i < " . ) the arrival instants of the type 1 customers, the exponentiality assumptions imply that the sequence _X2 = _X(T, 1 - 0) is a homogeneous embedded Markov chain. (Note that _X2 does not take into account the type 1 customer just arriving at T,~.) In order to determine its transition probabilities we assume that 0 = To i < T~ < "- ' . Since the state _X(t) between two GI arrivals is governed only by exponentially distributed times (Poisson arrivals, exponential service times), the dynamics of X(t) during the intervals [T. l, T.�89 is given by a Markov process. Let

p~,y_(s) = P ( X ( s ) = ylX(0) = x, Ti 1 > t) , 0 < s < t , (2.1)

be the transition probabilities of this Markov process, which we denote by _Y -- (_Y(s), s >_ 0). (Note that p~,y_(s) does not depend on t.) Let the state space X be numbered in lexicographical order, i.e. (0, 0), (0, 1), . . . , (0, N), (1, 0), . . . , (1, N - 1) . . . . , (N - 1, 0), (N - 1, 1), (N, 0), and let X, = {(k, 0) . . . . . (k, N - k)} be the states with k type 1 customers. The rate matrix Q = (qx._y) of _Y is an (N + 1)(N + 2)/2 x (N + 1)(N + 2)/2 matrix and has the following block structure

Q =

"D O (9 (9 ""

F 1 D 1 (9 ..

(9 F 2 D 2

FN-1

(9 . . . (9

(9

DN- 1 (9

FN DN

(2.2)

where D k (0 _< k _< N) are the N - k + 1 x N - k + 1 matrices containing the transition intensities from Xk into Xk, and Fk (1 _< k _< N ) are the (N -- k + 1) • (N -- k + 2) matrices containing the transition intensities from X k into Xk_ 1. (9 denotes the corresponding matrix with zero members�9 Since only type 2 cus- tomers arrive or leave the system during (To i, Tli), we find for D k zJ(k) = laid, 0 <_ i, j _< N k) and Y k - ,'it) -- = ( j ~ d , O < i < N - k , O < j < N - k + 1):

52 A. Brandt

-(k/~l + i#2 + 22)

- ( k ~ + (N - k)m)

d} k) = (Dk)i, j = i#2

22

0

fi(k)~-(Fk)i,j-~{kokll ,J

i f / = j

otherwise .

i f i = j and i r N - k

if i = j = N - k

i f / = j + 1

i f j = i + 1 and i r N - k

otherwise ,

The transition probabili ty matrix P(s) = (p~,_y(s), x, y e X) satisfies the Ko lmo- gorov equat ion

d N e ( s ) = e ( s ) Q . (2.3)

Let

~,,_ = e ( x _ L 1 = y lX_~. = ~)

be the transit ion probabilities of the embedded M a r k o v chain. Taking into account the dynamics of the system and (2.1) we obtain

ra_l,i),(t_k,j) = S po,i),~t_k,j)(s)dA(s) for l - 1 + i < N , I _> 1 , (2.4a) 0

O<_k<_l , l - k + j < _ N ,

rc~,i),~-k,j ) = ~ p~,i),~l-k,j)(s)dA(s) for l + i = N , 0 < k _< l , (2.4b) 0

1 - k + j < _ N ,

r(k,o~,,,, ) = 0 otherwise . (2.4c)

Clearly, the Markov chain _X, ~ is irreducible and since the state space X is finite, its s tat ionary distribution p~, x s X, exists and is given by

P-~ = Y~ P_,5.-~, x ~ x , y~ ~ = 1 . (2.5) y x

Individual Blocking Probabilities in the Loss System GI + M[/I~IN[0 53

S p e c i a l c a s e N = 1. In the special case of N = 1 one can solve (2.3) easily. D e n o t e the states (0, 0), (0, 1) and (1, 0) by 0, 1 and 2, respectively. T h e n

2 2 (1 - e -~z2+u2) ' ) , p o a ( t ) = 1 - P o o ( t ) - 22 + ]/2

Po2(t) = Pl2(t) = 0 ,

]/2 (1 e -(a~ +u2)t) , (2.6) Plo(t) = 1 -- Pl l ( t ) - 22 .~_ ]/2

P22(t) = e-#,, ,

P2o(t) = i ] / i e - U i S P o o ( t - s ) d s 0

22] /1 e-(a2+u2)~(1 -- e-(U,-u2-a2)t) _ ] /2 (1 - - e -~lt) + ( 2 . 7 ) 22 + ]12 22 "F ]/2 (121 - - ]/2 - - 2 2 ) '

p2~(t) = 1 -- P22(t) - P20(t) �9

In case of N = 1 e q u a t i o n (2.4) reads

too = r2o = ~ P2o(s)dA(s) , r i o = ~ Plo(s)dA(s) , ( 2 . 8 ) 0 0

roi = r2i = ~ P2i(s)dA(s) , r i l = S pil(s)dA(s), 0 0

to2 = r22 = S p22(s)dA(s) , r i 2 = 0 . ( 2 . 9 ) 0

The s t a t iona ry d is t r ibu t ion Pi, i = 0, 1, 2 is given by

(1 - - r 0 2 ) ( 1 - - r l l ) r o l r 0 2 ( 1 - - r l i ) = , P l - , P2 -

Po 1 + roa -- r l l 1 + roi - r i i 1 + roi - r l i (2.10)

If N > 1, then an explicit so lu t ion of (2.5) and a c o m p u t a t i o n of the s t a t iona ry d i s t r ibu t ion p_~ a long the lines as for N = 1 lead to e n o r m o u s difficulties, since

54 A. Brandt

the algebra and analytical computations become hopelessly complex. Thus, the p_~ have to be computed numerically.

It is well-known that P ( t ) = e ~t solves (2.3). We will use the well-known majorization construction for Markov chains (cf. e.g. [3]) in order to derive an appropriate algorithm. Let q* = sup(-q_~,_~: _x E X} and q > q* be fixed. Then

(qt)"0. P ( t ) = e -qt n! '

n=O

(2.11)

where

O = I + I Q = D ' + F ' , q

I i "'" 1 (9 . .

