On the Subexponential Properties in Stationary Single-Server Queues: A Palm-Martingale Approach

On the Subexponential Properties in Stationary Single-Server Queues: A Palm-MartingaleApproachAuthor(s): Naoto MiyoshiSource: Advances in Applied Probability, Vol. 36, No. 3 (Sep., 2004), pp. 872-892Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/4140413 .

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Adv. Appl. Prob. 36, 872-892 (2004) Printed in Northern Ireland

@ Applied Probability Trust 2004

ON THE SUBEXPONENTIAL PROPERTIES IN STATIONARY SINGLE-SERVER QUEUES: A PALM-MARTINGALE APPROACH

NAOTO MIYOSHI,* Tokyo Institute of Technology

Abstract

This paper studies the subexponential properties of the stationary workload, actual waiting time and sojourn time distributions in work-conserving single-server queues when the equilibrium residual service time distribution is subexponential. This kind of problem has been previously investigated in various queueing and insurance risk settings. For example, it has been shown that, when the queue has a Markovian arrival stream (MAS) input governed by a finite-state Markov chain, it has such subexponential properties. However, though MASs can approximate any stationary marked point process, it is known that the corresponding subexponential results fail in the general stationary framework. In this paper, we consider the model with a general stationary input and show the subexponential properties under some additional assumptions. Our assumptions are so general that the MAS governed by a finite-state Markov chain inherently possesses them. The approach used here is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The proof is essentially an extension of the M/GI/1 case to cover 'Poisson-like' arrival processes such as Markovian ones, where the stochastic intensity is admitted.

Keywords: Stationary work-conserving single-server queue; subexponential distribution; Palm-martingale calculus; stochastic intensity kernel

2000 Mathematics Subject Classification: Primary 60K25 Secondary 60G55; 60G70; 91B30

1. Introduction

Recently, heavy-tailed distributional properties appearing in queueing and insurance risk models have been studied with much interest in the literature (see e.g. [12], [18], [20]). This paper studies the subexponential properties in stationary work-conserving single-server queues. Namely, we study the proposition that when the equilibrium residual service time distribution is subexponential, the stationary workload, actual waiting time and sojourn time distributions in the queue also have subexponential tail asymptotics. So far, this kind of problem has been investigated in various queueing and insurance risk settings. When the interarrival time sequence and the service time sequence are mutually independent and both consist of i.i.d. random variables (i.e. GI/GI input), Pakes [16] showed the subexponential property for the stationary actual waiting time distribution. The corresponding risk model was studied in [11], where the subexponential asymptotics of the ruin probability were obtained. Due to the property of convolutions of subexponential distributions, it can be readily shown that the stationary sojourn time distribution also has the same asymptotics (see [19]). Asmussen et al. [3]

Received 22 October 2001; revision received 14 April 2004. * Postal address: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, 152-8552, Japan. Email address: [email protected]

872



Subexponential properties in stationary single-server queues 873

generalized the result to the risk model with Markov-modulated Poisson process arrivals and

state-dependent claim sizes (service times), and a similar Markov-modulated queueing model was studied by Jelenkovi6 and Lazar [13]. Asmussen et al. [4] further considered the risk model with a general ergodic arrival process and independent claim sizes and, under a fairly general assumption, showed the subexponential properties of the time-stationary and claim-

stationary ruin probabilities (corresponding to the stationary virtual and actual waiting time distributions). They also studied the model where the input process has a regenerative structure, which includes Markovian arrival streams (MASs). Moreover, Takine [21] considered the

single-server queue with an MAS input governed by a finite-state stationary Markov chain and showed the subexponential properties with only an assumption on the state-dependent service time distributions.

