Accurate formulae for the loss probability evaluation in a large class of queueing systems

Performance Evaluation 18 (1993) 125-132 North-Holland

125

Accurate formulae for the loss probability evaluation in a large class of queueing systems Andrea Baiocchi INFOCOM Department, University of Roma "La Sapienza", Via Eudossiana 18, 00184 Roma, Italy

Received 24 December 1991 Revised 9 October 1992 Abstract

Baiocchi, A., Accurate formulae for the loss probability evaluation in a large class of queueing systems, Performance Evaluation 18 (1993) 125-132. In this paper a very accurate approximation method is developed, that leads to explicit expressions for the loss probability of the M / G / 1 / K and G I / M / 1 / K queues. The results can also be generalized to the corresponding discrete-time queueing models, i.e. the G e o / G / I / K and GI/Geo/1/K queues. The approximations obtained by this method are asymptoticallycorrect as the buffer size increases. Moreover, they can be inverted, thus allowing a very effective buffer dimensioning to be done. Keywords: Queueing models; Poisson arrival; finite buffer; loss probability; asymptotic approximation; explicit buffer sizing.

I. Introduction A great interest has recently arisen in the study of finite buffer queues (e.g. in the communication field, in conjunction with the proposal of techniques such as the ATM). In particular, the analysis of the loss probability is becoming a central issue in the applications of the queueing theory. On the other hand, no simple closed form solutions are known to the author, unless for the M / M / 1 / K queue; quite cumbersome closed formulae have been derived for the M / P H / 1 / K and P H / M / 1 / K queues [4], but they lend themselves to little more than numerical computation, especially for large dimensionalities of the state space of the P H distribution. Even the well known M/G/1/K and G I / M / 1 / K queues can be analyzed only numerically, for example by using the E m b e d d e d Markov Chain (EMC) approach. Correspondence to: A. Baiocchi, INFOCOM Department,

University of Roma "La Sapienza", Via Eudossiana 18, 00184 Roma, Italy. Email: [email protected].

This p a p e r addresses the problem of determining explicit accurate approximations of the loss probability for the M / G / 1 / K and the GI/M/1/K queues. The adopted methodology can be straightforwardly extended to discrete-time models, such as the G e o / G / 1 / K [3] and the GI/Geo/I/K queues. The most interesting properties of these approximate formulae are that: (i) they are asymptotically correct, i.e. the ratio of the exact value of the loss probability to the approximate value tends to 1 as the buffer size increases [1]; (ii) they reduce to the exact expression of the loss probability in the case of the M / M / 1 / K queue (see the Appendix B); (iii) their coefficients can be easily computed from the main queue parameters; and (iv) they can be inverted, allowing a straightforward buffer dimensioning to be done. As for the organization of the paper, in Section 2 the main notations and general definitions used in the p a p e r are introduced. Section 3 deals with the approximation of the loss probability. Finally, in Section 4 some numerical examples are presented.

0166-5316/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

126

A. Baiocchi / Loss probability evaluation

2. General definitions and duality property Only the steady-state probability distributions of the EMCs will be considered. The embedded time epoches are chosen as the departure and the arrival time instants for the M / G / 1 / K queue and the G I / M / 1 / K queue, respectively. The Laplace-Stieltjes Transform (LST) of the Cumulative Probability Function (CPF) of the interarrival time and the mean arrival rate will be denoted by F * ( s ) and A, respectively. The LST of the service time CPF and the mean service completion rate will be denoted by H * ( s ) a n d / z , respectively. The buffer size is denoted by K; obviously, up to K + 1 customers are allowed into the queue. Further, the following items are defined: Ao mean offered traffic, equal to h//x; ai probability of the event v = i, i >~ 0, where v is the random variable representing the number of arrivals within a service time in the M / G / 1 queue; ~b(z) Probability Generating Function (PGF) of v; R least modulus root of the equation z = $ ( z ) , apart from the trivial root z = 1; t~ i probability of the event ~ = i, i >/0, where z; is the random variable representing the number of service completions within an interarrival time in the G I / M / 1 queue; ~b(z) P G F of 1;; tr least modulus root of the equation z = ~b(z), apart from the trivial root z = 1; ~-i(K) i-th element of the Queue Length Probability Distribution (QLPD) at the embedded epoches for the M / G / 1 / K queue, i = 0, 1. . . . . K;