I = "' D ' = "" F ' = "" t

0 D N �9

D~, = I + - O k , F~, = q q

(2.12)

,i] (2.13)

and Q is given by (2.2). From (2.4a, b) and (2.11) we obtain

r(l-l,i),(l-k,j) = ~ (On)(l,i),(t-k,j)t?l(n), n=O

l - l + i < N , l > l , k = 0 . . . . . l , l - k + j < N ,

r(l,i),(t-k,j) = (Q) ( , ,O , ( t - k , j )m(n ) , n=O l + i = N , k = 0 . . . . . 1 , l = k , . . . , N , l - k + j < N ,

(2.14)

(2.15)

where

re(n) = e -qt d A ( t ) . 0 (2.16)

Individual Blocking Probabilities in the Loss System GI + M I 2~[N]O 55

The coefficients re(n) are mixed Poisson probabilities, which can often be com- puted explicitly, e.g. if A ( t ) is an Erlang distribution with k phases and intensity # or a uniform distribution on [0, a] or deterministic.

The members of Q" can be computed by iterating Q. Since (~ is a stochastic matrix, (~" is stochastic, too. This offers to check numerical inaccuracies, or, by a renormalization, to make some reasonable corrections, since it is known that 0." tends to the vector of the stationary probabilities belonging to the Markov chain Q.

Now we are in a position to compute the transition probabilities of the embedded Markov chain given by (2.14), (2.15) and its stationary distribution.

R e m a r k : A different way to compute the members of (~" consists in taking advantage of the block structure of Q. However, we found that the corre- sponding algorithm exploiting the block structure of Q did not have any advan- tage in memory or computing time. Hence the details are not outlined here.

3 The Time Stationary Model. Time Stationary Probabilities

In this section we introduce the time stationary model and give a numerical algorithm for computing the steady state probabilities in continuous time, which are needed for the computation of the individual blocking probabilities in the next section.

Using the theory for processes with an embedded marked point process (PEMP), one can construct, by means of the inversion formula for PEMP's, cf. e.g. [4], Theorem 1.5.4 or [2], Theorem 4.2.5, a stationary P E M P ~ = IX(t), t _> 0, T] , where _X(t) = (Xi(t),-Xz(t)), ~ = (Tn, I,) and T, are the arrival instants of customers and i , indicates whether the n-th arrival is a type 1 (i, = 1) or a type 2 (i, = 2) customer. ~ describes the time stationary model of the system considered. Note that the arrival process ~ l = (~i), (~2 = (T a)) of type 1 (type 2) customers is again a stationary renewal (Poisson) process with inter arrival time distribution A(') (intensity 22) and that ~ i and ~2 are independent. The distribution

q~_ = P ( X ( t ) = x_) , x ~ X ,

is the steady state distribution of the model in continuous time (time stationary probabilities). If A(.) is a non-lattice distribution, one can show that

lim P ( X ( t ) = x ) = q~ ,

56 A. Brandt

for an arbitrary initial state of the system, i.e. q_~ is the limiting distribution in continuous time.

Consider the time stationary model. Then the intensity of the point process 05' is 2i. Denote the intensity of the points where the stationary process _X(t) jumps from _x to _y by 2_~,_y, i.e.

2 _ ~ , _ y = E # { t : 0 _ < t _ < l : ~ ( t - 0 ) = _ x , X( t )=Y} �9

(Note that X(t) is right continuous, i.e. X(t) = ~ ( t + 0)). Applying the intensity conservation principle, cf. e.g. [4], Sec. 1.6, we obtain for 0 _< i ___j _< N:

~(N-j,i),(N-j-l,i) "~- /~(N-j,i),(N-j,i-1) ~- /~(N-j,i),(N-j,i+l) -~ ~(N-j,i),(N-j+l,i) (3.1)

= )~(N-j,i-1),(N-j,i) -~ ~(N-j-l,i),(N-j,i) "~ ~(N-j,i+l),(N-j,i) ~- /~(N-j+l,i),(N-j,i) �9

If one of the indices lies outside X, we set the corresponding intensity zero, e.g. 2~o,o)m,-1) = 0. Since the interarrival times of the type 2 customers and the service times are exponentially distributed, it holds

~(N-j,i),(N-j,i-1) ~ " q(N-j,O" i" #2 ,

2(N-j,i),(N-j-*,i) = q(N-j,i)" ( N - j )" #2 , (3.2)

2(N_j,i),{N_j,i+l) = q(N_j,i)'.g2" l I {i < j } ,

where lI{ } denotes the indicator function of { }. From [4], Formula (1.2.6), it follows

2(N-j,i),~N-j+I,0 = P(N-j,O' 21" l I { i < j } �9

Combining (3.1)-(3.3) we conclude for 0 < i < j __. N:

(3.3)

q(N-j,O'(( N - - J ) ' # l + i#2 + )'2" 1I{i < j } ) + p(u_j , i ) '2~l I{ i < j }

= q(N_j,i_l)~2 q- p(N_i_l,i),~t q- q(N_j,i+l)(i "~ 1)# 2

+ q(N-j+l,i)(N - - j + 1)#1 �9 (3.4)

In particular, for i = j = 0 we derive from (3.4)

Individual Blocking Probabilities in the Loss System GI + MI-~INI0 57

qtN, o ) N # I = PiN_l,0)21 . (3.5)

For a fixed j ~ {1, . . . , N} let qt0 = (qtN-j,O), " " , qtN-j,j)) T" Then the system of equat ions (3.4) can be written as

AtJ)q tj) = d tj) + (N - j + 1 ) # ~ t/-~) , j = 1 . . . . . N , (3.6)

= (do;,, where O~J-1) = (qtN-j+~,o) . . . . . q(N-j+lj-1), 0) r = k 0 J ' " ' "

and

A (j) =

b<o j) c~ ~ 0 0 . . .

a r b~ ~ c~ ) 0 . . .

