Second-order properties of the loss probability inM/M/s/s +c systems

Queueing Systems 15 (1994) 289-308 289

Second-order properties of the loss probability in M/M/s/s + c systems *

Ant6n io Pacheco 1

School of Operations Research and lndustrial Engineering, E& TC Building, Cornell University, Ithaca, NY14853-3801, USA

Received 16 July 1992; revised 9 February 1993

The Erlang loss function, which gives the steady state loss probability in an M/M/s /s system, has been extensively studied in the literature. In this paper, we look at the similar loss probability in M/M/s/s + c systems and an extension of it to nonintegral number of servers and queue capacity. We study its monotonicity properties. We show that the loss probability is convex in the queue capacity, and that it is convex in the traffic intensity p if p is below some p* and concave if p is greater that p*, for a broad range of number of servers and queue capacities. We prove that the one-server loss system is the only M/M/s/s + c system for which the loss probability is concave in the traffic intensity in all its range.

Keywords: Queues; loss probability; monotonicity; convexity.

1. I n t r o d u c t i o n

The Er lang loss funct ion,

a~/s! R+ B(s,a) - ~iS__oai/i[, s~N , a~ , (1)

is one o f the m o s t f a m o u s funct ions in queueing theory. Er lang [9] used B(s, a) to represent the s t eady state probabi l i ty t ha t a cus tomer (call) which is a m e m b e r o f a Poisson s t ream o f ra te a, arr iving a t a group of s servers ( te lephone t runks) wi th un i t exponent ia l service t ime, will be rejected. Later , several a t t empts were m a d e to general ize the Er lang loss funct ion. I t has been f o u n d tha t the Ef l ang loss func t ion holds for M / G / s / s systems, i.e. B(s, a) is the s teady state loss p robabi l i ty in an M / G / s / s sys tem wi th offered load a and uni t mean-service time. This resul t was first given a r igorous p r o o f by Sevas t ' yanov [28].

* Research supported by Grant BD/645/90-RM from Junta Nacional de Investiga~o Cientifica e Tecnol6gica.

i On leave from: Departamento de Matem~tica, Instituto Superior T6cnico, Av. Rovisco Pals, 1096 Lisboa Codex, Portugal.

�9 J.C. Baltzer AG, Science Publishers

290 A. Pacheco / Loss probability in M / M / s / s + e systems

One of the nice properties of the Erlang loss function is that it can be extended to a transcendental function of two complex variables (see theorem 3 in Jagerman [14]), as follows:

B(s,a)- e -x 1 + dx (2)

Because of its importance, the Erlang loss function has received a great deal of attention in very diverse ways. The following brief review of some of the important work in this area provides a background for the results of the present paper. For a more detailed review of some of the early work on the Erlang loss function, see Tak/tcs [30] and references therein.

Aside from eq. (2), approximations of B(s, a) for a nonintegral number of servers s were considered using a variety of methods such as: the equivalent random method (Wilkinson [31]), Haywards approximation (Fredericks [10]), continued fraction procedures (Levy-Soussan [21] and Burke [5]), the decomposition method (Sanders et al. [27]), different types of interpolation (Akimaru et al. [2], Kortanek et al. [18] and Rapp [26]), and using asymptotic expansions (Jagerman [14] and Akimaru and Takahashi [3]).

The derivatives of the Erlang loss function have also been studied; they arise in some problems such as optimal trunk group size apportionment problems [I]. Akimaru and Nishimura [1] present tables of the derivatives of B(s, a) with respect to s and a, while Jagerman [14] and Akimaru and Takahashi [3] developed asymptotic functions and approximations for the same derivatives.

The computation of the Erlang loss function gives rise to some problems when the number of servers is large. In practice, a large number of trunks occurs in some problems such as in the study of certain satellite communication systems (Horing et al. [13] and Miller [23]). This fact suggests the need to obtain asymptotic approximations of the Erlang loss function. Work in this area was done by Akimaru and Takahashi [3], Descloux [7], Jagerman [14], Miller [23] and Newell [24]. As a sim- pler alternative to the use of the exact formula, Harel [11] gives lower and upper bounds for the Erlangloss function.

For the optimal design of queueing systems it is of primary importance to know the behaviour of certain performance measures of the design; among the most important such measures is the loss probability. This fact is probably the main reason for the attention the Erlang loss function has been recently receiving (for a more detailed review of the recent work on optimization on queues see Harel [12] and references therein).

The function B(s, a) decreases with the number of servers s, and increases with the offered load a, for s, a > 0. For a positive integer number of servers and fixed offered load a > 0, the Erlang loss function was first shown to be convex in the number of servers by Messerli [22], who used an argument based on optimal alloca- tion. At about the same time, algebraic proofs of the same result were given by Buchner and Neal [4] and Descloux [8]. More recently, Shanthikumar and Yao [29]

A. Pacheco / Loss probability in M / M / s / s + e systems 291

gave yet another proof of the same result by stochastic comparison, using the con- cavity of the throughput in M/M/s / s systems as a function of the number of servers. The first proof of the convexity of the Erlang loss function in the nonintegral number of servers was given in Krupp [20], but the proof of this result in Jagers and Van D o o m [16] seems to be better known. In Jagers and Van D o o m [17] it was proved that a generalization of the Erlang loss and Erlang delay functions is convex in the (nonintegral) "number of servers" and a correction of Jagers and Van D o o m [16] was made. For a fixed offered load a > 0, the inverse of the Erlang loss function was shown to be log-convex in the (nonintegral) number of servers by Jagerman [14]; another proof of this result was given in Jagers and Van D o o m [16].

