On the Quasi-Stationary Distributions of the GI/M/1 QueueAuthor(s): E. K. KyprianouSource: Journal of Applied Probability, Vol. 9, No. 1 (Mar., 1972), pp. 117-128Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/3212641Accessed: 03/06/2010 21:40

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J. Appl. Prob. 9, 117-128 (1972) frinted in Israel

? Applied Probability Trust 1972

ON THE QUASI-STATIONARY DISTRIBUTIONS OF THE GI/M/1 QUEUE

E. K. KYPRIANOU, University of Manchester

Abstract This paper studies the existence, in a stable GI/M/1 queue, of the limit as

t -- oo of the distribution of the virtual waiting time process at time t condi- tioned on the event that at no time in the interval [0, t] the queue has become empty. The conditional limit distribution obtained when the traffic intensity is strictly less than one is the weighted sum of an exponential and a gamma distribution. Similar conditional limit distributions are obtained for the queue size process and the waiting time process as defined by Prabhu (1964). GI/MI1 QUEUE; DUAL M/G/1 QUEUE; PRABHU'S WAITING TIME PROCESS; VIRTUAL WAITING TIME PROCESS; QUEUE SIZE; BUSY PERIOD; LAPLACE-

STIELTJES TRANSFORMS OF DISTRIBUTIONS; ASYMPTOTIC EXPANSIONS OF DIS-

TRIBUTION FUNCTIONS; QUASI-STATIONARY DISTRIBUTIONS

1. Introduction

It is well known that when the traffic intensity p of a queueing system is less

than, or equal to, the value one, the first busy period is finite with probability one. Yet such a period may be sufficiently long for a process associated with the queue, e.g., the queue size process Q(t), to settle to a state of statistical equilibrium within the first busy period. It would then be of great interest to investigate the limit distribution of such a process conditional on it still being in the first busy period. When this conditional limit distribution exists we shall call it the quasi-stationary distribution of the process. For example, when an insurance company deals with

positive risk sums only, its capital, in the period of prosperity, exhibits the same behaviour as the waiting time process Y(t), defined by Prabhu (1964), of the

GI/M/I queue, whereas if it deals with negative risk sums only, its capital, in the period of prosperity, behaves as the virtual waiting time process of the M/G/1 queue. In both cases the quasi-stationary distribution of the capital reserve not

only becomes of particular importance but is the natural limit distribution to

consider, for one is only interested in the performance of an insurance company before, and not after, bankruptcy.

In [6] we have considered the quasi-stationary distribution of the virtual

waiting time process of the M/G/1 queue when certain conditions are imposed on the service time distribution. The quasi-stationary distribution of the queue size

process of the M/G/1 queue under the same conditions has been considered in [5].

Received 14 May 1971.

117

118 E. K. KYPRIANOU

In the present work we investigate the quasi-stationary distribution of three processes associated with the GI/M/1 queue, namely, the waiting time process Y(t) defined by Prabhu (1964), the queue size process Q(t) and the virtual waiting time process W(t). Explicit results are obtained for these three quasi-stationary distributions.

We consider a GI/M/1 queueing system where the negative exponential service time distribution has parameter A and the inter-arrival times are assumed to have the common distribution dB(t), 0 < t < co. We shall let t(z) denote the Laplace- Stieltjes transform of the inter-arrival distribution, i.e.,

/(z) = e-z'dB(t)

convergent for R(z) > ar ( < 0). With the above notation the traffic intensity p of the GI/M/1 queueing system may be written as

(1) p = - { '(0)

2. The waiting time process Y(t)

For the GI/M/1 queue the virtual waiting time process W(t) is not Markovian as in the case of the M/G/1 queue. As such, its study becomes less manageable so, instead, we first investigate the quasi-stationary behaviour of a new process Y(t) through which we shall be able to obtain the quasi-stationary distribution of both W(t) and Q(t). The process Y(t), to which we shall refer as the 'waiting time process', was first introduced by Prabhu [7] and is defined as follows.

If at time t the queue is empty we take Y(t) = 0, otherwise Y(t) is defined to be the time which has elapsed since the arrival of the customer being served at time t. If at any instant t, Y(t) > 0 then it increases at unit rate as long as the customer receiving service continues to do so. But if t = t,n, where t,, is the epoch of departure of the nth customer, then

Y(t, + ) = max {(Y(t, -) -u,,+ 1,0},

where un,,+ is the interval of time between the arrival of the nth and the (n + 1)th customers. The Y(t) process is defined to be continuous on the left so that Y(t,,) = Y(t,,-). If at any instant t, Y(t) = 0 then it continues to be zero till the next arrival.