0 a <j) b~ ) ct~ ) : " . ".. " , "..

at J)

0

0

b, JJ-)2 cJ~2 0

at J) bJJ'__) 1 c~21

0 a (j) b) j)

The coefficients a t j), bi u), cl j) and d} J) are given by

(3.7)

at J) = - - "~2 ,

b~ j) = ( N - j ) ' # 1 + i ' # 2 + ~2" I{ i < j ) ,

c~ i) = --( i + 1). ~2 ,

d~ j) = --ptN_j,0"21" 1I{i < j } + ptN_j_l,0"2~ ,

O < i < j ,

O < i < j - - 1 ,

O < i < j .

(3.8)

In the Appendix, Lemma A2, the matrices A c j), 1 _< j < N, are shown to be regular, whereas the (N + 1) x (N + 1) matrix A <N) is singular. Looking at the structure of A IN) we see immediately that the rows up to the first one are linearly independent vectors, i.e. A <m has the rank N. The first row depends on the other rows. Since the qtN-j,0, 0 <_j _ N, i _< N - j , are a probabil i ty distribution, we have the addit ional equat ion

lV N - j

2 L qtN-j , i )= 1 . (3.9) j=o i=o

58 A. Brandt

We modify the N-th system of equations of (3.6) by replacing the first equation by (3.9), i.e. we consider the following system of equations:

~(N)q(S) = ~(s) + f i l l S ( N - i ) , (3.10)

where ~(N) = (d, d~N), . . . , d(NN)) T, ~(N-1) = (0, q(1, l) . . . . , q(i.N-i), O) T, and

~ ( N ) =

1 1 1

a(N) b?) ci m

0 a (N) b~ N)

. , ,

, , ,

, � 9

C(2 N) � 9

" . �9 � 9

a (N)

0

0

1 1

0 0

b ~ 2 c~-~ 0

a (N~ b ~ i c~2i 0 a (~) b~ N)

(3.11)

The coefficients a (N), b} m, cl N) and d} N) are given by (3�9 and d = 1 - ~-i j

~ q(N-j,o" In the Appendix, Lemma A3, it is shown that ~(N) is regular. j = 0 i=0

Thus (3.5), (3.6) with j = 1, . . . , N - 1, and (3.10) provide a recursive numerical algorithm for computing the time stationary probabilities q,,j), ( i , j ) e X. The recursion works in such a way that the stationary probabilities for the state sets XN, XN-1, XN-2 . . . . , Xo are computed successively.

4 Individual Blocking Probabilities

The probability Psi that an arbitrary arriving type i customer (i = 1, 2) gets lost, i.e. that he finds all customers busy, is called the individual blocking probability�9 We will show that the Psi can be computed by using the results of Sec. 2 and 3.

Let us consider the time-stationary model ~ and, besides this, also the model started with an arbitrary initial state, where the renewal process of the type 1 customers is not necessarily assumed to be stationary�9 In accordance with the notations given in Section 1 let

X n 1 = X ( L 1 - - 0 ) , - - 2 ~___ . S ( ~ n 2 __ 0 ) 1 X n _ , X n j = X ( T n 1 - - 0 ) ,

_x2 = _X(T~ ~ - o ) ,

Individual Blocking Probabilities in the Loss System GI + M I/~IN[0 59

where T, 2 are the arrival instants of the type 2 customers ( ' " < T0 z < 0 < 7"12 < " ' ) . Then, by means of coupling arguments as in [1] and [2] one can show that, if A(.) has a non-lattice distribution, it holds

p_~ = lim P(X 1 = x) = lim P(_X,~ = _x) ,

r_~ = lim P(X, 2 = x) = lim P ( ~ = _x) , n---~ o0 n--~ oo

where ~ is the stationary distribution of the embedded Markov chain, cf. (2.5); Px (zx) is the distribution of the number of customers in the system found by an arbitrary arriving type 1 (type 2) customer, respectively, cf. [-4], Theorem 1.3.12. Since (T 2) is a Poisson process, we conclude from the PASTA Theorem, cf. [13], that z_~ are just the time stationary probabilities, i.e. z_~ = q~. Hence,

N

P m = ~ P(i,N-i) , (4.1) i = 0

N

PB2 = Y', qt*,N-i) �9 (4.2) i = 0

The probability PB that an arriving customer gets lost is given by

~1 ' ~2

PB - 2i + 2z pm + 21 + ~ . ~ P B 2 �9 (4.3)

This follows by conditioning with respect to the two types of arrivals, cf. [4], Formula (1.2.6). The blocking probabilities can be computed numerically in the following way:

Step 1: Compute the stationary distribution p_~ according to (2.5) by using the algorithm given in Section 2.

Step 2: Compute the time stationary probabilities q~ by solving (3.5), (3.6) for j = 1 . . . . , N - 1 and (3.10).

Step 3: Compute the PB~ by means of (4.1), (4.2) and P8 by (4.3).

Special case N = 1, # = #1 = ~2. For N = 1 and # = #1 = #2 we have the model GI + MIM[I[0. As in Section 2 let 0, 1, 2 be the states (0, 0), (0, 1), (1, 0). One finds from (2.6) and (2.7)

Plo(t) = Pao(t) = # ~ e-tZ2+"'~lI{s -< t}ds . 0

(4.4)

6O

Thus (2.8) and (4.4) yield too = rio, which implies, by using (2.9),

A. Brandt

1 - - r l l = 1 - - r l l - - r 1 2 ~ rlO ~ r00 (4.5)

and

1 + /'01 - - /'11 = 1 + r o l - - l ' l i - - r12 = r i o "Jr" 1"Ol -= too + to1

= 1 - roz �9 (4.6)

Hence, from (2.10), (4.5), (4.6) we obtain

Po = too �9 (4.7)

F rom (4. i), (4.7), (2.8) and (4.4) we obtain for the individual blocking probabil i ty

Pm

1 - P m = Po = too-- ~ p2o(s)dA(s)= # ~ ~ e - ( ~ = + " l I { t < s}dtdA(s) 0 0 0

= # ~ e-(*~+")'(1 - A(t))dt 0

= # e-U'(1 - A(t))2 a e-a:"du dt . (4.8) 0

For the time stationary probabilit ies qo, ql, q2 one finds from (3.6), (3.10)

# - 2 1 p o 22 ~1 q o - # + 2 ; q l = qo , q 2 = , # ~-Po - (4.9)

Combining (4.2), (4.9) and (4.8) we get for the individual blocking probabil i ty PB2

- P ( 1 - 2~ ~ e-(a2+u)"(1-- A(u))du) . (4.10) 1 -- PB2 = qo # + 22 o

In view of 1 = -~1 ~ (1 - A(u))du, eq. (4.10) can be continued as

Individual Blocking Probabilities in the Loss System GI + M liV~ I N I0 61

co

- S (1 - e-(a~+u)")(1 - A(u))du # + 2 z o

= #2 i ~ S e-(12+U)tlI{t _< u } d t ( 1 - A(u))du 0 0

= p ~ e-"'21 ~ (1 - A(u))due-12'dt . (4.11) 0 t

The formulas (4.8) and (4.11) are a special case of the loss probabilities given in [12] for the G~ + "" + G, IM[ll0 system (with r = 2, F~(t) = A(t), Fz(t ) = 1 - e -z2t in the notations of [12]).