Since MASs can approximate any stationary marked point process (see [2]), it might be

conjectured that the same subexponential properties hold in the general stationary framework. However, this is not true and some counterexamples are illustrated in [4]. In this paper, we consider the single-server queue with a general stationary input and prove the subexponential properties under some additional assumptions. Our assumptions are so general that the MAS

governed by a finite-state Markov chain, which is considered in [21], inherently possesses them. The approach used here is the Palm-martingale calculus (see [5]), that is, the connection between the notion of Palm probability and that of stochastic intensity via Papangelou's formula (see [5], [9] and also [17]). In [15], the author derived a general formula for the stationary workload distribution of work-conserving single-server queues via the Palm-martingale approach, and this gives the starting point for the current work. The proof is essentially an extension of the M/GI/1 case to cover 'Poisson-like' arrival processes such as Markovian ones, where the stochastic intensity is admitted.

The paper is organized as follows. As a preliminary, the definition and some properties of

subexponential distributions are stated in Section 2. The single-server queueing model with a

general stationary input is described in Section 3, where the input process is given as a stationary marked point process associated with the stochastic intensity kernel. The result obtained in [15] is also reviewed there. Section 4 provides the main result, that is, the subexponential properties are derived under some general assumptions. The proof of a key lemma is postponed to Section 5. A concluding discussion is made in Section 6.

2. Subexponential distributions

In this section, we state the definition and some properties of subexponential distributions which are used in the following sections. For more details on such distributions, the reader is referred to, for example, [12, Appendix A3], [18, Section 2.5] and also [19].

For a distribution function F of a nonnegative random variable, let F denote its tail, that is, F(x) = 1 - F(x) for x e R+. Also, let F*n denote the n-fold convolution of F with itself, that is, F*l = F and

F*(n+l)(x) =- F(x - y) dF*n(y), n = 1, 2..

The corresponding tail is denoted by F*n (x) = 1 - F*n (x). A distribution function F (and the

corresponding random variable) is said to be subexponential if F(x) > 0 for all x > 0 and

F*2(x) lim - = 2. x-~o F(x)



874 N. MIYOSHI

The following properties of the subexponential distributions are well known and their proofs can be found in, for example, [12], [18], [19].

Lemma 2.1. (i) If F is subexponential, then

F(x + y) lim = 1 for all y R. (2.1) x -O F(x)

(ii) Let F and G be two distribution functions on [0, oc) such that limx + G(x) / F (x) = c for some constant c E (0, cxo). Then F is subexponential if and only if G is subexponential.

(iii) Let F be a subexponential distribution function and G be a distribution function on [0, 00) such that limx~,0 G(x) / F (x) = c for some constant c E [0, o•).

Then

lim _ G(x - y) dF(y) = c, (2.2)

x-0oo F(x) o

lim _ F(x - y) dG(y) = 1. (2.3)

x+O F(x)

The property (2.1) is sometimes presented as the definition of long-tailed distributions (see e.g. [19]). Lemma 2.1(i) then says that any subexponential distribution is long-tailed. The property (2.2) is used to show that

limx-, F * G(x)/F(x) = 1 + c since F * G(x) =

F(x) + fox G(x - y) dF(y), where F * G denotes the convolution of F and G. Probably, (2.3) may not be found in the literature. It is, however, immediate from (2.2) since

fox

rx

o F(x - y) dG(y) = G(x) - F(x - y) dG(y)

= G(x) - jG(x - y)dF(y)

= F(x) - G(x) + G(x - y) dF(y).

Let Fe denote the equilibrium residual lifetime distribution of F, that is,

Fe (x) = I F(z) dz,

f0 9

where i = -

xfo x dF(x) = fo F(z) dz is the mean value with respect to F. Note that

Fe(x) = t fx F(z) dz. The following is also well known and will be used often in the discussions in Sections 4 and 5.

Lemma 2.2. If Fe is subexponential, then

F(x) lim = 0, (2.4)

X- 00 Fe(x) that is, the tail of Fe is heavier than that of F.

In fact, (2.4) holds if Fe is long tailed (see [19, Corollary 3.3]).