The Erlang loss probability was shown to be convex in the service rate by Harel [12]; in the same paper it was shown that the loss rate (throughput) of M/G/s/s systems is jointly convex (concave) in the arrival and service rates (another proof of this result was given in Krishnan [19]) and the expected number of customers in the system is concave in the arrival rate but it is neither jointly concave nor jointly convex in the arrival and service rates. The steady state loss probability in G/M/s/s + c systems was shown to be decreasing convex in the number of servers, as well as in the service rate, by Chang et al. [6] using stochastic comparison. This result is an extension of the result of the convexity of the Erlang loss function on the number of servers, proved in Messerli [22], and on the service rate, proved in Harel [12]. An algebraic proof of the particularization to M/M/s / s + c systems of the convexity of the loss probability in the service rate was given in Pacheco [25]; in the same paper it was shown that the loss rate (throughput) of M/M/s / s + c systems is jointly convex (concave) in the arrival and service rates but neither the steady state mean number in the queue nor in the system is either jointly convex or concave in the arrival and service rates.

For a fixed positive integer number of servers, Harel [12] showed that B(s, sp), p > 0, is convex in the traffic intensity p if p is below some p* and concave if p is greater than p*.

In this paper we consider the generalized Erlang loss (GEL) function

o (a)c B(s, a, c) s[ = seN, a~IR +, c~N0, (3)

k = 0 " i=1

which gives the steady state loss probability for the system M/M/s / s + c, in which there is a waiting room with capacity c. The classical Erlang loss function (1) is then identical with B(s, a, 0). It is useful not to constrain s and c to be integers. We can extend B(s, a, c) to a continuous function on (0, +00) 2 x [0, +co), with B(s, a) given by formula (2), as follows:

292 A. Pacheco / Loss probability in M / M / s / s + c systems

B(s,a,c) - (a/s)C with a r s. (4) g _ l ( s , a ) +.Ix- (a/s) c+1'

1 - a/s

The advantage of the GEL function over the classical Erlang loss function is that the GEL function, because it has one more parameter associated with the queue capacity, gives more flexibility in terms of the design of queueing systems. By changing the queue capacity c from 0 to + ~ in the GEL function, we take into consideration the fraction of lost customers in a broad class of systems ranging from the Erlang loss system to the Erlang delay system. In our study we pay attention to two different parametrizations of the GEL function, namely B(s, a, c) and B(s, sp, c). In the first parametrization, the offered load a is one of the parameters of interest, and in the second the traffic intensity p is considered instead; both cases are of practical interest. In the rest of the paper we assume, unless explicitly stated, s, a, p e (0, +oo) and c e [0, +c~).

We start, in section 2, by deriving some preliminary results that will be used later in the paper. In section 3 we study monotonicity properties of the GEL function as well as the limit values of the same function when its parameters go to 0 and +OO.

In section 4 we first prove the convexity of the GEL function with respect to the queue capacity parameter c, for c >~ 0 and positive integer number of servers s; this extends the equivalent result for integer values of the queue capacity obtained by Shanthikumar and Yao [29]. Next we prove that for s e N, c s No, B(s, sp, c) is convex in the traffic intensity p ifp is below some p~,c and concave ifp is greater than p~,~, if s ~< 3 or (4 ~< s ~< 100, 0 ~< c ~< 2000). We show that the one-server Erlang loss system (system M / M / 1 / 1 ) is the only M / M / s / s + c system where the steady state loss probability is concave with respect to the traffic intensity for any traffic intensity.

In section 5 we present tables of the zeros of the second derivative of the GEL function with respect to the traffic intensity and make some comments about these values.

2. Prel iminary results

We start this section by obtaining results on the monotonicity of the Erlang loss function B(s, a). The reason for this is that we base the monotonicity results for the GEL function B(s, a, c) on similar results for B(s, a) and the relationship between these two functions, as given by eq. (4).

LEMMA 1 + For s, a, p e IR , the functions B(s, a) and B(s, sp) have the following properties:

increases with a; B(s, a) ---* O; B(s, a) --~ 1. (i) B(s, a)" " a--'O§ ~ + ~

A. P aeheco / L o s s p robab i l i t y in M / M / s / s + e s y s t e m s

� 9 . S - - + O + 8 - - + + o o (11) B(s, a) decreases with s; B(s, a) --+ 1; B(s, a) --+ O. . . . . . " " p~O + " " p--*+oo

(111) B(s, sp) increases with p; B(s, sp) ~ 0; B(s, sp) ---, 1. (iv) B(s, sp) decreases with s;

s~o+ s-~+o~ ( 0 p < 1, B(s, sp) ~ 1;B(s, sp) ~ I I - 1 / P p>~l.

2 9 3

Proof We have, for x, s, a, p ~ ~+, (1 s �9 s a -*0+ s a-*+~ | +x /a ) decreaseswltha;( l+x/a) ., +oo; ( l+x /a ) ----+ 1. ( 1 s . �9 s s - + 0 s s - ~ + o o | +x /a ) mcreaseswl ths; ( l+x/a) , l '~( l+x/a) ---+ +oo.

| (1 + x/sp) s decreases withp; (1 + x/sp) s p~o +oo; (1 + x/sp) s PZ-~ ~ 1. | (1 + x/sp)~increaseswiths;(1 + x/sp)**-~~ l ;(1 + x/sp)~-++~ ~ eX/C From this, eq. (2), and by monotone and dominated convergence, the desired results follow. []

In terms of queueing systems, lemma 1 states that in loss systems the loss probability decreases with the number of servers, for fixed offered load or fixed traffic intensity, and increases with the offered load and the traffic intensity, for a fixed number of servers.

The next lemma contains results that will be used in subsequent sections, and is therefore presented separately. We remark that the result (i) of this lemma has been established by other authors (e.g. lemma 1 in Harel [12]).

L E M M A 2

For (s, p, c) ~ (]R+) 2 x ~+, (i) (1 - p)n-l(s, sp) +/9>0. (ii) For A(s, p, c) = (1 - p)B-l(s, sp) + ( p - pC+l) wehave

�9 A ( s , p , c )>(= ,<)Oc~ ,p<(= ,>) l . | A(s,p, c) + pc+l >0.