Let D(t) denote the number of departures from the queue up to time t. We then define the process X(t) to be

X(t) = ut + 112 + ? + tD(t).

Let also Y(0) = y > 0. If Y(z) > 0 for 0 ? z ? t it is then easy to see that Y(t) can be expressed in the form

On the quasi-stationary distributions of the GI/M/1 queue 119

(2) Y(t)= y + t - X(t).

We now let

(3) S(y) = inf {t Y(t) = 0, Y(O) = y}

denote the busy period initiated by a waiting time Y(O) = y > 0 so that from (2)

(4) S(y) = inf{t I X(t) - t ? y}.

Prabhu [7] has shown that

(5) Pr{S(y)< co} = 1 if pl1,

< 1 if p>1.

In view of (4) investigations in the existence of the quasi-stationary distribution of

Y(t), i.e., in the limiting behaviour of the conditional probability

(6) L(x, t) = Pr{Y(t) ? x S(y) > t),

O < x < oo, y $ 0, are meaningful only in the cases when p ? 1. The conditional

probability L(x, t) can also be expressed as

Pr {Y(t) > x; S(y) > t} (7) L(x, t)= 1 -

Pr (S(y) > t} where in view of (2) and (4)

Pr{Y(t)>x;S(y)>t} = Pr{ inf {y+z-X(z)} >0;y+t-X(t)>x) (8) oj<_t = Pr{ sup {X(z)- z} < y; x + X(t) - t < y}.

Prabhu [7] pointed out that, in view of the fact that the virtual waiting time

process of an M/G/1 queue initiated by a waiting time W(0) = x is given by

W(t) = max ( sup { () - }, x + (t)- t },

where f(t) = v1 + v2 + + VA(t) is the random sum of the service times and

A(t) is the number of arrivals up to time t, the expression on the r.h.s. of (8) gives the probability

Pr{W(t) < y W(0) = x} = F(x; y -, t)

of the dual M/G/1 queue in which the Markovian input has parameter A and the service times have distribution dB(t), 0 < t < f , so that its traffic intensity is given by j = p-'. Therefore

(9) Pr {Y(t)> x; S(y)> t} = F(x; y -,t)

and consequently

120 E. K. KYPRIANOU

(10) Pr {S(y) > t} = Pr {Y(t) > 0; S(y) > t} = F(0; y -, t).

Substituting (9) and (10) in (7) we have that

F(x; y -, t) (11) L(x, t) = 1 F(; y -, t)

F(0; y -, t)

so that the problem of obtaining the quasi-stationary distribution of Y(t) in the

GI/M/1 queue with p ? 1 reduces to one of determining the limit as t -- co of the ratio F(x; y, t) /F(0; y, t) of the virtual waiting time distribution functions of the dual M/G/1 queue with traffic intensity p > 1. However,

lim_,,,F(x; y, t) = 0 for

all x if ?> 1, so, to determine the limit as t -* co of (11), we propose to replace F(x; y -, t) and F(O, y -, t) by their asymptotic expansions before allowing t to go to infinity, The asymptotic expansion of F(x; y, t) for large t is obtained through its double transform R(x; z, w) where

R(x; z, W) = e-wt e-ZdyF(x; y,t) dt 00+

(12) = e-wt-zYdvF(x; y,t)dt - e

- wtF(x; 0, t) dt

= D*(x; z, w) - F*(x; 0, w)

for A(z) > 0 and A(w) > 0. Bene' (1957) showed that

(13) D*(x; z, w) =eX - z((w))

1 e (w w - z + A - kO(z)

and

e xr(w) (14) F*(x; 0, w) =

e(- ,

where c(w) is the unique root of w = z - 1 + 1O(z) in the region R(z) > 0 when /(w) > 0. Inserting (13) and (14) in (12) we have

e-x(w 2(z) - 1 - w e-- x (15) R(x; z, w) = + e ?(w) w - z + . - z() w - z + A - ,O(z) We shall now need the following lemma.