3. The stationary single-server queue

We consider a work-conserving single-server queueing system, where all random elements are defined on a common probability space (Q, , P). On (Q2, T), a family of measurable shift operators {OA}teR is defined and satisfies Po 071 = P for all t E R, that is,

{Ot}tR is

stationary in P. Let N denote a point process on (R, S (R)) representing the time epochs at which customers arrive at and enter the system, and let {Tn }nE be the corresponding point sequence satisfying ... < To < 0 < T1 < ... and limn,,, Tn = +oo, that is, N is P-almost

surely simple and locally finite. Also, for n E Z, let Sn (E R+) denote the service time required by the customer who arrives at Tn. We assume that {(Tn, Sn)}nc is compatible with

{Ot}t~R in

the sense that {(Tn, Sn)}nEZ o t = {(Tn - t, Sn)}nEz for all t E R. Due to the stationarity of the shift operators, {(Tn, Sn)}nOz then forms a stationary marked point process on the real line with the mark space (R+, B(R2+)). We further assume that the intensity of N is positive and finite, that is, X = E[N((0, 1])] e (0, oo). Then the Palm probability with respect to (N, P, Ot) is defined by

Po (A) = E 11 o OtN(dt) , Ae

•, (3.1)

where 1A is the indicator of the event A (see e.g. [5, Section 1.2]). The expectation with respect to PO is denoted by EO?. It is known that Po (To = 0) = 1 and EO?[Tn+I - Tn] = 1/X for all n Z. The traffic intensity of the queue is given by p 1 EO [So].

In the discussions throughout the following sections, the stochastic intensity kernel associated with {(Tn, Sn)}n)Z plays an important role. Let {YF }t=R denote a history to which the marked point process {(Tn, Sn)}n,, is adapted, that is, for t E R, 3T is a sub-ao-field of F such that F• C Ft whenever s < t and fc 1K(So) o OsN(ds) is Tt-measurable whenever C x K e

B((- oo, t] x R+). We assume that {}Ft ER is compatible with {Ot heR, that is, Ot Fs = Fs-t. We also assume that the point process N admits the {IF}-stochastic intensity

{.(t)}tER that is

P-almost surely locally integrable and satisfies

E[N((s, t])I | s] = E j (u)du s , s < t,

where we can assume that t{(t)}t ER is {IF}-predictable (see [8, Section 11.4]). Let Fo denote the conditional service time distribution with respect to Fo- = Ut<o ,, that is,

Fo(x) = PoN(So < x I o-). (3.2)

This allows the service time distribution of an arriving customer to depend on the history until his or her arrival. In this setting, the {3F -stochastic intensity kernel associated with {(Tn, Sn)}nIZ is given by {1)(t) dFo o O}tl}tR and X(0) dFo is Fo--measurable. A useful result in the Palm-

martingale calculus, which is known as Papangelou's formula (see [5, Section 1.9.2] or [9] and also [17]), is a connection between the notion of Palm probability and that of stochastic intensity, that is, the Radon-Nikodm derivative of PO with respect to P on

Fo- is given by

d PO /d P

(0-

= )(0)/k. In other words, for any FO_-measurable and nonnegative random variable X, E[LX(O)X] -

X E [X]. Now, we define the processes of contents in the queueing system. Let V(t) (e R+)

denote the workload in the system at time t. Without any loss of generality, we assume that {V(t)}tER is right-continuous with left-limits. Since we consider work-conserving service disciplines, the workload process between two successive arrival epochs is described



876 N. MIYOSHI

by Lindley's equation V(t) = (V(Tn-) + S, - (t - Tn))+ for t E [Tn, Tn+1), P-almost

surely, where x+ = max(x, 0). It is well known that, under the stability condition p < 1, there exists a unique and P-almost surely finite workload process which is jointly stationary with {(Tn, Sn)}nEZ (see [14]), and in the following, we take {V(t)}tER as such a process. Then, it is also known that P(V(O) = 0) = 1 - p. As in [15], we also use a supplementary variable L(t) (E Z+) called the phase at time t. The phase process {L(t)}tER is defined as L(t) = 0 whenever V(t) = 0, and L(t) steps up by 1 at every customer's arrival epoch and steps down by 1 when the workload process first goes back to the level which is found just before the customer's arrival epoch, that is,