(iii) The continuous function, given by

1 - pC+l f (p ,c) pC(1 - p) ' p r I ,

is nonincreasing in p, for fixed c.

Proof (i) This result is equivalent to B(s, sp) > 1 - 1/p which in turn is an immediate

consequence o f l emma 1 (iv). (ii) Not ing that B -1 (s, sp) - 1 > O, V(s, p), by lemma 1 (iii), and writing

A(s,p,c) = (1 - p)[B-l(s, sp) - 1] + (l - pC+l),

2 9 4 A. Paeheco / Loss probability in M / M / s / s + e systems

the first part of (ii) can be easily verified. The second part of the same result is true by (i), since

A(s,p,c) + pC+1 = (1 - p)B-l(s, sp) + p.

(iii) We note first that

0 1 - pC+l - c + (c + 1)p - pc+l

Op pC(1 - p) pc+l(1 _ p)2

here, since the denominator of the last expression is positive for p r 1 and the numerator is a concave function of p which is maximized for p = 1 with the maximum value equal to 0, the result follows if p r 1. If p = 1, the result follows using L'H6pital's rule. []

3. Monotonic i ty properties of the GEL function

In this section we study the monotonicity properties of the GEL function. Lemma 3 is based on equation (4), which relates the GEL function with the Erlang loss function, on lemma 1, which describes the monotonicity properties of the Erlang loss function, and on lemma 2.

L E M M A 3

For (s, a, p, c) ~ (N+) 3 x N +, B(s, a, c) and B(s, sp, c) have the following properties: (i) B(s, a, c) increases with a; B(s, a, c) ~o+ O; B(s, a, c) a-Z-~ ~ 1. (ii) B(s, a, e) decreases with" s; B(s, a, c) s--,o+~ 1; B(s, a, e) s--,+~ O.

�9 p--+O + . . p - - + + e ~ (iii)B(s, sp, c) increases with p; B(s, sp, c) ---?. O; B(s, sp, c) ~ 1.

. - ~ 1 , ~ / ~ s"~U~" p C ( l - p ) (iv) B(s, sp, c) aecreases wires; •[s, sp, c) ---+ ~ ;

s-++~o f 0 p < l , B(s, sp, c) [ 1-1/p p> l.

(iv) B(s, sp, c) decreases with c;

~+~o f 0 p < l , B(s, sp, c)

L 1 - 1 / p 1.

Remarks For fixed traffic intensity p we can decrease B(s, sp, c), the fraction of lost custo-

mers, either by increasing the number of servers s or by increasing the queue capacity c; in the limit (large number of servers or big queue capacity) we get similar losses (close to 1 - l /p, ifp~> 1, and to O, if p< 1). Similarly, if we keep the offered

A. Pacheco / Lossprobability in M/ M/s/s + c systems 295

load constant, we can decrease the fraction of lost customers, B(s, a, c), either by increasing the number of servers or the queue capacity; in both cases the fraction of lost customers will go to 0.

The above observations explain in part why it can be important to play with the queue capacity in order to reduce the loss probability. As we see, the queue capacity competes in economic terms with the number of servers to reduce the fraction of lost customers when keeping the load (or traffic intensity) constant. This is a very important practical reason to use the GEL function instead of the classical Erlang's loss function.

Proof (i) and (iii) These two results are equivalent, as can be easily verified. To prove

(iii), we write B(s, sp, c) in the following form:

Note that:

B(s, sp, c)= fB- I ( spp) - I 1 - pc+l.] -1

c(-i _ p ) / " (5 )

| B -1 (s, sp) - 1 is nonnegative and decreases with p, by lemma 1 (iii), | ( 1 - pr ~[pC (1 - p)] is nonincreasing in p, by lemma 2(iii); this and eq. (5) imply that B(s, sp, c) increases with p. Similarly, the limit results in (iii) follow from eq. (5) and the limit results in lemma 1 (iii).

(iv) This is a consequence ofeq. (5) and lemma 1 (iv). (ii) For 0 < Sl < s2, we have

( a ) ( a ) ( a ) B(s~,a,c)=B Sl,Sl--,Csl >B s2,sa--,csl >B Sa, S2~,c =B(s2,a,c), (6)

where the first inequality follows from (iv) and the second inequality from (iii). From (6), it follows that B(s, a, c) decreases with s. The limit results in (ii) follow directly from lemma 1 (ii) and eq. (4).

(v) We have

pC(1 - p) _pC(1 - p) g(s, sp, c) = (1 - p)g-l(s , sp) q- ( p - p c + l ) A(s,p ,c) ' (7 )

with A (s, p, c) as in lemma 2. This gives

0B(s, sp, c) _ pC(1 - p) ln p[A(s, p, c) + pC+Z] Oc AE(s, p, c) <0, (8)

where the last inequality follows from lemma 2(ii) (and L'H6pital's rule, if p = 1). From the inequality in (8), we conclude that B(s, sp, c) is (strictly) decreasing in c. The limit result in (v) follows from eq. (5), using L'H6pital's rule. []

As can be seen from the previous proof, the fact that B(s, a, c) decreases with s is a consequence that B(s, a, c) increases with a and B(s, sp, c) decreases with s.

296 A. Pacheeo / Loss probability in M / M / s / s + e systems

For M / M / s / s + c systems in steady state, this means that the statement that the loss probability decreases with the number of servers, for fixed traffic intensity, implies that the same loss probability decreases with the number of servers, for fixed offered load.

This last result is intuitively true since: if we add more servers while keeping the traffic intensity fixed we keep the maximum service rate constant (we just distribute the load by more servers), while if we keep the offered load fixed as we acid more servers the maximum service rate turns out to be proportional to the number of servers (we add more identical servers).

4. Second-order properties o f the G E L funct ion

In this section we study second-order properties of the GEL function. Our first result concerns the convexity of the GEL function B(s, a, c) in the queue capacity c, for fixed integer number of servers s and fixed offered load a. As was mentioned in the introduction, the convexity in the integers of the GEL function in the queue capacity was proved by Shanthikumar and Yao [29, corollary 1]; our result extends that result to nonnegative real numbers.