5 Numerical Results

The numerical algorithm based on (2.4a), (2.4b), (2.4c), (2.11), (2.14), (2.15), (3.5), (3.6) and (3.10) for computing the individual loss probabilities was implemented in a C-Program. For the computation of P(t) we used (2.11) and simple iteration of (~. The numerical experiments were carried out on a PC 486 DX 33. A simulation program was written for checking the stability of the numerical algorithm. We found that the numerical algorithm was stable up to N = 30 on a 486 DX 33.

Now we give two examples of the numerical computation of the individual blocking probabilities treated in Kuczura [8] for the GI + MIM[N]O system. Let 0i = 2i/#i, i = 1, 2.

Example 1: D + M[J~IN[0. Let a = 1/21 be the determinstic interarrival time of type 1 customers. The coefficients re(n) in (2.16) are the Poisson probabilities

m(n) = e -~a(qa)" n! " It is easy to check that for given 01 + 02, 0~/02 and #~/#2 the

individual blocking probabilities do not depend on the particular choice of #l, i.e. we can choose / t i := 1. In Tables 5.1 and 5.2 the individual blocking probabilities for N -- 5 and N = 25 are given for various parameters. The row / / 1 / ] / 2 = i corresponds to the D + MIM]N]O system treated in Kuczura [8]. The numerical results show that the different service parameters #l and #2 may have considerable influence on the individual blocking probabilities.

In the second example below we will assume the GI stream to be an overflow stream. Overflow processes are often approximated by interrupted Poisson processes (IPP), cf. [-7]. An IPP is a Poisson process of intensity z that is

Tab

le 5

.1.

Indi

vidu

al b

lock

ing

prob

abil

itie

s fo

r th

e sy

stem

D +

M I

A41

5[0

a) 0

1 "]

-02

=2

01/0

2 =

1/19

0

1/0

2

=

1/4

01/0

2 =

1/1

01/0

2 =

4/1

0~/0

z =

19/1

~1/~

2 Pa

l PB

2 P

nl

PB2

Pnl

Pn2

PB1

Pn2

Pal

PB

2

1:4

1:2

1:1

2:1

4:1

0.03

13

0.03

65

0.03

13

0.03

65

0.03

13

0.03

65

0.03

13

0.03

65

0.03

11

0.03

65

0.02

10

0.03

38

0.02

08

0.03

38

0.02

04

0.03

37

0.01

97

0.03

36

0.01

86

0.03

35

0.01

30

0.02

64

0.01

26

~02

63

0.01

20

0.02

61

0.01

13

0.02

60

0.01

07

0.02

60

0.00

64

0.01

72

0.00

62

0.01

72

0.00

59

0.01

71

0.00

57

0.01

71

0.00

55

0.01

71

0.00

32

0.01

18

0.00

32

0.01

18

0.00

31

0.01

18

0.00

31

0.01

18

0.00

30

0.01

18

b) 0

1 +

02 :

01

/02

=

1/19

01

/02

= 1/

4 01

/02

= 1/

1 01

/02

= 4/

1 01

/02

= 19

/1

#1/P

2

1:4

1:2

1:1

2:1

4:1

01/~

2

PB

t PB

2

0.09

75

0.10

99

0.09

75

0.10

99

0.09

74

0.10

99

0.09

70

0.10

98

0.09

60

0.10

98

PB1

P1~2

0.07

78

0.10

75

0.07

62

0.10

75

0.07

35

0.10

73

0.06

97

0.10

70

0.06

56

0.10

70

PB1

PB

2

0.06

15

0.10

07

0.05

87

0.10

06

0.05

53

0.10

04

0.05

23

0.10

05

0.05

01

0.10

06

PB1

PtJ2

0.04

60

0.09

20

0.04

40

0.09

23

0.04

23

0.09

25

0.04

11

0.09

26

0.04

04

0.09

27

PB1

PB2

0.03

68

0.08

74

0.03

62

0.08

77

0.03

57

0.08

78

0.03

54

0.08

79

0.03

53

0.08

79

C)

01 +

02

:4

01/0

2 =

1/19

01

/02

= 1/

4 01

/02

= 1/

1 01

/02

= 4/

1 01

/02

= 19

/1

PB1

PB

2

1:4

1:2

1:1

2:1

4:1

PB1

PB2

0.18

16

0.19

91

0.18

15

0.19

91

0.18

12

0.19

91

0.17

97

0.19

91

0.17

66

0.19

90

PH1

PB2

0.15

82

0.19

93

0.15

39

0.19

94

0.14

74

0.19

93

0.13

94

0.19

94

0.03

13

0.19

98

PB1

PBZ

0.13

85

0.20

07

0.13

19

0.20

16

0.12

48

0.20

26

0.11

89

0.20

36

0.11

50

0.20

45

0.12

21

0.20

75

0.11

78

0.20

98

0.11

45

0.21

14

0.11

24

0.21

25

0.11

13

0.21

31

PB1

PB2

0.11

39

0.21

71

~11

27

0.21

83

0.11

19

0.21

90

0.11

15

0.21

93

0.11

12

0.21

95

,-.t

Tab

le 5

.2. I

ndiv

idua

l bl

ocki

ng p

roba

bili

ties

fo

rth

e sy

stem

D +

MI5

4125

10

a) Q

1 +

Q2

-- 1

5

O~/

Q2

= 1

/19

Qt/O

2 =

1/4

O

l/Q

2 =

1/1

01/Q

2 =

4/1

Q1/

Q2=

19/

1

~t/

~2

P

ut

PB

2 PB

1 P

B2

Pnl

P

B2

Pnt

P

B2

P~I

P

nz

l:4

l:

2

1:1

2:1

4:1

0.00

37

0.00

48

0.00

37

0.00

48

0.00

36

0.00

48

0.00

35

0.00

48

0.00

34

0.00

47

0.00

28

0.00

39

0.00

27

0.00

39

0.00

25

0.00

39

0.00

24

0.00

39

0.00

22

0.00

38

0.00

14

0.00

23

0.00

13

0.00

23

0.00

12

0.00

23

0.00

12

0.00

22

0.00

11

0.00

22

0.00

05

0.00

10

0.00

05

0.00

10

0.00

04

0.00

10

0.00

04

0.00

t0

0.00

04

0.00

09

b) 0

1+

02

=2

0

Q1/

Q2

= 1/

19

0~/0

2 =

1/4

O

l/O2

= 1

/1

e~/Q

z=4/

1

1:4

1:2

1:1

2:1

4:1

PB1

PB2

0.04

31

0.04

97

0.04

23

0.04

97

0.04

11

0.04

96

0.03

95

0.04

95

0.03

75

0.04

94

Pal

P

B2

0.03

78

0.04

78

0.03

60

0.04

77

0.03

39

0.04

74

0.03

17

0.04

72

0.02

97

0.04

72

Pal

P

B2

0.03

03

0.04

34

0.02

84

0.04

33

0.02

66

0.04

32

0.02

52

0.04

32

0.02

43

0.04

33

PB1

PBz

0.02

27

0.03

80

0.02

17

0.03

82

0.02

09

0.03

84

0.02

04

0.03

84

0.02

02

0.03

84

0.00

02

0.00

05

0.00

02

0.00

05

0.00

02

0.00

05

0.00

02

0.00

05

0.00

02

0.00

05

Q1/

Q2=

19/

1

PB1

PB2

0.01

85

0.03

50

0.01

83

0.03

52

0.01

81

0.03

53

0.01

80

0.03

53

0.01

79

0.03

53

c) 0

1 +

Q2

=2

5

Q1/

Qz=

1/1

9 Q

1/Q

2= 1

/4

Q1/

Q2=

1/1

Q

1/Q

2=4/

1 Q

1/Q

2= 1

9/1

Pl/

~2

PB

1 PB

2 PB

1 PB

2 PB

1 Pn

2 PB

1 Pn

2 PB

1 PB

2

0.11

15

0.14

85

0.10

59

0.15

01

0.10

07

0.15

16

0.09

67

0.t5

29

0.09

41

0.15

38

0.10

50

0.15

95

0.10

21

0.16

24

0.10

00

0.16

44

0.09

88

0.16

56

0.09

81

0,16

63

1:4

1:2

1:1

2:1

4:1

0.13

20

0.14

38

0.12

93

0.14

38

0.12

55

0.14

37

0.12

04

0.14

37

0.11

43

0.14

36

0.12

18

0.14

44

0.11

63

0.14

46

0.t1

00

0.14

48

0.10

35

0.14

50

0.09

76

0.14

54

0.10

33

0.17

15

0.10

26

0.17

30

0.10

22

0.17

38

0.10

19

0.17

42

0.10

18

0.17

44

.<

G

fro o"

o"

+

64 A. Brandt

al ternatively on for an exponent ial ly distr ibuted t ime (mean 1/7 ) and then off for an exponent ial ly distr ibuted t ime (mean 1/~o). An I P P is a renewal process, cf. e.g. [7], i.e. it is a G I s t ream with the interarrival distr ibution

Al(t ) = kl(1 - e-*,t) + k2(1 - e-.~,) . (5.1)

where

1 = x( r + co + 7 + ~/(~ + co + ?)2 _ 4~o) / ' I /2 Z

(5.2)

kl - r - r2 , k 2 = 1 - k 1 . (5.3) / '1 - - / '2

Assuming, that the overflow s t ream is specified by its mean m and variance v as defined in [10], we obta in by match ing the m e a n and variance of the I P P to m and v and by using a s tandard approximat ion :

r =mz + 3z(z--1) , co= C 1 1 , 7= - 1 o9, (5.4)

where z = v/m is the peakedness. The details are given in [7].

Example 2: IPP + M I M I N [ 0 . Fo r a given peakedness z, 01 + 02, 01/02, #1/#2 and #2 one can identify the paramete rs 7, o , r of the I P P in view of ffl = m and (5.4). F r o m (5.1)-(5.4) we find for re(n), cf. (2.16):

m(n)-- klrl ( q ) n q+/'l U # ( + q+r2

Tables 5.3, 5.4 present numerical results for the individual blocking probabil i t ies for var ious values of Q1 + 02, 01/02, #1/#2, where #2 = 1 is fixed and the over- flow s t ream is assumed to have peakedness z = 2. The row #1/#2 = 1 corre- sponds to the IPP + M[MtN[O system treated in [8]. The results illustrate the influence of the pa rame te r #1/#2 on the individual blocking probabili t ies. We have confirmed the general principle of regularity, cf. [8]: the overflow calls arriving less regular than the Poisson calls are more likely to be blocked.