#i(K)

i-th element of the Q L P D at the embedded epoches for the G I / M / 1 / K queue, i = 0, 1. . . . . K + 1. H(K) loss probability of M / G / 1 / K queue; /7(K) loss probability of G I / M / 1 / K queue. According to the above definitions, $ ( z ) = H * ( A - Az) and ~b(z) = F*(/x - / . , z ) . The result proved in the Appendix A shows that: (i) the equation z = $ ( z ) has no roots with modulus less than or equal to max{l, R}, apart from 1 and R themselves; a perfectly analogous result holds for the equation z = ~b(z), replacing R with ~r; (ii) if p , denotes the radius of convergence of $ ( z ) , then A 0 < 1 =, R ~ (1, p,), A o = I~R=I and A 0 > l ~ R ~ ( 0 , 1 ) ; ( i i i ) i f p * d e notes the radius of convergence of ~b(z), then A0 l

o" e (1, p,~). Finally, a duality definition for the M / G / 1 / K and G I / M / 1 / K queues is introduced. To this end it is convenient to display the one-step transition probability matrix of the EMC of the M/G/1/K queue at the service completion epoches:

oo

S0

~1

~2

~K-1 i=K oo

£g0

O/1

O~2

O~K- 1

p=

i=K oo

0

Ol0

Ol1

OlK_ 2 i=K-1

0

0

0

ao

1 -a o (1)

Andrea Baioechi was born in 1962. He received the Laurea degree in electronics engineering and the Ph.D. in information and communications engineering in 1987 and 1992, respectively, both from the University of Roma "La Sapienza". In 1991 he spent one year at the Department of Mathematical Methods and Models for Applied Sciences of the University of Roma "La Sapienza", where he held courses in numerical analysis. In July 1992 he joined the INFOCOM Department of the same university as a researcher in communications. His main scientific interests lie in the field of multiaceess network protocols, traffic modeling and performance evaluation in broadband communications networks.

A. Baiocchi / Loss probability evaluation

where ai=J0

,~ (At) i - - ~v. e - a t d H ( t ) '

i>~0.

(2)

Analogously, the one-step transition probability matrix of the E M C of the G I / M / 1 / K - 1 queue at the arrival epoches is given by

&o

1 -&o

•K-2

0

0

0

&K-3

40

0

127

In the following, a subscript d will distinguish the variables referring to the dual queue of a given one. The definition of duality entails that 7 r i ( g ) = ~ d , K _ i ( g -- 1), i = 0, 1. . . . . K, and ~ i ( K ) = T r d , K + l _ i ( K + 1), i = 0 , 1 . . . . , K + 1. Such property will be exploited in the next section, where the approximate formulae are derived. It is to be emphasized that the duality property holds only for the queues with finite buffer sizes.

3. Approximation of the loss probability

i=K-1

p=

CCK-1

6K-2

i=K

~1

&0

oo

~0 i-K

(3) where f ~(~t)i

&i=L

- - - -w~ . e

-tzt

dF(t),

i>I0.

(4)

Note the different buffer sizes of the two queues, required for the matrices P and /; to have the same size K + 1. Let us renumber the states of the E M C of the M / G / 1 / K queue in reversed order, so that the state i becomes the ( K - i)-th one, i = 0 . . . . . K. Then, comparing the eqs. (1)-(2) with the eqs. (3)-(4), it can be recognized that the one-step transition probability matrix of this new Markov chain has just the same structure and the same entries as /; in eq. (3), provided that the role of the arrival and service processes are interchanged with respect to the original M / G / 1 / K queue. The G I / M / 1 / K 1 queue so obtained will be referred to as dual of the original M / G / 1 / K queue. Analogously, the dual queueing system of a G I / M / 1 / K queue is defined as a M / G / 1 / K + 1 queue, in which the role of the arrival and service processes are interchanged and the i-th state of the Markov chain embedded at the departure epoches corresponds to the ( K + 1 - i)-th state of the E M C of the original queue, i = 0, 1. . . . . K + 1. Again the relation between the EMCs of the two dual queues can be highlighted by comparing the eqs. (1)-(2) with the eqs. (3)-(4)•