Lemma 1. For t^ > 1 the equation w - z + A - A1d(z) = 0 has the unique root z = 5(w) in the region M(z) > ( when M(w)> 4, where ( is the real non-negative root of 1 + ,W'(z) = 0 and q =r - + 2W(() 0. If 6 = 1 then ( = ~= 0. On the line M(w) = 4, w = 4 is the only singularity of the root 5(w) and this singularity is a branch point of order two. In the region M(w) ? ? and for w $

q

/ _>-

(w)) Ww I -

On the quasi-stationary distributions of the GI/M/1 queue 121

Proof. The results of this lemma are comparable with those of Lemma 1 in

[6]. However, the latter was proved under the condition that O(z) is a meromorphic function over the extended z-plane whilst no condition is imposed on O(z) in this lemma.

Consider the function

(z) = Z - A + 1(Z) defined for R(z) >

r,. Clearly 0(0) = 0 and 0'(0) = 1 + A1/'(0) = 1 - 3 < 0. Further, for z restricted to the real interval or < z < co, 0"(z) = foot2e -ztdB(t) > 0, so that O(z), a, < z < 00, is a convex function passing through the origin and having a negative first derivative there. Since also O(z) -- oo as z -* oo it follows that there always exists just one point z = C ? 0 such that 0'(C) = 0 and

0"(C) > 0. Let k = 4(O). It then follows that given e there exists a 6 such that for

Iw - 4 1< 6 the equation w = O(z) has only two roots inside the region I - I < and the two roots coalesce at w = 0, i.e., w = 0 is a branch point of order two for each of the two roots. Clearly since 0'(O) = 1 -

A = 0 when A = 1 it follows that in this case 0 = 0 and 0 = 0.

Since the constant C is such that 0'(C) = 0 and 0"(C) > 0, it is always possible to find a C6 such that

(16) 5 C < *(w) + - + 1( )

for R(w) > 4. Also choose a sufficiently large M such that M > R(w) - 4. Let z, and z2 be the points of intersection of the line p(z) = C. and the circle C given by w - z + 1 = A(C) + M. Then consider the contour e consisting of the line

R(z) = C, for J(z1) YJ(z) ? J(z2) and the part of the circle C with W(z) > (. On this contour, when W(z) > Ca, we have Iw - z + A = 2/(()

+ M > A1(•)

> 2I A(z) I and when W(z) = we have, in view of (16), that Iw - z +) > I (w)

- ra + A > 2)(C) I A(z) . Therefore the inequality I w - z + I > I . A(z) I holds

over the whole contour e. Hence, by Rouch6's theorem, w - z + A and w - O(z) have the same number of zeros inside W. Since C, can be made arbitrarily close to C and M can be made arbitrarily large it follows that for W(w) > 4 there is only one unique root ?*(w) of w = O(z) in W(z) > C. Since it is already known that when R(w) > 0, w = O(z) has the unique root ?(w) in W(z) > 0 it follows that ?*(w) - ?(w) in a(z) > C. Further, since O(z) is analytic in W(z) > C we have that ?(w) is analytic in R(w) > 4. Also from the first part of the proof it is obvious that w = 4 is a branch point of order two for ?(w).

Consider now the function O(z) as a mapping from the z-plane to the w-plane. Under this mapping any other zero C' of 0'(z) = 0 (C' $ C) with R(5') = C will be transformed onto the point w = 4' which must necessarily be a branch point of 5(w). Since however

(') = - 2 + 1 e- ' cos(J(Y')t)dB(t)

< kb,

122 E. K. KYPRIANOU

such a singularity b' of ?(w) will be situated strictly to the left of in the w-plane. It therefore follows by the continuity of ?(w) that on the line R(w) = , w = / is the only singularity of ?(w).

Finally since ?(w) satisfies the equation w = 0(?(w)) we have that

(17) J(w) = IJ(+((w))) I IJ((w)) I + A((((w))),

and since for R(w) ? (w 0 )), R(?(w)) > C ? 0 we have from (17)

1J(w? fW_))_? I +A5 and this completes the proof.

3. The asymptotic expansion of F(x; y, t)

We are now in a position to obtain, for large t, the asymptotic expansion of the Laplace-Stieltjes transform

f(x; z,t) = e- ZdvF(x; y, t) 0+

of the distribution function F(x; y, t).