L(t) = L(t-) + 1, t = Tn, n E Z, L(t) = L(t-) - 1, t =

inf{s > Tn : V(s) < V(Tn-)}, n E Z,

where we assume that {L(t)l} t is right-continuous with left-limits. Note that {L(t)} tR corresponds to the queue length process under the preemptive-resume last-come, first-served (LCFS-PR) discipline. By the definition, {(V(t), L(t))I}t R is clearly adapted to the history {Yt})teR-

In the setting described above, the following fundamental result was derived in [15].

Proposition 3.1. (Miyoshi [15].) The stationary workload distribution is given by 00

P(V (0) <x) = (1 - p) G(k)(x), (3.3) k=O

where G(0) = 1[,O,0) and G(k), k = 1, 2 .. ., is recursively given by

G (k)(x) = x

-E[(O)Fo(z) I L(O) = k - 1, V(O) = y] dz dG(k•-)(y).

Furthermore, the stationary distribution of the phase is given by

k-Pl(O

P(L(0) = k) =

(1 - p) E[h(0)Fo(x) I L(0) = i]dx, k = 0, 1,2,..., (3.4)

i=0

where, by convention, H- O = 1.

Note that (3.4) can also be written as

k-i

P(L(O) = k) = (1 - p)H E[R(O) I L(O) = i]EoN[So I L(O-) = i], k = 0, 1, 2,... i=O

(3.5) Indeed, by the definition of the conditional service time distribution Fo in (3.2), the integrand in the right-hand side of (3.4) reduces to

E[X(O)Fo(x) 1{L(O)=i}] XPN(So > x, L(O-) = i)

P(L(O) = i) P(L(O) = i) _ Po%(L(0-) = i) Po(So > x I L(0-) = i)

P(L(O) = i)

= E[X(O) I L(0) = i]Po?(So > x I L(O-) = i),

where Papangelou's formula is used in the first and third equalities.




The formula (3.3) is a generalization of Bene''s formula for M/GI/1 queues (see [6], [10]). Indeed, if E[X(0)Fo(z) I L(0) = k - 1, V(0) = y] =

•F(z), where Fi(z) = Po (So > z) =

E [Fo(z)], (3.3) readily reduces to Bene"'s formula P(V(0) < x) = (1 - p) 0 pkFe*k(x) Similarly, (3.5) as well as (3.4) is a generalization of the geometric queue length distribution for LCFS-PR M/GI/1 queues.

4. The result

In this section, we first give two assumptions and then present the main result under such assumptions. In the following, F denotes the stationary distribution of the service times, that is, F(x) = PO (So < x) = EO [Fo(x)] and /-,1 Eo [So]. Accordingly, Fe denotes the equilibrium residual service time distribution. The first assumption concerns the limit and the uniform integrability of Fo(x) / F (x).

Assumption 4.1. (i) For all x > 0, F(x) > 0 and there exists a limit fo* such that

lim Fo(x) = f P-a.s. (4.1) x-+oc F(x)

(ii) We have

Eo[ Fo

(x) Eo sup < x0. Lx>OF

-(x )

Assumption 4.1 (i) says that the tail of almost every realization of Fo is equivalent to or lighter than that of F. This assumption corresponds to Equation (4.4) of [3] and also Assumption 1(2) (or its equivalent, Assumption 2(2)) of [21]. Define the event A = {lim

x++ Fo(x)/F(x) =

fo*}. The compatibility ofF twith {Or tItz implies that -1 A = A up to the Po?-null event. Thus, Assumption 4.1(i) implies that the event A holds P-almost surely too (see [5, Section 1.6.2]). Due to the _Fo-measurability of Fo, the limit fo* in (4.1) is also Fo_-measurable and, under Assumption 4.1, it follows readily that E [fo*] = 1 by the dominated convergence theorem since EO [Fo(x)] = F(x).

The next assumption is a kind of mixing property, that is, asymptotic independence.