Since a single server queueing system has a throughput upper bound equal to 1 Erl, the loss probability tends to 0 for A 0~< 1 and to (A 0 - 1 ) / A o , for A 0 > 1, as the buffer size increases; that is to say H ( ~ ) = / I ( ~ ) = max{0, 1 l/A0}. Firstly, the approximation for the M / G / 1 / K queue is derived. The loss probability can be expressed in terms of the probability of having the queue empty at a generic time point, Pr{empty}, by means of a simple flow conservation argument, equating the mean flow into the queue, i.e. A [ 1 - H(K)], and the mean flow out, i.e. I * [ 1 - Pr{empty}]. Since it can be shown [2] that Pr{empty} is given by r r o ( K ) / [ A o + ~r0(K)], then = 1

H(K)

1 7to(K) + A ° .

(5)

According to a well known result, holding for the embedded Q L P D of the M / G / 1 / K queue, the ratios "rri(K)/Tro(K) , i = 0, 1. . . . . K, do not depend upon K [2]. Formally, it can be stated that ~ i ( K ) = cirro(K), i = O, 1 . . . . . K , for any K >/0. The sequence {ci} is the solution of the linear equation system given by o~ic0+ E~+__I 1 Oli+l-jCj = Ci, i >t 0; therefore the c,'s can be computed recursively, starting with c o = 1, according to the relation: 1 -% Cl

oL0

(6) Ci+l =

Ci--Ol i --

ai+a_jC j ,

i>~ l,

with ~i, i >/0, given by eq. (2). From the normalization condition of the QLPD, it follows that 7 r o ( K ) d ~ = 1, where d~ = Y'.iK=oCi, K >10. The

128

A. Baiocchi / Loss probability evaluation

generating function D ( z ) of the sequence {d r} can be easily derived, yielding D ( z ) = O ( z ) / For A 0 ¢ 1, the two least modulus poles of D ( z ) are distinct, both simple and equal to 1 and R. They represent the dominant poles of D(z). Hence, D ( z ) can be approximated by its partialfraction expansion with respect to such two poles, obtaining 1

D(z) =

(1 - A o ) ( 1 - z ) 1 -

[~b'(R) - 1](1

-z/R)

(7)

It is worth mentioning that the resulting expression of D ( z ) turns out to be exact if and only if D ( z ) is a rational function with only two poles, namely 1 and R. In Appendix B it is shown that this happens if and only if the service time probability distribution is negative exponential, i.e. only for the M / M / 1 / K queue. From the right-hand side of the eq. (7) the inverse transform of D ( z ) can be easily found: it is equal to the sum of a constant and of an exponential term. Reminding that rr0(K) = 1 / d K, from eq. (5) the following approximation can be deduced for A 0 < 1:

1

Ao , 1 - A o C1R-K

0"(1) ' K + C3+ - 34,"(1)

/I(K)-/7(m)

---[max{l, 1/Ao}] 2 × [//a(K+

1) -

(12) As a consequence of the duality, 0d(Z) reduces to ~b(z) and therefore R d = or. Then from eqs. (8)-(12) the approximate formulae for this case can be so obtained: C1 o-K+I

d, K+ ' 1 -A~

Ao
C2 RK Ao

H ( K ) -/7(00) •

(11)

where C 3 - 0"(1)/2. As for the G I / M / 1 / K queue, it suffices to note that, thanks to the duality p r i n c i p l e / I ( K ) = 7 ? r + l ( K ) = % 0 ( K + 1) and Ado = 1/.4 o. Then, using the eq. (5), it can be easily seen that the ratio between / I ( K ) - / I ( ~ ) and lld(K + 1) //a(~) tends to [max{l, l/A0}] 2, as K ~ ~. Therefore

(8)

where C 1 = ( 1 - A o ) 2 / [ ~ ' ( R ) - 1]. In the case A 0 > 1, subtracting out H(oo) from both sides of eq. (8), some elementary algebraic manipulations yield the following approximation:

1

C3

H(K) -

1

C1 R -K

II( K ) ~-

hence d K -~ [2/~b"(1)I[K + W"(1)/3~b"(1)] and from the eq. (5) and the equality r r 0 ( K ) = 1 / d K it follows that

,

(9)

1

A0Ao> 1

K + 1 + C3 +

¢"(1)

A o - 1 C2RK A0=I,

where C 2 = [ 1 - O'(R)]/A~. Finally, for A 0 = 1, the function D ( z ) has a double pole at z = 1. This is the least modulus pole of D(z), i.e. the dominant one. Thus, approximating D ( z ) by means of the partial fraction expansion relevant to the pole z -- 1, after some tedious algebra, we get: 2 D(z) = 0"(1)(1-z)

[

z ~

0"(1) ] + 30"(1---~ ;

(10)

(13) where C1.--- 1 - 4)'(00, C2 = (1 - A o l ) 2 / [ ~ b ' ( ( r ) - 1] and C 3 = ~b"(1)/2. It is to be stressed that the results obtained above depend only on: (i) the validity of eq. (5); (ii) the particular structure of the one-step transition probability matrix of the Markov chain embedded at the departure epoches of the M/G/1/K queue; (iii) the duality principle.

A. Baiocchi / Loss probability evaluation Therefore, extension to other queueing models of the previous results can be envisaged. F o r e x a m p l e , t h e r e a s o n i n g l e a d i n g t o eqs. (8)-(13) can be also applied to the Geo/G/1/K and GI/Geo/1/K q u e u e s , s i n c e p o i n t s (i), (ii) a n d (iii) a b o v e a r e s a t i s f i e d . M o r e i n d e t a i l , l e t t h e t i m e axis b e d i v i d e d i n f i x e d size u n i t s ( s l o t s ) . As regards the Geo/G/1/K q u e u e , if p is t h e p r o b a b i l i t y o f a n a r r i v a l i n a s l o t a n d H ( z ) is t h e PGF of the number of slots required for servicing a customer, then O(z) = H(1 -p +pz) and A 0 = pH'(1). As for the GI/Geo/1/K q u e u e , if q denotes the probability of a service completion in a s l o t a n d F ( z ) is t h e P G F o f t h e n u m b e r o f s l o t s r e q u i r e d f o r a n a r r i v a l t o t a k e p l a c e , t h e n ~b(z) = F ( 1 - q + qz) a n d A 0 = 1 / q F ' ( 1 ) .

4. Numerical results T h e a i m o f t h i s s e c t i o n is t o a s s e s s t h e a c c u r a c y o f t h e a p p r o x i m a t e e x p r e s s i o n s o f t h e loss probability discussed in the previous section. Through the evaluation of some numerical examp l e s , it is s h o w n t h a t t h e e r r o r s i m p l i e d b y s u c h approximations are negligible in the whole range o f v a l u e s o f K a n d A 0. Only the M/G/1/K queue formulae are evaluated, since the accuracy of the GI/M/1/K q u e u e f o r m u l a e is e x a c t l y t h e s a m e . M o r e o v e r , o n l y t h e f i r s t f e w v a l u e s o f t h e b u f f e r size n e e d b e considered, since the approximations are asympt o t i c a l l y c o r r e c t as K ~ oo. As an example, the M/D/1/K q u e u e is c o n s i d e r e d , s o t h a t q ~ ( z ) = e "40~z-1) T h e n u m e r i c a l results given in Tables 1-3 highlight the extreme accuracy of the proposed formulae in this case. The data in Table 1 and 2 refer to the comparison between the exact and approximate values of

129

Table i Comparison between exact and approximate values of the loss probability of the M / D / 1 / K queue for various values of K and A o < 1 A 0 = 0.8

A 0 = 0.9

Buffer Exact ssize solution 4.4444E- 1 1.9957E- 1 1.0330E- 1 5.8764E-2 3.5290E-2 2.1857E-2