THEOREM 1 B(s, a, c) and B(s, sp, c) are convex in c, for fixed s ~ 1~, a, p ~ ~+.

Proof It suffices to prove the result for B(s, sp, c). Using eq. (8) we get, after some

easy steps,

O~czB(s, sp, c ) pC(lap)Z(1 - p) A---~(s,-~,~ [A(s,p,c) + pc+l][A(s,p,c) + 2pC+l] > 0 ,

where the last inequality follows from lemma 2(ii) (and L'H6pital's rule, ifp = 1). []

We next study the second order properties of the GEL function with respect to the traffic intensity (or, equivalently, with respect to the offered load). Assume s ~ N; we recall that in Harel [12] it is shown that the Erlang loss function is convex in the traffic intensity p if p is below some p* and concave if p is above p*. We consider what happens if c ~ 1~10.

To start with, we write

(sp)* p~ s! = r B(s, sp, c) = s i �9 c s+c ' (9) , - , (sp) (sp) "--" d

2 _ . , T + - 7 - . F_, i=0 i=1 i=0

A. Pacheco / Lossprobability in M / M / s / s + c systems 297

where

sill! i = O, 1 , . . . , s - 1, ~ ( s ) = s~/s ! i > s .

From this, the first derivative ofB(s, sp, c) with respect to p can be expressed by

0 Ots+c(g)ps+c-l[~Si+=~ Oq(S)(S q- C -- i)p i] _ _ lOpB~s, sp, c) = r~,s+~ ~ "s" i~2

[2..M=0 it )P J

For fixed s and c define the polynomial Ps, c (P) as follows:

e,,c(p) := ~ ~(~,~/p k := ~ ~k,,(~,cl pk + k=0 k=0 k=s+c

where

(10)

(11)

s+c ] i=k-s-c

(12)

~k.,(~, c) := ~ , (~)~k- , ( , ) (~ + c - i)(~ + c - 1 + 3 i - 2k) . (13)

After some straightforward calculations, it can be shown that the second derivative of B(s, sp, c) with respect to p is the product of a positive function by Ps, c(P), namely,

Oa B(s, sp, c) = as+c(s)P~+~-2 (14) Opz s+c i 3 Ps, c(P) ; [E,=0 ~,(s)p ]

thus, the sign of0aB(s, sp, c)/Op 2 is simply the sign ofPs,~(p). We now consider the sign of the coefficients of Ps,c (p) with a view to determining

the sign of P~,~ (p) itself. The next two lemmas contain relevant results.

L E M M A 4

For any s ~ N, c ~ No, we have (i) F o r k = 0 , 1 , . . . , s + c - 1,

k

~ ( s + c - i)(s + c - 1 + 3 i - 2k) --- (k + 1)(s + c)(s + c - 1 - k) . i=0

(ii) For s + c ~< k ~< 2s + 2c,

s+c

Z ( s + c - i ) ( s + c - l + 3 i - 2 k ) = - ( 2 s + 2 c - k ) ( 2 s + 2 c . + l - k ) . i=k-s-c

Proof (i) F o r k = 0 , 1 , . . . , s + c - 1,

298 A. Pacheco / Loss probab ility in M / M / s / s + c systems

k y~(s + c - i)(s + c - 1 + 3 i - 2k) i=0

k = Z [ ( s + c)(s + c - 1 - 2k) + i(2s + 2c + 2k + 1) - 3i21

i=0

= ( k + 1 ) { ( s + c ) ( s + c - 1 - 2 k ) + k [ ( 2 s + 2 c + 2 k + 1) - ( 2 k + 1)1 )

= ( k + 1 ) ( s+ c)(s+ c - 1 - k).

(ii) Fo r s + c<~k<~2s + 2c, s+c 2s+2c-k

Z ( s + c - i ) ( s + c - l + 3 i - 2 k ) = Z j ( 4 s + 4 c - l - 3 j - 2 k ) i=k-s-c j=0

2s+2c-k 2s+2c-k

= ( 4 s + 4 c - 1 - 2 k ) ~ j - 3 y~ j2 j=O j=O

= (2s + 2c - k)(2s + 2c + 1 - k) (4s + 4c - 1 - 2k) - (4s + 4c + 1 - 2k) 2

= - ( 2 s + 2c - k ) ( 2 s + 2c + 1 - k ) . D

LEMMA 5

For any s ~ N, c e No, with % (s, c) defined in eqs. (12) and (13), we have:

(i) "yo(S,C)=(s+c)(s+c- 1)/>0. (ii) 7~(s,c)>>.O,k= 0 , 1 , . . . , s + c - I. (iii)%(s, c) = -a~(s)2(2s + 2c - k)(2s + 2c + 1 - k) <0, 2s + c - 1 <<.k <<. 2s + 2c - 1. (iv) '~2~+2c(S, c) = 0.

(v) I f 7k(S, c) changes sign at mos t once for k e {s + c, s + c + 1 , . . . , 2s + c - 1}, then 3p~,~/> 0 s.t. 02B(s, sp, c)/Op 2 > ( <)0 r p < ( > )p~,~.

Proof (i) This is a direct consequence of a0 (s) = 1, Vs, and the definit ion of 7k(S, c) in

eq. (12). (ii) It is useful to express Ps,c(P) in another form. By taking the/3k,i(s, c) terms at

pairs f rom bo th ends of the summat ion defining the coefficients 7k(S, c) of the poly-

nomial Ps,c(P), in eq. (12), we get

es, c(p)= Y~ %(s,c)P~= Z tUk,i(s,c) pk+ Z q/k,i(s,c) pk, k=0 k=0 i=0 k=s+c Li=k-s-c

(i5)

A. Paeheco / Lossprobability in M / M / s / s + c systems 299

where

with

k,i(s, c) A,;(s, c) + c) = 2 - �89 c), (16)

g~,i(s,c)= [2(s + e)(s + c - 1 ) - k(2s + 2 c - l + k) + 6 i ( k - i)], i = 0 , 1 , . . . , k .