Assumption 4.2. For any A, B E bo,

lim P(A n Ot-1B) = P(A) P(B). t-+oo

With the assumptions given above, we have the main result in the paper.

Theorem 4.1. For the queueing system described above, let Fe be subexponential. Then, under

Assumptions 4.1 and 4.2,

lim P(V(O) > x) p lim) - 1 (4.2)

x 00 Fe (x) 1 - p

By Lemma 2.1(ii), Theorem 4.1 says that, if Fe is subexponential, then the stationary workload distribution is also subexponential under the imposed assumptions. The formula (4.2) gives the same result as elsewhere in the literature (see e.g. [3], [4], [11], [13], [16], [21]), where the single-server queues and the corresponding risk models in various different settings have been investigated. Theorem 4.1 is derived from the following lemma.



878 N. MIYOSHI

Lemma 4.1. Suppose that Fe is subexponential. Then, under Assumptions 4.1 and 4.2, we have the following:

(i) For k = 1, 2, ... and any Fo-measurable, nonnegative and P-integrable random variable X,

k-1

lim E[X l{L(O)=k,V(O)>x}] P

EoE[fo* 1{L(-)=i)] E[X 1{L(O)=k-l-i}]. (4.3) x-+Coo Fe(x) 1 - pi=

(ii) For some y E (0, 1) and any r E (0, 1], there exists some constant Ks > 0 such that, for k=l, 2 .1..

P(L(O) = k, V(O) > x) F Ke(1 + r)kyk. (4.4)

Fe (x ) The proof of Lemma 4.1 consists of several sub-lemmas and is postponed to the next section.

Once Lemma 4.1 is provided, Theorem 4.1 can be verified as follows.

Proof of Theorem 4.1. Since y E (0, 1) in the inequality (4.4), we can choose e so small that (1 + e)y < 1. Hence, applying the dominated convergence theorem and then (4.3) with X=l,

P(V(O) >x) '

P(L(O) = k, V(O) > x) lim _ = mli

x ->D a Fe (x k=l Fe (x) oo k-i

P E [fo* 1{L(O-)=i)] P(L(0) = k - 1 - i) k=l i=O

= E[fo* 1L(0o-)=

i}] P(L(O) = k - 1 - i) i=O k=i+l

1-p

where we use the fact that E [fo*] = 1 in the last equality.

Moreover, Theorem 4.1 leads to the subexponential properties for the stationary distributions of actual waiting times and sojourn times under the FCFS (first-come, first-served) discipline.

Corollary 4.1. If Fe is subexponential, then, under Assumptions 4.1 and 4.2,

P?,(V(O-) > x) Po (V(0) > x) p lim = lim (4.5) x•00 Fe(x) x-0- Fe(x) 1 - p

Proof First, we generalize Theorem 4.1 as follows: for any Fo-measurable, nonnegative and P-integrable random variable X,

SE[X 1{v(0)>x)}] P lim - [X]. (4.6)

x ->. Fe(x) 1 - p

To prove this, let X1 = X A 1 for 1 > 0, where a A b = min(a, b). Since E[XI 1{v(O)>x}] =

J;-1 E[X 1I{L(O)=k,V(O)>x}] and E[XI1 {L(O)=k,V(O)>x}] < I P(L(0) = k, V(0) > x), similar

to the proof of Theorem 4.1, it follows that, for any e > 0, there exists an xo > 0 such that

E[Xi l{v(0)>x}] P Fe(X )

p E[XI] < for all x > xo. Fe(x) 1

- p




Then, since XI f X as 1 - oo, (4.6) follows by the monotone convergence theorem and the P-integrability of X.