Approximarion

Exact solution

Approximation

5.6180E- 1 2.0427E- 1 1.0320E- 1 5.8586E-2 3.5180E-2 2.1785E-2

4.7368E- 1 2.3464E- 1 1.3844E- 1 9.1356E-2 6.4362E-2 4.7203E-2

5.8824E- 1 2.4033E- 1 1.3912E- 1 9.1648E-2 6.4554E-2 4.7340E-2

A 0 = 0.95 Bur- Exact fer solution size

Approximation

Buf- Exact fer solution size

0 1 2 3 4 5

5.5188E- 1 2.4708E- 1 1.5332E- 1 1.0795E- 1 8.1307E-2 6.3854E- 2

6 7 8 9 10 15

4.8718E-1 2.5192E- 1 1.5714E- 1 1.1022E- 1 8.2789E-2 6.4906E- 2

Approxirnation

5.2380E 4.3159E 3.6120E3.0596E2.6165E1.3131E-

5.1591E 4.2543E 3.5624E 3.0188E2.5824E1.2964E -

Ao=l Buf- Exact fer solution size

Approximation

0 1 2 3 4 5

6.0000E-1 2.7273E- 1 1.7647E - 1 1.3043E - 1 1.0345E- 1 8.5714E-2

5.0000E-1 2.6894E- 1 1.7634E - 1 1.3044E- 1 1.0345E- 1 8.5714E-2

Bur- Exact fer solution size 6 7 8 9 10 15

Approximation

7.3171E-2 6.3830E- 2 5.6604E - 2 5.0847E- 2 4.6154E-2 3.1579E-2

7.3171E-2 6.3830E- 2 5.6604E - 2 5.0847E- 2 4.6154E-2 3.1579E-2

queue and its

A o = 1.1

/ - / ( K ) - H(oo) (//(~o) = 4.76E - 2)

H ( K ) - H(~) (H(oo) = 9.09E - 2)

Buffer size

Exact solution

Approximation

Exact solution

Approximation

0 1 2 3 4 5

4.6548E2.3806E1.4826E 1.0408E7.8292E 6.1484E -

5.5991E2.4144E1.4834E 1.0407E 7.8286E 6.1479E -

4.3290E2.1119E1.2466E 8.2775E 5.8770E4.3469E -

5.2421E2.1427E 1.2474E 8.2769E 5.8767E 4.3467E -

1 1 1 1 2 2

1 1 1 1 2 2

2 2 2 2 2 2

Table 2 Comparison between exact and approximate values of the loss probability of the M / D / 1 / K queue for various values of K and A o = 1

Table 3 Comparison between exact and approximate values of the difference between the loss probability of the M / D / I / K limiting value H(oo), for various values of K and A 0 > 1 A o = 1.05

2 2 2 2 2 2

1 1 1 2 2 2

1 1 1 2 2 2

130

A. Baiocchi / Loss probability evaluation

the loss probability for A 0 < 1 and A 0 = 1, respectively. As for Table 3, it reports both the exact and approximate values of the difference between the loss probability and its asymptotic value, i.e. /7(K)-/7(oD), for A 0 > 1. This has been done to highlight the accuracy of the approximation of the significant part o f / 7 ( K ) , once the known additive term /7(00) is subtracted out. It can be noted that, the higher is A 0, the larger are the errors relevant to the first few values of the buffer size. In particular, a large relative error systematically occurs at K = 0. On the other hand, / 7 ( O ) = A o / ( 1 + A o) for an M / G / 1 / O queue, w h i l e / I ( 0 ) = &0 = F * ( / z ) for a G I / M / 1 / O queue; so there is no need of an approximate formula for K = O. Such results clearly show that the approximate expressions can be safely used instead of the exact value and therefore they can be powerful tools for the loss performance analysis of a finite buffer queue of the types considered in this paper.

Acknowledgement The author would like to thank Dr. Nicola Bl6fari-Melazzi for his helpful suggestions.

References

Appendix A Let us consider an equation of the form

f?e b'(z-" d X ( t )

Theorem. The equation z = X * ( b - b z ) has at most one real positive root ~ with modulus less than 1 + 3,/b, apart from the trivial root z = 1. When ~ exists, it is also the least modulus root of the equation z = X * ( b - b z ) , apart from z = l . Sufficient conditions for the existence of ~ are: (i) p < 1, 3, < oo and X * ( - 3 , - ) > 1 + y / b ; (ii) p < 1 and 3, = oo; (iii) p >1 1. Moreover ~ E (1, 1 + 3,/b) = 1

p
s¢~ (0, 1),

p>l.