(17)

It is easily seen that, for i = 0, 1 , . . . , k,

�9 gk, i(s, c) = [2(s + c)(s + e - 1) - k(Zs + 2c - 1 + k) + 6i(k - i)] decreases as

li - k/21 increases.

�9 ai(S)ak-i(s) is positive and decreases as li - k/21 increases. Therefore, we have for k = 0, 1 , . . . , s + c - 1,

k k

Z g k , i(s, c)>10 =~ 71c(s, c) = �89 ~ ai(s)a~-i(s)gk,i(s, c)>t0. i=0 i=0

The result (ii) now follows since, using lemma 4(i), for k = 0, 1 , . . . , s + c - 1,

k k

Z g k , i ( s , c ) = 2 y ~ ( s + c - i)(s + c - 1 + 3 i - 2k) i=O i=0

= 2 ( k + 1 ) ( s + c ) ( s + c - 1 -k )>~O.

(iii) and (iv) Note that ak(s) = as(s), for k>>.s - 1. Using lemma 4(ii) we have, for k = 2s + c - 1,2s + c , . . . , 2s + 2c,

s+c

7k(s, c) = ~ ai(s)ak_i(s)(s + c - i)(s + c - 1 + 3i - 2k) i=k-s-c

= -as(s)2(2s + 2c - k)(2s + 2c + 1 - k) ,

and the required results follow directly.

(v) Note that if 7~(s, c) changes sign at most once for k E{s + c, s + c + 1, . . . , 2s + c - 1} then, using (i)-(iv), we conclude that 7k(s, c) changes sign at most once (in all its range: {0, 1, . . . , 2s+2c}) . This and eq. (14) imply, since %(2s + 2c - 1) < 0 and%(2s + 2c) = 0, that

�9 >~ ~ * 3Ps,c~.O, such that: 0f2 B(s, sp, c) > ( < )O r P< ( > )Ps,c " []

Lemma 5(v) gives a sufficient condition for the steady state loss probabili ty in an M / M / s / s + c system to be convex in the traffic intensity p i fp is below some Ps, c and concave if p is greater than p*c. In order to check the validity of that sufficient

300 A. Pacheco / Lossprobability in M / M / s / s + c systems

condit ion s coefficients of the polynomial Ps,~(p) have to be evaluated; this is a very simple task (at least using a computer program).

Theorem 2 relies heavily on lemma 5; it essentially states that, for a broad range of values of s and c, B(s, sp, c) as a function of the traffic intensity p is convex if p is below some P*s,c and concave ifp is greater than Ps,~.

THEOREM 2

For s e n and ceN0, if s e {1,2, 3} or (4<~s~< 100, 0 <~c~<2000), then the steady state loss probability in an M / M / s / s + c system, as a function of the traffic intensity p, is convex ifp is below some P*s,c( t> 0) and concave ifp is greater than Ps, c.

Proof Suppose s e N and c e No, with s e { 1,2, 3} or (4 ~< s ~< 100, 0 ~< c ~< 2000), and let

As,c := {s+ c , s + c + 1 , . . . , 2 s + c - 1}.

Using lemma 5(v), to prove the desired property we just need to show that

%(s, c) changes sign at most once for keA, ,c . (18)

Now, since AI,~ --- {c + 1} and A2,c = {c + 2, c + 3}, condition (18) is trivially satisfied i f s = 1 or s = 2. I f s = 3 then A3,c = {c + 3, c + 4, c + 5} and, since by lemma 5(iv) 7~+5 (3, c) < 0, condition (18) is obviously satisfied if %+4(3, c) < 0. In order to show that 7c+4(3, C) <0, note that a0(3) = 1, Oq(3) = 3 and cxi(3) = 9/2, for i~>2. Now, using eqs. (12) and (13) and lemma 4(ii),

c+3 %+4(3, c) = ~ ai(3)ac+4-i(3)(c + 3 - i ) ( 3 i - c - 6)

i=1 c+3

= c~2(3) S [ ( c + 3 - i ) ( 3 i - c - 6)1 i=1

= - (9)2(c + 2)(c + 3) + ~ 9(c + 2)(c + 3)

_ 27 3 ) < 0 . - - : ( c + 2)(c +

Thus, the claim is true ifs = 3. For (4 <~ s ~< 100, 0 ~< c <~ 2000), condition (18) was checked, by a computer pro-

gram, to be true for all values ofs and c, thus implying the validity of the claim for (4~s~< 100, 0 ~< c ~<2000). []

We investigate next some interesting properties of P~,c values. As it is well known, the Erlang loss function is concave for the one-server case (proposition 2(a) in Harel [12]); this special feature is lost when we go to the GEL function. More-

A. Pacheco / Lossprobability in M/ M/s /s + c systems 301

over, the one-server loss system is the only M / M / s / s + c system for which the steady state loss probability is concave in the traffic intensity in all its range. These facts and more are stated in the next theorem.

T H E O R E M 3

Assume that s e N, c ~ No and let

p~,~ -- inf p > 0 s . t . - -~B(s , sp, c) <0 .

Note that, with s and c as in theorem 2 (in particular for s = 1, 2, 3),

o 2 op2 B(s, sp, c) > (<)0 ,~ p < ( > )pL.

We have: (i) p*c = O C:~ [s = l A c = O].

* < c < ( = , >)3. (ii) Pa,c (=, >)1 r (iii)p~,~ < (>)1 r c<(>~)3.

(19)

(20)

Proo f (i) Note that, if [s = 1 A c = 0] then B(1,p,0) is a concave function of p since

P1,0 (p) is negative Vp > 0 because, by results (i), (iii) and (iv) oflemma 5,

7o(1,c) = 72(1,c) = 0>71(1,c) ;

thus, P~,0 = 0. Moreover, if (s, c) # (1,0) then 7o(s, c) = (s + c)(s + c - 1) >0 and this implies that 02B(s, sp, c)/Op2>O for sufficiently small p>0; thus Ps, c >0 if (s, c) # (1,0). This concludes the proof of(i).