Tab

le 5

.3. I

ndiv

idua

l bl

ocki

ng p

roba

bili

ties

for

the

IP

P +

M

I M

1510

sys

tem

wit

h pe

aked

ness

z =

2

and/

~2

= 1

a) 0

1 +

02 =

2

0~/0

2= 1

/19

O~/

O2=

1/4

O

~/O

2= 1

/1

0~/0

2=4/

1 0~

/02=

19/

1

gl/g

2 Pa

l PB

2 P

al

Pa2

Pal

Pa

2 P

al

PB2

PBa

PB2

1:4

1:2

I:1

2:1

4:1

0.18

34

0.03

76

0.17

34

0.03

77

0.15

68

0.03

79

0.13

30

0.03

80

0.10

60

0.03

80

0.18

31

0.04

05

0.17

30

0.04

09

0.15

67

0.04

15

0.13

46

0.04

20

0.11

07

0.04

21

0.18

36

0.04

64

0.t7

33

0.04

76

0.15

76

0.04

91

0.13

80

0.05

03

0.11

85

0.05

05

0.18

50

0.05

17

0.17

45

0.05

38

0.15

94

0.05

62

0.14

17

0.05

79

0.12

54

0.05

83

0.18

58

0.05

40

0.17

53

0.05

66

0.16

05

0.05

93

0.14

36

0.06

12

0.12

85

0.06

18

go

t~

0"

b) 0

1 -[

-02

=

01/0

2 =

1/19

01

/02

= 1/

4 01

/02

= 1/

1 0

1/0

2=

4/1

0

1/0

2=

19

/1

fll/

~2

PB1

PB

2 P

~1

Pn2

P

al

PB

2 P

al

Pa2

P

at

Pa2

1:4

1:2

1:1

2:1

4:1

0.28

80

0.10

89

0.27

87

0,10

92

0.26

29

0.10

95

0.23

94

0.10

99

0.21

06

0.11

03

0.28

31

0.10

62

0.27

39

0.10

71

0.25

91

0.10

84

0.23

87

0.11

00

0.21

60

0.11

14

0.27

50

0.10

32

0.26

58

0.10

56

0.25

23

0.10

88

0.23

59

0.11

23

0.21

99

0.11

50

0.26

90

0.10

24

0.25

98

0.10

65

0.24

72

0.11

15

0.23

33

0.11

62

0.22

11

0.11

96

0.26

66

0.10

21

0.25

75

0.10

72

0.24

53

0.11

31

0.23

24

0.11

83

0.22

14

0.12

17

t-~

Q

r~

+

C)

01 §

02

=4

01/0

2 =

1/19

0

1/0

2=

1/

4 01

/02

= 1/

1 0

1/0

2=

4/1

01

/02

= 1

9/1

~1/

~2

1:4

1:2

1:1

2:1

4:1

Pal

P

a2

0.38

27

0.19

59

0.37

46

0.19

61

0.36

06

0.19

66

0.33

94

0.19

72

0.31

27

0.19

79

PB

1 P

B2

0.37

51

0.18

67

0.36

73

0.18

79

0.35

48

0.18

96

0.33

78

0.19

18

0.31

91

0.19

40

Psi

PB

2

0.36

00

0.17

13

0.35

25

0.17

44

0.34

17

0.17

87

0.32

91

0.18

36

0.31

73

0.18

77

Pal

P

B2

0.34

65

0.15

98

0.33

90

0.16

54

0.32

92

0.17

24

0.31

89

0.17

92

0.31

03

0.18

43

Pal

PB

2

0.34

05

0.15

50

0.33

30

0.16

23

0.32

36

0.17

06

0.31

42

0.17

80

0.30

67

0.18

32

Tab

le 5

.4. I

ndiv

idua

l bl

ocki

ng p

roba

bili

ties

for

the

IP

P +

M[1

~125

10 s

yste

m w

ith

peak

edne

ss z

= 2

an

d #

2 =

1

a)

~o I

+ ~z

=

15

O',

O~

Q~/

~2 =

1/

19

Qa/

Q2

= 1/

4 ~1

/02

= 1/

1 Q

a/Q

2=4/

1 Q

1/~

2=

19

/1

ill/

P2

Pal

Pn

2 P

al

PB2

Pgl

PB2

P~l

Pg2

Pal

Pn

2

1:4

1:2

1:1

2:1

4:1

0.01

69

0.00

59

0.01

56

0.00

59

0.01

39

0.00

58

0.01

20

O.0

057

0.01

04

0.00

56

0.02

O4

O.0

O83

0.

0187

0.

0082

0.

0168

0.

0080

0.

0151

0.

0079

0.

0139

0.