= X * ( b - bz)

(A.1)

where b is a real positive number, X ( t ) is the CPF of a non-negative non-degenerate random

(A.2)

Proof. By virtue of the hypotheses, the function f ( z ) = X * ( b - bz) is analytic within the circle of radius r = 1 + 3,/b > 1 centred at the origin of the complex plane, so that the equation z = f ( z ) can be considered for [ z [ < 1 + 3,lb. We can simplify the equation z = f ( z ) by removing the trivial root at z = 1, which always exists. It is in fact known that the LST Y * ( s ) of the CPF of the residual-life random variable associated to X ia given by Y*(s) =

[1] A. Baiocchi, Asymptotic behaviour of the loss probability of the M/G/1/K and GI/M/1/K queues, Queueing Systems 10 (1992) 235-248. [2] D. Gross and C.M. Harris, Fundamentals of Queueing Theory (Wiley, New York, 1985). [3] A. Gravey,J.-R. Louvion and P. Boyer, On the Geo/D/1 and Geo/D/1/n queues, Performance Evaluation 11 (2) (1990) 117-125. [4] M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach (Johns Hopkins Univ. Press, Baltimore, MD, 1981).

z =

variable X and X * ( s ) the corresponding LST. Let 3' = sup{x >/0 [ X * ( - s ) = E[exp(sx)] is analytic for Re[s] - 3 , . In the following we assume that 3' > 0. Moreover, let a = E[X] and p =ab. We can state the following

1-X*(s)

(A.3) as The function Y * ( s ) inherits the analiticity properties of X * ( s ) , so that sup{x > / 0 [ Y * ( - s ) is analytic for Re[ s] < x} = 3,. Therefore, the equation z = X * ( b - b z ) can be simplified to 1/p=Y*(b-bz)-g(z), for [ z l < r . The function g ( z ) has the following properties (a) g ( z ) is analytic in I z l < r ; it is real and positive for any real z; (b) 0 < g ( 0 ) = Y * ( b ) = [1 - X * ( b ) l / p < l / p ; (c) g ( 1 ) = 1; (d) g ' ( z ) > O , V z • ( O , r); i.e. g ( z ) is a monotonously increasing function of z, for z ~ (0, r); (e) g " ( z ) > O. V z ~ (0, r); i.e. g ( z ) is a strictly convex function of z, for z ~ (0, r). These properties show that the equation 1 / p = g ( z ) has at most one real positive root in the region [ z l < r, denoted by ~:. Thanks to properties (a)-(e), such a root surely exists and belongs

A. Baiocchi / Loss probability evaluation

to the interval (0, 1) when p > 1, while it equals 1 when p = 1. For P < 1, if ~ exists, then ~ ~ (1, r); a sufficient condition for the existence of ~ is that we can find ~, ~ (1, r) such that g(~,) > 1/p. In terms of the LST of X(t), this condition amounts to the existence of ~ ~ ( - T, 0) such that X*(g) > 1 - g/b. This is surely possible if y < oc and X * ( - T - ) > 1 + y/b. There remains to verify that such inequality is always verified when 3' = oo. In fact, according to Jansen's inequality, we have X*(s) = E [ e x p ( - s x ) ] > / e x p ( - s E { x ] ) = e x p ( - s a ) , which is definitely strictly greater than 1 - s / b for s ~ -oo. The considerations above show that there cannot be any other real positive root of the equation g(z) = 1/p, apart from ~. There remains to prove that ~ (when it exists) is the least modulus root of eq. (A.1) apart from the trivial root z = 1, i.e. the least modulus root of the equation g(z) = 1/tg. First, for any 6 ~ [0, 27r), it can be easily verified that c~

fo

oo

Ig(u e'~)l ~< f0 eVt("c°s~ l ) d Y ( t ) 4 dY(t)=g(u)=lg(u)l,

u ~ [0, r ) ,

(A.4)

the equality holding only for 3 = 0. This implies that the modulus of g(z) attains its unique maximum on the positive real axis, when evaluated on a circumference centred at the origin.