(ii) Since O~ i ( 1 ) = 1, Vi, using eqs. (12) and (13) and lemma 4(i)-(ii),

2c+2 c 2c+2

7k(1,c) = Z ( k + 1)(c+ 1 ) ( c - k ) - Z ( 2 c + 2 - k ) ( 2 c + 3 - k ) k=0 k=0 k=c+l

c c c+l

= c ( c + 1) ~ ( k + 1 ) - ( c + 1) Z ( k + 1 ) k - ~"~fl(j + 1) k=0 k=0 j=0

= �89 1)2(c+2) - l c ( c + 1 ) 2 ( c + 2 ) - � 8 9 1 ) ( c + 2 ) ( c + 3)

= ~(c+ 1)(c+2)2(c - 3).

Thus,

2c+2

pb<(=, >)1 r Z 7~(1,c)<(=, >)0 ~ c<(=, >13. k=0

(iii) Since a0(2) = 1 and ai(2) = 2,Vi~>l, using eqs. (12) and (13) and lemma 4(i)-(ii),


c+l c+1 k

Z % ( 2 ' c ) = 4 Z Z (c + 2 - i)(c+ 1 + 3 i - 2k ) - 3(c+2)(c+ 1) k=O k=O i=0

c+l c+l

- 2 Z ( c + 2)(c + 1 - 2k) - 2 ~ ( c + 2 - k)(c + 1 + k) k=l k=l

c+1 c+l

---4 y ~ ( c + 2)(k + 1)(c + 1 - k ) - 2(c + 2) Z ( c + 1 - 2k) k=O k=O

e+l

- ( c + 2 ) ( c + 1) - 2(c+ 1)2(c+2) + 2 Z k ( k - 1) k=l

c+l c+l

=4(c + l)(c + 2 / ~-'~(k + 1 ) - 4 ( c + 2) Z ( k + ilk k=O k=O

- ( c + 2 ) ( c + 1) - 2 ( c + 1)2(c+2) +~c(c+ 1 ) ( c + 2 )

= (c + l)(c + 2) [6(c + 2)(c + 3) - 4(c + 2)(c + 3) - 3 - 6(c + 1) + 2c] 3

and 2c+4

k=c+2

therefore

_,c(+2) (c+ 1)(2c 2 + 6 c + 3), 3

2c+4 c+2

7 k ( S , C ) = 4 ~ Z ( C + 2 i ) ( c + 1 + 3 i - 2 k ) - 2 ( c + 2 ) [ c + 1 - 2 ( c + 2 ) ] k=c+2 i=k-c-2

2c+4

= -4 ~ (2c + 4 - k)(2c + 5 - k) + 2(c + 2)(c + 3) k=c+2

c+2

= -4 Z j ( j + 1) + 2(c + 2)(c + 3) = j=O

(c + 2)(c + 3)[-4(c + 4) + 6] 3

_ 2 ! e l 2 ) ( c + 3 ) (2c+ 5),

2c+4

k=O

(c+ 2)[(c + 1)(2c 2 + 6c + 3 ) - 2(c + 3)(2c + 5)] 3

(c + 2) (2c3 + 4c 2 _ 13c - 27). 3

A. Paeheeo / Loss probability in M / M / s / s + e systems 303

Table 1 Zeros of GEL's second derivative with respect to p.