Now, for any 3-_-measurable, nonnegative and Po?-integrable random variable Y, we have by Papangelou's formula and (4.6) that

Eo?[Y l{v(o-)>x}] _ E[X.(O)Y l{v(o)>x}] lim -lim

x -c Fe (x) x X ?. Fe (x) p E[i.(O)Y]

1-p X

= Eo [Y]. (4.7) 1-p

The first part of the corollary is just (4.7) in the case when Y - 1. Next, we show the second part of (4.5). Since V (0) = V (0-) + So, Po -almost surely,

PO (V(O) > x)= 1 - PO (V(O) < x)

= 1 - Eo x l{so<x-y) 1{V(0-)edy}

Here, noting that V(0-) is Fo_-measurable and by using the conditional distribution Fo in (3.2), we have

P? (V(O) > x) = 1 - EO Fo(x - y) 1{V(0-)edy}

= P%(V(0-) > x) + E? Fo(x - y) 1{V(0-)edy} o YIx0

Thus, from the first part of the corollary, it suffices to show that

lim EO Fo(x - y) 1{V(O-)edy} 0-.

(4.8) x n Fe (X) o

By Lemma 2.2, for any 8 > 0, there exists an xo > 0 such that

Fo(x) F(x) Fo(x) S =c - , < Kf for all x > xo, P?-a.s., Fe (x) Fe(x) F(x)

where Kf = supx>o Fo(x)/F(x). Then, for any x > xo,

E[f Fo(x - y) 1{V(0-)edy]

< E EO Kf Fe(x - y) 1{ (0-)dy}

+ PO(V(0-) E (x - xo, x]) f 0x

< Fe(x - y) EN[Kf l{V(0-)edyl} p o(V(0--) > x - xo) - PoN(V(O-) > x).

(4.9)

By the first part of the corollary and Lemma 2.1(i), the second and third terms on the right-hand side of (4.9) divided by Fe(x) cancel each other as x -+ oc. Consider the first term on the right- hand side of (4.9). Note here that Eo [Kf l{v(0-)<x}]/ EO [Kf] is a proper distribution function Ilcrlu31U V \~7/.IY L~11N[Kf ] is a proper distribution functionlr~ U1LLIULIII LUIC?LV L



880 N. MIYOSHI

since E [Kf] is positive and finite under Assumption 4.1. Also, since Kf is Fo_-measurable, (4.7) implies that lim,,, EO?[Kf 1{v(o)>x}]/Fe(X) = p Eo[Kf]/(1 - p). Thus, it follows from (2.3) in Lemma 2.1(iii) that

1 [x lim Fe(x - y)E[Kf l{V(0-)edy}] = Eo[Kf] < 00.

x--oo Fe(x)

Therefore, since r is arbitrary, (4.8) holds.

Remark 4.1. We have seen that Assumption 4.1(i) corresponds to the assumption in [21]. When the input process is given as an MAS governed by a finite-state Markov chain as in [21], Assumption 4.1(ii) is natural since, in such a case, the number of possible realizations of Fo is finite. Furthermore, if the Markov chain has the stationary distribution, Assumption 4.2 is automatically satisfied, too. Indeed, the limiting distribution of such a Markov chain is independent of the initial distribution and coincides with the stationary distribution.

Remark 4.2. In the case that the service times {Snnez are i.i.d. and independent of the arrival point process N, Theorem 3.1 of [4] derives the same results as (4.2) and the first part of Corollary 4.1 under a fairly general assumption. Asmussen et al. [4] showed that a kind of mixing property is a sufficient condition for their assumption (our Assumption 4.1 is not needed in their case). Since our Assumption 4.2 is another kind of mixing property, it might be possible to say that some kind of mixing condition (asymptotic independence) is required for the subexponential results like (4.2) and (4.5).

5. Proof of Lemma 4.1

In this section, for simplicity of notation, we write K(k)(x) -

1{L(O-)=k,V(O-)_x}

and 7 (k)(x) = X(k)((C) - X(k)(x) = 1lL(O-)=k,V(O-)>x) for k e Z+ and x E I+. Define

r (x) = inft >0:t - Sn l(o,t](Tn) x

, x > 0. (5.1)

nEZ

Note that r (x) can be interpreted as the time at which the workload level first goes down by x given that the initial workload is not smaller than x at time 0. In other words, r (x) also corresponds to the first hitting time to the empty state given that the initial workload is just equal to x at time 0. The proof of Lemma 4.1 begins with verifying the following lemma.