O(z)=

1

131

f ' ( 1 ) = p, show that f ' ( ~ ) > 1, f ' ( ~ : ) = 1,

p
f ' ( ~ ) < 1,

p>l.

(A.5)

The result proved in this Appendix can be applied to the study of the M / G / 1 / K queue, by letting X(t) = H(t) and b = 3., whence p = A 0, = R. In case of a G I / M / 1 / K queue we choose instead to let X(t) =-F(t) and b =/z; as a consequence p = 1/.4 o, while ~ = ~r. In particular, eq. (A.5) guarantees that the coefficients of the asymptotic expressions of the loss probability C~, C2, C 1 and C2 are surely positive.

Appendix B Let us assume that the equality holds in eq. (7), i.e. that D(z) is a rational function having only the two poles 1 and R. Assume for the time being that R 4:1 or equivalently A 0 4: 1. Then 1 D(z) = ( 1 - A o ) ( 1 - z )

M + (1-z/R)'

where the residual relevant to the pole R has been briefly denoted by M = - [ O ' ( R ) - 1 ] -1. Since ~b(z)= z D ( z ) / [ D ( z ) - 1], eq. (B.1) implies that:

z [ ( M ( 1 - A 0 ) + 1 / R ) z - ( M ( 1 - A 0 ) + 1)] [ R-A°] -A°z2+ M(1-A0)+A 0 z-(M(1-A0)+A0) R R

Let us suppose that there exist some point x 4:~ such that [xl~< ~ and g(x) = 1/p. Then, using property (d) and applying eq. (A.4) to g(x) with the strict inequality, since x cannot be real and positive, it follows I g ( x ) l < g(I x I) ~
(B.1)

(8.2)

By means of a rather tedious algebra it can be verified that 0 ( 1 ) = 1, 0 ' ( 1 ) = A o , 0 ( R ) = R and O'(R) = ( M - 1 ) / M . The two unknowns R and M are determined by the behaviour of O(z) for z = 0 and z ~ - ~ . In fact, since 0(0) = a0 > 0, it follows that O(z) cannot have a zero in z = 0 and hence it must be M ( 1 - A 0 ) + A 0 = 0 (see eq. (B.2)). Moreover, from

O(z) = H * ( & - ) L z ) = f e A'(z-') d H ( t ) , Finally, simple geometrical considerations on the plot of f(z), together with the fact that

"0

(8.3)

132

A. Baiocchi / Loss probability evaluation

it follows that qt(z) is analytic is a half-plane of the z-plane containing the negative real semi-axis: therefore we can take the limit of ¢J(z) as z -oo. Equation (B.3) readily shows that such a limit is zero, provided H(0) = 0. This means that the service time probability density function cannot have an impulse at t = 0. The requirement that the limit of ~O(z) as z ~ -oo be zero leads to M(1 - A 0) + 1 / R = 0 (see eq. (B.2)). The two conditions above allow the calculation of M and R, yielding A0 M =

-

A0-1 1

(B.4)

Ao"

Substituting the expressions in eq. (B.4) into eq. (B.2) yields ~b(z)= 1/(1 + A 0 - A o z ) and

hence, reminding that A 0 = A / / z , it follows H * ( s ) = t~/(s + t~), i.e. the service time distribution must be negative exponential. In the special case A 0 = 1 the same reasoning as above can be repeated, starting with eq. (10) in which the = is replaced by exact equality and performing the same calculations as in the case A o 4: 1. It can be derived that ~O(z)= 1 / ( 2 - z) and hence, reminding that in this case A = tz, we find H * ( s ) = I z / ( s + Iz). The considerations made up to now prove that the service time probability distribution must be negative exponential, if the formulae given in Section 3 are to be exact. It is easy to verify that also the reverse holds, i.e. the approximations given in Section 3 are exact for the M / M / 1 / K queue. Therefore it can be concluded that the proposed approximate formulae turn out to be exact if and only if they are applied to the M / M / 1 / K queue.