Lemma 2. As t -+ o

rqe-Ix(Cx + 1)(0 + A - A)f(z)){2/rCS2(4(z)- )1-'"t-- + O(e)'t-2)

f(x; z, t)= if p <l,

L(2 - AI(z))( 1rn)(z))-'t-- + O(t-') if p = 1

for R(z)> C and 0 < x < 00, where 4 and ( are as defined in Lemma 1. The constants q and i are given in (24) and (29) respectively.

Proof. The asymptotic expansion of f(x; z, t) for large t and for p ? 1 is

given by the behaviour of its Laplace transform R(x; z, w) near its singularity situated furthest to the right of the complex w-plane, which, as can be seen from

(15) and Lemma 1, is the branch point w = 4 of the root ?(w). We first note that for any a > 4

(18) f(x; z,t) = ew(x; z, w) dw.

We also have, by Cauchy's theorem, that

(19) f ew (x; z, w)dw = 0,

since it can easily be deduced using Lemma 1 that ewte(x; z, w) is analytic within the contour C(e,R) of Figure 1.

On the quasi-stationary distributions of the GI/M/I queue 123

S+ iR I a + iR

+iR a-iR

Figure 1 Contour C(e, R)

However

O+iR a

(20)] ewt' (x; z, w)dw <? I (x; z, u + iR) Ieutdu-+ 0 Ja+iR I

as R -+ oo since by Lemma 1 JI (.(s

+ iR)) ? R - A. By similar arguments a-iR

(21) eWt' (x; z, w) dw -+ 0

as R -+ oo. Further, in the case when a > 1, since the singularity at w = 4 is only a branch point, it follows that

(22) lim exp{Obt + tee'0} (x; z, + ee'i)iee'edO = 0. e-+O J r/2

In the case when P = 1, (x; z, e) = o(e-+) so that (22) is valid for 1 = 1 as well. Therefore allowing .

-+ 0 and R -+ 0o in (19) we obtain using (20)-(22) and (18)

(23) f(x; z,'t) - e'a(x; z,4 + iv) dv.

We shall now show that in some small neighbourhood S(6, ) = {w: I w - < 6 of 4 the function a(x; z, w) can be written in the form

124 E. K. KYPRIANOU

(A(w) + (w - 4)BBl(w) if p > 1,

w(x;z,) = A2(w) + w-vB2z(w) if = 1,

where the Ai(w) and Bi(w), i = 1,2, are all analytic functions in S(6, 0), so that the following lemma can be used to obtain the asymptotic expansion of (23) as t - 00o.

Lemma 3. Let g(y) be N + 1 times continuously differentiable for

ye(-(- oo, a - e] u [a + e, oo) and g(y)= h(y)+ (y - c)l/mk(y) for a - e < y < a + e, where e > 0 is arbitrarily small, m - 1,0, 1 is an integer, (y - c)l/m takes its principal value and h(y) and k(y) are N + 1 times continuously differentiable for a - e < y < a + e. Then, provided g(y) - 0 as J y - cc,

eityg(y)dy = GN (t)+ O(t-N-U)

as t -+ oo, where

= - 2sin(m) F(n + 1 + 1/m)in(n-1/m)/2 k(n)()t-n- 1

-meit, n=O n!

k(") is the nth derivative of k and U = 0 if m < - 1 and 1 if m > 1. Lemma 3 is an extension of the results of Erd6lyi ((1956), Section 2.8, pp.

46-51) and its proof is omitted. Now for A > 1 and for w e S(6, 0) the root ?(w) has the expansion (cf. Hille

(1959), p. 265)

(w= + (w - (w),

where (w - 4)+ takes its principal value and the function •(w) is analytic in S(6, 0) with

(24) 1

,

(z - ()d((z) - 0)1/dz 27i -jl=r O(z)

- Here r is taken sufficiently small so that inside the circle I z - = r, 4(z)

- 0 has no zeros apart from z = C. Therefore in the region S(6, 0)

(25) exp { - ?(w)x} = e-~x{cos(ix (w)(w - )-) + (w - O)ip(w)}

and

(26) (w) - ' = '1 {(2(2 _ (w - 4)2(w))-1 + (w - q(w),

where p )

- {xf(w

)}2n+I/

1 n0 w))2n+1 2p(w) =? !(w -

•, -)", q(w) = -

(w--)" so that

On the quasi-stationary distributions of the GI/M/1 queue 125

(27) p(4) = - xq and q(0) -

We note that since in the region S(6, 0), e-("w) and ?(w)-' on the 1.h.s. of (25) and (26) respectively each has the branch point at w = 0 as its only singularity, whereas in the same region cos{ix?(w)(w - 0)+} and {{2 - (w -

_ )2(w)}-1 on

the r.h.s. of (25) and (26) respectively are analytic, it follows that both p(w) and q(w) are convergent and analytic in S(6, /). Therefore from (25) and (26) and for we S(6, 0) we have that

(28) exp { - x•(w)} /((w)) = a(w) + (w - 0)*b(w),

where clearly a(w) and b(w) are analytic in S(6, 4) and in view of (27)

e-gx b(o) = - (?x + 1).