Lemma 5.1. For any nonnegative random variable X and for k = 1, 2, ....

E[X. -(k)(x)] = E[ (0) xJx-yc (X, z)dzdx (k-1)(y) , (5.2) 00 f

where Q>(X, z) is Fo_ -measurable and is given by

D (X, z) = E?[X o T(w-z) I So = w, Fo-] dFo(w). (5.3)

Furthermore, for k = 1, 2, ...

E[X 1{L(0)=k}] =

] E[(0))(X, z) l {L(O)=k-1}] dz. (5.4)




Lemma 5.1 is a generalization of Proposition 3.1. Indeed, when X - 1, noting that ( (1, z) = Fo(z), we inductively obtain (3.3) and (3.4) from (5.2) and (5.4), respectively, by using P(L(O) = 0) = 1 - p.

Proof The proof follows similar lines to that of Proposition 3.1 (Theorem 1 in [15]). The difference is the existence of X and this is handled by exploiting {r(x)}x>o in (5.1). Since the phase L(t) is equal to the queue length at time t under the LCFS-PR discipline, we discuss the proof in the context of this discipline. Let R(t) denote the residual service time of the customer in service at time t when L(t) > 0, and let R(t) = 0 otherwise. Note that, under the LCFS-PR discipline, the customer in service is the one who arrived most recently among the customers in the system. For k E Z+ and x, y e R+, let

A(k, x) = {L(0-) = k, V(0-) < x} e o0-,

B(k, x, y) = {L(0-) = k, V(0-) - R(0-) < x, R(0-) < y} E Fo-.

Also, let Nk-1,x denote the sub-point process of N counting the number of the customers' arrival epochs at which the queue is in A (k - 1, x), that is,

Nk-l,x(C) =

fcIA(k-l,x)

o OtN(dt), C E ((R).

k Ix k-lx k-l x The corresponding point sequence {Tn~-l,xneZ satisfies ... < Tk-1,x 0 0 < Tk- ...

by convention. By the definition of the Palm probability (3.1), the intensity of Nk-1,x is given as 4k-l,x = E[Nk-1,x((O, 1])] = k Po(A(k - 1, x)).

Here, if Po (A (k- 1, x)) = 0, there is no arrival while the system is in A (k - i, x) and clearly P(B(k, x, y)) = 0 for all y E R+. Now, suppose that Po%(A(k - 1, x)) > 0 and let Po-0,x denote the Palm probability with respect to Nk-1,x. The corresponding expectation is denoted by Eo_1,x . Applying the Palm inversion formula (see [5, Section 1.2.4]) to E[X 1B(k,x,y)], we have

E[X 1B(k,x,y)] = 4k-1,x Ek-l,x o ,Tk-1,x (X lB(k,x,y)) o O dt.

Note that under the LCFS-PR discipline, for t E [0, Tk-lx) on {Tk- = 0} = {To = 0} n A(k - 1, x),

Ot-1B(k, x, y) = {customer 0 is in service and his or her residual

service time is not greater than y at time t}, Pl1,x-a.s.

Namely, using {r(x)}x~R in (5.1),

J

_

(X 1B(k,x,y)) o Ot dt = X o y

(so-z) dz, Po-,x-a.s. 10, Tik-1x)

(see Figure 1). Thus, since Po-1,x ()

= PO (. I A(k - 1, x)),

E[X 1B(k,x,y)] = Xk-1,x EoI-,x[X o Or(so-z) ({so>z}] dz

0- j

E?[X o Or(so-z) 1{So>z} 1A(k-l,x)] dz. (5.5) 0 V l, ,



882 N. MIYOSHI

{L(t) = k, V(t) - R(t) < x, R(t) < y}

0= Tk - 1,x

FIGURE 1: A sample path of the workload process in the work-conserving single-server queue.