No. of Queue capacity servers

0 1 2 3 4 5 10 50 200

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 35 40 45 50 60 70 80 90

100 150 200 250 300 400 500

0.00000 0.53209 0.84268 1.00000 1.08235 1.12694 1.17379 1.08496 1.03042 0.36603 0.71631 0.92644 1.04061 1.10296 1.13748 1.17251 1.08440 1.03036 0.55200 0.80951 0.97066 1.06229 1.11376 1.14271 1.17117 1.08397 1.03032 0.66321 0.86695 0.99865 1.07602 1.12043 1.14570 1.16987 1.08360 1.03028 0.73756 0.90628 1.01813 1.08553 1.12490 1.14751 1.16863 1.08329 1.03024 0.79097 0.93504 1.03253 1.09249 1.12803 1.14862 1.16744 1.08300 1.03021 0.83127 0.95705 1.04361 1.09779 1.13030 1.14929 1.16631 1.08274 1.03018 0.86280 9.97446 1.05241 1.10193 1.13198 1.14965 1.16524 1.08249 1.03016 0.88817 0.98858 1.05956 1.10524 1.13323 1.14980 1.16421 1.08227 1.03013 0.90902 1.00027 1.06547 1.10792 1.13417 1.14979 1.16322 1.08206 1.03011 0.92647 1.01010 1.07044 1.11012 1.13487 1.14968 1.16227 1.08185 1.03008 0.94128 1.01848 1.07466 1.11195 1.13539 1.14949 1.16136 1.08166 1.03006 0.95401 1.02570 1.07828 1.11349 1.13577 1.14923 1.16049 1.08148 1.03004 0.96506 1.03199 1.08142 1.11478 1.13603 1.14893 1.15964 1.08130 1.03002 0.97475 1.03750 1.08416 1.11587 1.13620 1.14860 1.15883 1.08114 1.03000 0.98330 1.04238 1.08656 1.11680 1.13629 1.14823 1.15804 1.08097 1.02999 0.99090 1.04672 1.08869 1.11759 1.13633 1.14785 1.15728 1.08082 1.02997 0.99771 1.05060 1.09058 1.11827 1.13631 1.14745 1.15654 1.08067 1.02995 1.00382 1.05408 1.09226 1.11885 1.13625 1.14704 1.15582 1.08052 1.02993 1.00935 1.05723 1.09377 1.11934 1.13616 1.14663 1.15513 1.08038 1.02992 1.01894 1.06269 1.09634 1.12012 1.13590 1.14578 1.15380 1.08011 1.02989 1.02697 1.06724 1.09844 1.12068 1.13555 1.14493 1.15253 1.07985 1.02986 1.03376 1.07107 1.10017 1.12107 1.13515 1.14408 1.15133 1.07960 1.02983 1.03958 1.07434 1.10161 1.12133 1.13470 1.14324 1.15019 1.07937 1.02980 1.04461 1.07714 1.10280 1.12148 1.13423 1.14241 1.14909 1.07914 1.02978 1.05459 1.08263 1.10502 1.12154 1.13298 1.14041 1.14656 1.07862 1.02971 1.06193 1.08658 1.10645 1.12129 1.13169 1.13852 1.14426 1.07813 1.02966 1.06751 1.08950 1.10737 1.12085 1.13040 1.13674 1.14216 1.07769 1.02960 1.07184 1.09169 1.10793 1.12030 1.12914 1.13507 1.14023 1.07727 1.02955 1.07801 1.09462 1.10839 1.11902 1.12674 1.13200 1.13677 1.07651 1.02946 1.08206 1.09635 1.10829 1.11764 1.12452 1.12926 1.13373 1.07583 1.02938 1.08480 1.09734 1.10790 1.11625 1.12246 1.12680 1.13104 1.07521 1.02930 1.08670 1.09786 1.10734 1.11488 1.12055 1.12456 1.12861 1.07463 1.02923 1.08802 1.09808 1.10667 1.11356 1.11879 1.12251 1.12641 1.07410 1.02916 1.09031 1.09705 1.10292 1.10777 1.11155 1.11435 1.11771 1.07188 1.02887 1.08983 1.09490 1.09937 1.10312 1.10612 1.10838 1.11140 1.07014 1.02863 1.08856 1.09262 1.09624 1.09931 1.10180 1.10371 1.10648 1.06869 1.02842 1,08708 1.09046 1.09350 1.09610 1.09824 1.09990 1.10247 1.06744 1.02824 1.08409 1.08663 1.08893 1.09094 1.09261 1.09394 1.09620 1.06537 1.02793 1.08137 1.08340 1.08526 1.08689 1.08827 1.08938 1.09142 1.06368 1.02766


Table 2 Zeros of GEL's second derivative with respect to p.

Queue No. ofservers capadty

1 2 3 5 10 15 20 50 200

0 I 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 3O 35 40 45 50 60 70 80 90

100 150 200 250 300 400 500

0.00000 0.36603 0.55200 0.73756 0.90902 0.97475 1.00935 1.07184 1.08983 0.53209 0.71631 0.80951 0.90628 1.00027 1.03750 1.05723 1.09169 1.09490 0.84268 0.92644 0.97066 1.01813 1.06547 1.08416 1.09377 t.10793 i.09937 1.00000 1.04061 1.06229 1.08553 1.10792 1.11587 1.11934 1.12030 1.10312 1.08235 1.10296 1.11376 1.12490 1.13417 1.13620 1.13616 1.12914 1.10612 1.12694 1.13748 1.14271 1.14751 1.14979 1.14860 1.14663 1.13507 1.10838 1.15139 1.15654 1.15874 1.16011 1.15860 1.15568 1.15268 1.13872 1.10998 1.16454 1.16665 1.16717 1.16662 1.16305 1.15923 1.15573 1.14066 1.11099 1.17109 1.17145 1.17099 1.16936 1.16470 1.16046 1.15675 1.14134 1.11151 1.17366 1.17298 1.17196 1.16974 1.16454 1.16015 1.15640 1~14111 1.11162 1.17379 1.17251 1.17117 1.16863 1.16322 1.15883 1.15513 1.14023 1.11140 1.17242 1.17078 1.16927 1.16658 1.16115 1.15685 1.15326 1.13888 1.11091 1.17011 1.16827 1.16668 1.16395 1.15860 1.15444 1.15099 1.13721 1.11020 1.16723 1.16530 1.16368 1.16097 1.15578 1.15178 1.14848 1.13532 1.10933 1.16402 1.16206 1.16045 1.15780 1.15279 1.14897 1.14582 1.13328 t.10833 1.16064 1.15869 1.15711 1.15454 1.14973 1.14609 1.14309 1.13116 1.10723 1.15719 1.15527 1.15374 1.15125 1.14665 1.14318 1.14033 1.12899 1.10606 1.15373 1.15186 1.15038 1.14799 1.14360 1.14030 1.13759 1.12680 1.10483 1.15031 1.14850 1.14707 1.14479 1.14060 1.13746 1.13488 1.12462 1.10357 1.14697 1.14522 1.14385 1.14166 1.13766 1.13468 1.13223 1.12245 1.10229 1.14370 1.14202 1.14071 1.13862 1.13481 1.13196 1.12963 1.12032 1.10100 1.13749 1.13593 1.13473 1.13282 1.12935 1.12677 1.12465 1.11619 1.09841 1.13169 1.13026 1.12916 1.12741 1.12425 1.12190 1.11997 1.11224 1.09585 1.12632 1.12501 1.12399 1.12239 1.11950 1.11735 1.11559 1.10851 1.09335 1.12135 1.12014 1.11920 1.11773 1.11508 1,11311 1.11150 1.10499 1.09093 1.11674 1.11563 1.11477 1.11341 1.11097 1.10916 1.10767 1.10167 1.08860 1.10664 1.10572 1.10501 1.10390 1.10189 1.10040 1.09917 1.09420 1,08318 1.09821 1.09743 1.09684 1.09591 1.09423 1.09298 1.09195 1.08775 1~07834 1.09107 1.09042 1.08991 1.08912 1.08769 1.08663 1.08575 1.08216 1.07401 1.08496 1.08440 1.08397 1.08329 1.08206 1.08114 1.08038 1.07727 1.07014 1.07506 1.07463 1.07430 1.07378 1.07283 1.07213 1.07155 1.06914 1.06353 1.06736 1.06702 1.06676 1.06635 1.06560 1.06504 1.06458 1.06266 1.05811 1.06120 1.06093 1.06071 1.06038 1.05977 1.05932 1.05894 1.05737 1.05360 1.05615 1.05593 1.05575 1.05547 1.05497 1.05459 1.05427 1.05296 1.04978 1.05193 1.05174 1.05159 1.05136 1.05093 1.05061 1.05034 1.04923 1.04651 1.03813 1.03803 1.03796 1.03784 1.03762 1.03745 1.03731 1.03673 1.03526 1.03042 1.03036 1.03032 1.03024 1.03011 1.03000 1.02992 1,02955 1.02863 1.02545 1.02541 1.02538 1.02533 1.02524 1.02517 1.02511 1.02486 1.02422 1.02196 1.02193 1.02191 1.02187 1.02180 1.02175 1.02171 1.02153 1.02105 1.01735 1.01733 1.01732 1.01730 1.01726 1.01723 1.01720 1.01709 1.01680 1.01442 1.01441 1.01440 1.01438 1.01436 1.01434 1.01432 1.01424 1.01404