0077

0.02

70

0.01

24

0.02

48

0.01

23

0.02

27

0.01

21

0.02

11

0.01

18

0.02

01

0.01

17

0.03

32

0.01

58

0.03

04

0.01

58

0.02

82

0.01

56

0,02

66

0.01

54

0.02

56

0.01

52

0.03

61

0.01

73

0.03

31

0.01

73

0.03

07

0.01

72

0.02

90

0.01

70

0.02

81

0.01

68

b) Q

l+Q

2=

20

QI/

02

=

1/19

Q

1/Q

2 =

1/4

QI/

Q2

=I/

I Q

1/Q

2 =

4/1

Q1/

~2=

19/

1

#1/~

2 P

al

Pa2

Pal

P

B2

Pal

P

B2

Pnl

PB

2 P

al

PB2

1:4

1:2

1:1

2:1

4:1

0.09

29

0.05

10

0.09

04

0.05

11

0.08

68

0.05

13

0.08

24

0.05

14

0.07

82

0.05

15

0.09

59

0.05

33

0.09

37

0.05

37

0.09

13

0.05

42

0.08

93

0.05

47

0.08

80

0.05

51

0.10

19

0.05

69

0.09

97

0.05

78

0.09

79

0.05

88

0.09

67

0.05

96

0.09

61

0.06

02

0.10

72

0.05

91

0,10

45

0.06

06

0.10

23

0.06

18

0.10

09

0.06

26

0.10

01

0.06

31

0.10

95

O.0

596

0.10

64

0.06

16

0.10

39

0.06

29

0.10

23

0.06

37

0.10

14

0.06

41

c) Q

l+Q

2=

25

01

/02

=1

/19

01

/Q2

= 1/

4 01

/02

= 1/

1 0

1/~

2

= 4/

1 O

a/O

2= 1

9/1

fll/

fl2

Pal

P

B2

Pal

P

n2

Pal

P

B2

PB

I P

B2

Pal

P

B2

1:4

1:2

1:1

2:1

4:1

0.20

89

0.14

27

0.20

65

0.14

29

0.20

29

0.14

31

0.19

88

0.14

35

0.19

53

0.14

38

0.20

87

0.13

92

0.20

79

0.13

98

0.20

74

0.14

06

0.20

77

0.14

16

0.20

88

0.14

24

0.20

71

0.13

21

0.20

66

0.13

35

0.20

66

0.13

51

0.20

69

0.13

65

0.20

74

0.13

76

0.20

45

0.12

45

0.20

28

0.12

74

0.20

17

0.12

98

0.20

11

0.13

15

0.20

08

0.13

25

0.20

28

0.t2

04

0.20

05

0.12

45

0.19

89

0.12

74

0.19

78

0.12

91

0.19

73

0.13

01

Individual Blocking Probabilities in the Loss System GI + MI2V~INI0

6 The G I + GIIMIl l0 System

67

In this section we will consider the single server loss system where the input is a superposition of two independent renewal processes with interarrival distribu- tions Al ( t ) and Az(t) , respectively, having the means 2;-1= ~ ~ A~(t))dt. Again, the two types of customers require exponentially distributed service times with parameters #1 and/~2, respectively. We assume that the A~(t) have derivatives a d o = A'i(t) and, hence, the failure rates r i ( t )= ai(t)/-~i(t), A i ( t ) = 1 - A~(t), are given. The state X( t ) = X ( t + 0) of the system is given by

if the system is empty at time t

if a type 1 customer is served at time t

if a type 2 customer is served at time t .

Analogously as in Section 3, let ~ 1 = (T1) and ~ 2 = (,F:) be the two independent stationary versions of the renewal input processes (on the whole real line) and ~ = (T,, In, S,) the complete arrival process of all customers, where T,, i , are as in Section 3 and the S, are the desired service times of the customers. Using the method of renewing epochs, cf. e.g. [2], one can construct the stationary state process X(t) (in continuous time), which can be constructed from the "past", i.e. there is a measurable mapping such that

m m

x(t) = h((_~,o](O, ~ ' ) ) ,

where Ot denotes the shift operator and (-oo,o]~ the restriction of a point process realization ~ to (-0% 0], cf. e.g. [4], [2]. Then (~g, (X(t): t s N), is a time station- ary model of the system.

Denoting by Ri(t) the backwards recurrence-time of the last type i customer before t, i.e. Ri(t) = min{t - ~i: ~ / < t}, we find that (X(t), Rl(t), RZ(t)) is a Markov process with state space {0, 1, 2} x N+ x R+. By the stationarity, the

9 2 _ _

densities pt(i, u, v) = ~ v P ( X ( t ) = i, R l ( t ) < u, RZ(t) < v), i = 0, 1, 2, are inde-

pendent of t, i.e. p(i, u, v) = pt(i, u, v). Furthermore, from the independence of ~1 and ~2 and since the ~* are stationary, it follows

p(u, v) = p(O, u, v) + p(1, u, v) + p(2, u, v) = 21(1 -- Al(u))22(1 - Aa(u)) . (6.1)

68 A. Brandt

The Ri(t) can be considered as supplemented variables. Using the method of supplemented variables (cf. e.g. [6] pp. 86-98) one can derive the following system of partial differential equations for the p(i, u, v)

~ p(O, u, v) + ~p(O, u, v) = -p(O, u, v)(rl(u) + r2(v))

+ p(1, u, v)#l + p(2, u, v)#2 ,

~uP(1, u, v) + ~vP(1, u, v) = --p(1, u, v)(#1 + rl(u) + rz(v)),

~--uP(2, u,v) + ~p(2, u ,v)= -p(2, u,v)(#2 + rl(u) + rz(v)) ,

(6.2a)

(6.2b)

(6.2c)

with the boundary conditions

p(1, 0, v) = ~ (p(0, u, v) + p(1, u, v ) ) r l ( u ) d u , 0

p(2, u, 0) = ~ (p(0, u, v) + p(2, u, v))r2(v)dv , 0

p(1, u, 0) = ~ p(1, u, v)rz(v)dv , (6.3) 0

p(2, 0, v) = ~ p(2, u, v)rl(u)du , 0

p(0, u, 0) = p(0, 0, v) = 0 .

The system of the partial differential equations (6.2a-c) with the boundary conditions (6.3) is of a complex structure. It seems to be impossible to give a simple explicit solution. Using coupling techniques one can clearly show that X(t) ~ X(0) as t ~ 0% where ~ stands for convergence in distribution and X(O is the state of the system at time t if it was started with an arbitrary initial condition. Writing down the evolution of the system from arrival to arrival and using the inversion formula for point processes with an embedded marked point process one can give, in principle, an expression for the distribution of X(0) (i.e. an explicit solution of (6.2a-c)) in terms of a series of convolution powers. This approach is similar to the limit theorems and iterative approximations known for piecewise linear Markov-processes (cf. e.g. Gnedenko's Ergodic Theorem [-6]

Individual Blocking Probabilities in the Loss System GI + M]/fi~lN[0 69

p. 102-103 or [5] p. 174, p. 181). However, the resulting expressions will be of such a complexity that they seem to be irrelevant for practical purposes; hence we have not outlined the details.

Knowing the densities p(i, u, v), the individual blocking probabilities PBi are given by

p. l = 2; 1 ~ ~ (p(u, ~) - p(O, u, ~))~(.)aua~ 0 0

P,2 = 221 ~ ~ (p(u, v) - p(O, u, v))r2(v)dudv 0 0

(6.4)

and the total blocking probability p~ by (4.3).