Substituting (28) in (15) we have for w e S(6, 0),

a(x; z, w) = (w - 0(z))- {(01(z) - 2 - w)a(w) + e -zx

+ (w - 0)(A (z) - A - w)b(w)}.

Further, since by Lemma 1 IJ((( + iy))I > I y I - 1, it follows that j(x; z,/ + iy) -+ 0 as I y -+ oo. Therefore j(x; z, 0 + iy) satisfies all the conditions of Lemma 3. Hence applying this lemma to (23) with a = 0, m = 2 and

k(y) = (0 + iy - d(z))-1('1(z)- - - - iy)b(4 + iy)ec/4,

we obtain for the asymptotic form of f(x; z, t)

f(x; z,t) = x (Cx + 1)4(Ao(z) - A- b)(b - O(z)) t-'e't 1 + O(e4'tt-2).

When 1, = 1, = 0 and 0 = 0 so that in the region S(6,0) = {w: w < 6} the root ?(w) has the expansion

?(w) = w0(w), where wI takes its principal value and '(w) is analytic in the region S(6, 0) with

(29) (0) = =i =zd {4(z)}/d(z dz.

Therefore in the region S(6,0)

exp { - x (w)} /((w) = {j(w) + w cos(ixw4((w))} /((w),

where, using arguments similar to those used for the case j> 1, f(w) can be shown to be analytic in S(6,0). Consequently when $ = 1,

126 E. K. KYPRIANOU

j(x; z,w) = {(w)(w- -(z))-(w)((z) - w) + e-x

+ w-i(2f(z) - - w)cos(ixw15(w))}

in S(6, 0), in which case Lemma 3 can be applied to (23) with c = 0, m = - 2 and k(y) = {P(iy)(iy -

L(z))}-l{(2 (z) - 2

- iy) cos(ix(iy)*'(iy))}e -' i/ to

obtain, for large t,

f(x; z, t) =( + O(t ).

This completes the proof of the lemma. From Lemma 2 we see that the asymptotic behaviour of dF(z; y, t), 0 < y < co,

as t -+oo can be obtained from that of f(x; z, t) provided that the function

(O(z) - )-', (z) > represents the Laplace-Stieltjes transform of some

function zO(y), 0 < y < oo. This, in fact, is the case, for in the region M(z)> >

(OWz-0n= /)-=(z-)- 1o z-

where the series on the right is absolutely convergent; further, (1 - /i(z)) /(z - 0) is the Laplace-Stieltjes transform of

do(y) = e4'ydy - dB(y - x)e4xdx, O < y < o0

so that (cf. Doetsch (1958), p. 186)

4,(y) = 1:2" n.,4(y

- x)eexdx, 0

< y < oC n=0 0

where ar,n(y) is the nth convolution of qa(y) with itself. Writing

s,(y) = ( + 4)z(y) - (y - x)dB(x), 0 < y < oo

we have the following lemma as an immediate consequence of Lemma 2.

Lemma 4. With the same notation as in Lemma 2 and for 0 < y < oo and

t -+• o

F(x; y, t) le-(x + 1)(2C2 /'t)-ls,(y)eO'tt-

+ O(e"'t-2)

if p < 1, F(x; y, t) =

(/t)- l'so(y)t-- + O(t- 1) if p = 1.

4. A limit theorem

Theorem. For a GI/M/1 queueing system with mean service time 2-' and whose inter-arrival time distribution has Laplace-Stieltjes transform i(z), if Y(t) is

the time spent in the queue by the customer served at time t, Q(t) is the queue size

On the quasi-stationary distributions of the GI/M/1 queue 127

process, W(t) is the virtual waiting time process and S(y) = inf{t I Y(t) = 0; Y(0) = y > 0} then the limits

(i) L(x) = lim Pr {Y(t)? x S(y)> t}, 0 < x <co t - 00oo

(ii) qj = lim Pr{Q(t) =j IS(y) > t}, j = 1,2,...