A. Pacheco / Lossprobabiligv in M / M / s / s + e systems 305

Now, for c e No, we have

2c+4

p~,~ < (>)1 <=> ~ %(2, c )< (> )0 r 2c3+ 4c2 -13c -27 < ( > )0r <(I> )3. k=0

[]

5. Numerical illustrations

In this section we consider P*,c as defined in eq. (19) and we assume (conjecture) that eq. (20) holds for any number of servers s and queue capacity c.

Tables 1 and 2 give the values p~,c for a broad range of number of servers s and queue capacity c, spaced appropriately with a view to giving a good idea as to how P~,c changes as we change the number of servers (table 1) and the queue capacity (table 2).

All values of p~,~, which are the zeros of the second derivative of the steady state loss probability in M / M / s / s + c systems with respect to the traffic intensity, were computed based on eqs. (4) and (2) and using the NAG package for numerical inte- gration.

From tables 1 and 2 we can see that p~,~-+ 1, as s-+ + c~ or c-+ + oo; this was to be expected from lemma 3(iv) and (v) since, as s--+ + oo or c - + + oo, B ( s , sp , c)

as a function ofp tends to a function which is null for p < 1 and concave for p ~> 1. Suppose s ~ N and c ~ No. From the analysis of tables 1 and 2 it seems that the

values p~,~ described in theorem 2 are, for fixed number of servers (queue capacity), increasing in the queue capacity (number of servers) if the queue capacity (the number of servers) is less than some cs (Sc), and decreasing if the queue capacity (the number of servers) is greater than cs (Sc). Given s ~ N and c ~ No, it makes sense to consider

�9 { , } cs : &,cs = m a x Ps,c : C ~ No , (21)

�9 { , } Sc : &c,c = m a x &,c : S S N . (22)

Table 3 Greatest zeros of GEL's second derivative with respect to p.

Queue Maximum value Associated Queue Maximum value Associated capacity e of the root P~,,c # servers sc capacity c of the root p~o,c 4/: servers s~

0 1.090335 158 5 1.149795 9 1 1.098107 105 6 1.160119 6 2 1.108402 63 7 1.167167 3 3 1.121564 33 8 1.171446 2 4 1.136326 17 9 1.173659 1

306 A. Pacheco / Loss probability in M / M / s / s + e systems

Table 4 Greatest zeros of GEL's second derivative with respect to p.

Numberof Maximumvalue Associated Numberof Maximumvalue serverss oftherootp~,~ queuecap.cs serverss oftherootp~,~

1 1.173794 10 22 1.155390 8 2 1.172984 9 24 1.154097 8 3 1.171965 9 26 1.152864 8 4 1.170857 9 28 1A51685 8 5 1.169737 9 30 1.150556 8 6 1.168635 9 35 1.147929 8 7 1.167562 9 40 1.145543 8 8 1.166544 8 45 1.143357 8 9 1.165614 8 50 1.141343 8

10 1.164701 8 60 1.137735 8 11 1.163809 8 70 1.134579 8 12 1.162939 8 80 1.131775 8 13 1.162091 8 90 t,129257 8 14 1.161265 8 100 1,126973 8 15 1.160461 8 150 1,118044 9 16 1.159679 8 200 1.111622 9 17 1.158917 8 250 1.106631 9 18 1.158175 8 300 1.102571 9 19 1.157452 8 400 1.096238 9 20 1.156747 8 500 1.091424 10

Associated queue cap. c~

Table 3 gives the values se and P~c,c for fixed integer queue capacity c. Similarly, table 4 gives the values cs and P~,c, for fixed integer number of servers s. It is evident from table 4 that p~:, is decreasing in the number of servers, and from table 3 that sc is decreasing in the queue capacity.

Finally, table 5 presents for fixed number of servers (queue capacity) the smallest c (s) for which p*: ~> 1.

Table 5 Smallest number of servers and queue capacity for which the zeros of GEL's second derivative with respect to p are greater than or equal to 1.

Number of min {c eN0 ". PJ,c* ~'>-1} Queue servers s capacity c

ra in{seN" * >~1}

l~<s~<4 3 0 19 5<<.s<~9 2 1 10 lO~<s~<18 1 2 5 s>~19 0 c~>3 1

A. Pacheco / Loss probability in M / M / s / s + c systems 307

Acknowledgements

The a u t ho r wishes to t h a n k Professor M.F . R a m a l h o t o for her e n c o u r a g e me n t a n d for f rui t ful shar ing o f ideas. M a n y thanks are also due to Professors N . U . P rabhu , A. Desc loux and Dr. Arie Hare l for helpful assistance a n d to an anon- y m o u s referee for commen t s tha t resulted in an improved presenta t ion .

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Documents

Second-order properties of the loss probability inM/M/s/s +c systems