Special case ]2 = ]21 = ]22" In this case the residual service time of a customer does not depend on its type. Taking this and the dynamics of the loss system into account, we obtain:

p(0, U, V) ---- 21-,41(U)22Az(V)(l -- e -u(u^v)) , (6.5)

where u ^ v = min(u, v). (It is easy to check that (6.5) satisfies (6.2a) with P = #1 = ]22- Note that this is the key observation in [12], dealing with the G 1 + " " + G,]MI1]0 System). Using (6.1), (6.4) and (6.5) we conclude for the individual blocking probability PB1:

PB1 = ~ 1 ~ S (p(u, V) -- p(O, hi, v))~'l(U)dudl) o o

= 2; 1 ~ ~ 2~Al(u)2z32(v)e-~'~^V)rl(u)dud v 0 0

= ~ ~ 22al(u)A2(v)e-U(~^~)dudv 0 0

Ei ] = 2aal(u ) A2(v)e-U("^~)dv du 0

'dv)du

70 A. Brandt

- 2z i S2(v)e-U"lI{u <- v}dv)du

= 1 - ~ al(u)22 ~ .42(v)(1 - e - ~ ' l I { v < u} - e-U"lI(u < v})dvdu 0 0

= 1 -- ~ al(u)22 ~ .42(v)((1 -- e-U~)lI{v _< u} + (1 -- e-U")lI{u _< v})dvdu 0 0

=l-ial(u)~2~A-2(v)(lI{v<u}#io lI{t<-v}e-Utdt

+ lI{u __< v}# i lI{t __< u}e-U~dt)dvdu

0 0 0

+ lI{u E v}lI{t < u})dudvdt. (6.6)

In view of

lI{t ~ v}lI{v < u} -4- lI{u_< v}lI{t_< u} = lI{t < u}lI{t < v}

we can continue (6.6)

o t t

1 - - ~ ~ e-fzt( i - - A l ( t ) ) ~ 2 ~ A 2 ( v ) d l A d t o t

Formula (6.7) was proved in [12] for stationary ergodic arrival processes ~'. The derivation given here is an alternative proof in case of ~ being renewal processes.

Individual Blocking Probabilities in the Loss System GI + MIJ~INI0

Appendix

71

Denote by IFI the determinant of a matrix F.

Lemma AI : Le t a, b, c, d be real numbers. Then we have fo r the determinant IO~l o f the matrix

d d d "."

- c a + b a ""

0 - c a + 2b ".. ". . " , . " . ,

d d

a a

�9 ~

a 4- ( j -- 2)b a a

- c a + ( j - 1)b a

0 - c a + j b

(A.1)

the following recursion:

j - 2

IDjl = (a + jb)LDj-I[ + a ~ cJ-i- l lDil 4- cJ-ld(a 4- c) , i=1

j _ > 2 ,

I d d 1011= [ _ c (a + b) = d(a + b + c) . (A.2)

Proof: The proof can be obtained by using Laplace's rule for determinants (applied to the last row):

IOjl = (a +jb)lDi_l] + c. 1C~_11 , (A.3)

where

C j--1

d d d d "" d d-

- c a + b a a "'" a a

0 - c a + 2 b a : :

~ ".o " o ". . a a

- c a + ( j - 2)b a

0 "" 0 - c a

72

Applying again Laplace's rule (to the last row) we find

I f j - l l = atDj-2l + c. Ifi_21 .

F rom (A.3) and (A.4) we obtain

IDjl = (a + jb)lO~_l[ + aclDj-2l + cZlfj-21

and applying (A.4) successively

j - 2

IOjl = (a +jb)lOj-l l + a ~, cJ-i-~lOll + #-~lCxl , i=1

whichy ie ld s (A .2 ) inv i ewof lC~ l =1 d-c ad[ = d(a + c).

A. Brandt

(A.4)

L e m m a A2: The matrices A ~i), 1 < j < N, given in (3.7) are regular, whereas A ~N) is singular�9

Proof: With a = (N - J ) / / 1 , b = #2, c = 2 2 the matrices A (j) read as follows:

AO) =

a + c - b 0 0 0

- c a + b + c - 2 b : :

0 - c a + 2 b + c

a + ( j - - 2)b + c - ( j -- 1)b 0

- c a + ( j - 1 ) b + c - j b

0 . . . 0 - c a + j b

Adding the ( j + 1)th row to the j - th row and then the j - th to the ( j - 1)th row and so on, we obtain:

I ACJ) I =

a a a a . . . a a

- -c a + b a a "" a a

0 - c a + 2b a " "

0 0 - c a + 3b �9 . . , .

". ". ". a a

- -c a + ( j - 1 ) b a

0 0 "~" 0 - c a + j b

(A.5)

Individual Blocking Probabilities in the Loss System GI + MI/QINI0 73

The determinant [A(J) I is the determinant of a matrix of the type (A.1) with d = a. In case 1 < j < N we have a, b, c, d > 0 and thus L e m m a A1 provides by induct ion that IDjl > 0, i.e. A (j) is regular. I f j = N, then a = 0 and from (A.5) we obtain I aim[ = 0 immediately.

Lemma A3: The matrix ~tN) 9iven by (3.11) is regular.

Proof: With b =/22, c -~- 2 2 the matrix .~lm reads

1 1 1 1 . " 1

- c b + c - 2 b 0 "" 0

0 - c 2b + c - 3 b

0 0 - c 3b + c - 4 b : ".. ".. ".. "..

- c ( N - 1 ) b + c - N b

0 ' " 0 - c Nb

Adding now the (N + 1)st row to the N- th row and then the N- th to the (N - 1)st row and so on up to the second row we obtain

I~lN)l =

1 1 1 . . . 1 1 1

- c b 0 "" 0 0 0

0 - c 2b " ' . "., "..

...

- c ( N - 2)b 0 0

0 - c ( N - 1)b 0

0 0 - c N b

Applying Lemma A1 (with d = 1, a = 0) yields, in view of b, c > 0, by induct ion that IDjl > 0, i.e. ~ s ) is regular.

Acknowledoement: The author is grateful to S. Fuhge for contributing ideas to Section 2 and to H. Pohl for implementing the numerical algorithm as well as for making extensive numerical studies.

74 A. Brandt

References

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Verlag, Berlin [6I K6nig D, Stoyan D (1976) Methoden der Bedienungstheorie. Akademie-Verlag, Berlin [7J Kuczura A (1973) The interrupted poisson process as an overflow process. Bell Syst Tech J

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[10] Wilkinson RI (1956) Theories for toll traffic engineering in the USA. Bell Syst Tech J 35:421- 514

[11] Willie H (1988) Individual call blocking probabilities in the loss systems S M + M / M / N and G + M / M / N . EIK 24/11/12:601-612

[12] Willie H (1990) Individual blocking probabilities in the loss system G1 + "" + GNIM[IlO. Queueing Systems 6:109-112

[13] WolffR (1982) Poisson arrivals see time averages. Operations Research 30:223-231

Received: December 1994 Revised version received: January 1996