(iii) Q(x) = lim Pr{ W(t)? x IS(y) > t}, 0< x<00 t -* o0

exist independently of the initial conditions. (a) If p = 1 then, L(x) = 0, Q(x) = 0, O < x < co and q, = 0 allj > 1. (b) If p < 1 then,

(i) dL(x) = C2xe- xdx, 0 < x < co (ii) q = (1 - v)(A/)(1 - I /)j-+ v(AIA)2(j - )( - ) j = 1,2, (iii) dQl(x) = (1 - v)lie-"xdx + v/2xe-"xdx, 0 < x < co

where ( is the real positive root of 1 + A'(z) = 0, / = A(1 - tfr()) and v = C /y.

Proof. (i) From (6) and (11) we have that

L(x, t) = Pr{Y(t) x I S(y) > t} = 1 - F(x; y -, t)/F(O; y -, t) so that from Lemma 4,

(30) L(x) = lim L(x,t) = {-e-x"(x+1) if p<l,

t- +t if p =1.

(ii) We observe that Q(t), the number of customers present in the queue at time t is one more than the number of customers who arrived during the time Y(t). Therefore, for j = 1,2, -..

Pr{Q(t) = j I Y(t) = x} = Bjl(x) -

Bj(x),

where Bj(x) is the jth convolution of the inter-arrival time distribution with itself. Therefore

qj(t) = Pr{Q(t) =j I S(y) > t}

(31) = fPr{Q(t) =j I Y(t) = x}dxPr{Y(t) < x IS(y) > t}

= {Bj-

1(x) - Bj(x)}dxl(x,

t),

where the function l(x, t) is defined as

(x,t) = {Pr{Y(t)?xIS(y)>t} if x t,

O if x > t.

Allowing t-- oo in (31) we obtain, by making use of the dominated convergence theorem,

128 E. K. KYPRIANOU

qj = lim qj(t) =

limr dxl(x, t) {Bj (x) - Bj(x)} t- o0o t- o0

00

= ~xe-(X{Bj,(x) - Bj(x)} dx

(32) d

S --( , _ (_))j--1(?) + + -(1

- ( _

2(/)

= (1 - v)(It/2)(1 - I_/2)j-I + v(P/)2(j (_ - )j-2

if p< 1 and qj=0 if p= 1, j 1.

(iii) The virtual waiting time W(t) at time t is the time it takes to serve all

customers present in the queue at time t. Since the service time distribution is

Markovian, the distribution of the residual service time of the customer receiving service at time t is still negative exponential. Therefore

(33) Pr{W(t) ? xlS(y)

> t} = I Pr{Q(t) =J S(y) > t} JvJl-e-A"dv/(j-1)! j= 1

If we allow t to go to infinity in (33) then by the dominated convergence theorem

and (32) we have

n(x) = d 2_j1()}/ ;-fe-f dv/(j - 1)!

(34) = , 2 i O - 1) fe-v(1 ) dv1/e

= 1 -(1 + x)e -x(1- (0)

if p > 1, and QI(x) = 0 if p = 1,0 < x < ox. Therefore, for p < 1, we have from (34)

dQ(x) = (1 - v)ple- clx + vY2xe-"xdx

with ( = (1 - t(/)) and v = 1/up. This completes the proof of the theorem.

Reference

[1] BENE', V. E. (1957) On queues with Poisson arrivals. Ann. Math. Statist. 28, 670-677.

[2] DOETSCH, G. (1958) Einfiihrung in Theorie und Anwendung der Laplace-Transformation. Birkhauser Verlag, Basel.

[3] ERDELYI, A. (1956) Asymptotic Expansions. Dover, New York.

[4] HILLE E. (1959) Analytic Function Theory. Vol. I, Blaisdell.

[5] KYPRIANOU, E. (1970) On Limit Distributions in Queues and Inventories. Ph.D. Thesis, Manchester University.

[6] KYPRIANOU, E. (1971) The quasi-stationary distribution of the virtual waiting time in

queues with Poisson arrivals. J. Appl. Prob. 8, 494-507.

[7] PRABHU, N. U. (1964) A waiting time process in the queue GI/M/i. Acta Math. Acad.

Sci. Hung. 15, 363-371.