Note that the last expression holds even when P? (A(k - 1, x)) = 0 (the discussion in [15] is a little too rapid in this point). Now, we consider the transformation of the last expression in terms of the Palm probability PO into an expression in terms of the time-stationary probability P via the stochastic intensity kernel. Using the conditional service time distribution Fo in (3.2), we have

E%[X o Or(so-z)llso>z I O] = N EO[X o O(W-z) I So = w, YO-] dFo(w)

= D(X, z).

Therefore, since D(X, z) 1A(k-1,x) is 0_$-measurable, substituting the above into (5.5) and then applying Papangelou's formula, we have

E[X 1B(k,x,y)] =- E[X(0)D(X, z) 1A(k-1,x)] dz. (5.6)

Hence, taking the convolution of V(0) - R(0) E dy and R(0) < x - y over y E (0, x], we obtain (5.2). Also, by letting both x and y tend to infinity in (5.6), we obtain (5.4).

The next lemma says that, under a mixing condition (Assumption 4.2), the first hitting time from large initial workload to the empty state asymptotically conditions on the system being idle.

Lemma 5.2. UnderAssumption 4.2, for any A E FTO and any Fb-measurable, nonnegative and

P-integrable random variable X,

lim E[X o 0r(x) 1A] = E[X I L(0) = 0] P(A), (5.7)

x-> oo

lim EN[X o Or(x) 1A] = E[X I L(0) = 0] P?(A). (5.8)

Proof Before verifying (5.7) and (5.8), we first show that under Assumption 4.2, for any A, B E Fo,

lim P(A n 0-I B) = P(A) P(B I L(0) = 0), (5.9) x--

-00 i (x)




where i (x), x > 0, denotes the time at which the total number of idle periods from the origin reaches x, that is,

i(x) = inf t >: 0: 1{L(s)=0} ds >x . (5.10)

In other words, (5.9) says that events observed only when the queue is idle are asymptotically independent of the initial distribution and according to P(. I L(O) = 0). By the definition of

i(x) in (5.10), for any x > 0, there exists a t > 0 such that L(t) = 0 and f 1{L(s)=0) ds = x P-almost surely. Thus,

PA

-1B)-

P(An 0-1B n

{L(t) = 0, fot 1{L(s)ds = x}) (5.11) P(A n i(x) P(L(t) =

0, fo l{ 1L(s)=O} ds = x)

Define the event E(t, x) = {fo 1{L(s)=O} ds = x} for t, x > 0. Then, due to the stability under

p < 1, we have P(E(t, x)) -- 1 as (t, x) -- (oo, oo). Noting this, for the numerator of (5.11),

P(A n t-B

n {L(t) =O 0} E(t, x))

= P(A n Ot1B n {L(t) = 0})

- P(A n 0t B n {L(t) = 0} n E(t, x)C),

where, under Assumption 4.2, P(A n OIB n {L(t) = 0}) - P(A) P(B n {L(0) = 01) as t -+ oo. Also, for the denominator of (5.11), we have by the stationarity

P({L(t) = 0} n E(t, x)) = P(L(0) = 0) - P({L(t) = 0l n E(t, x)C).

Hence, taking both x --* o and t - oc simultaneously in (5.11), we obtain (5.9) since t does not appear on the left-hand side of (5.11).

We now prove (5.7). We can clearly see that, for any t > 0 such that L(t) = 0,

f {L(s)=o l ds = t - V(0) - Sn l(o,t](Tn) P-a.s.,

0 e

and comparing the definition of i(x) in (5.10) with that of r(x) in (5.1), we have r(x) =

i(x - V (0)) for x > V (0) P-almost surely. Thus, since V (0) is P-almost surely finite under

p < 1, we obtain from (5.9) that, for A, B E FO,

lim P(-1 B-1A) = lim P(1 B n A) = P(B I L(0) = 0) P(A),

lim P(Ox(x)B

n A)= lim P(Oi(xV(O))

that is, (5.7) in the case of X = 1B. The case where X is Fo-measurable, nonnegative and

P-integrable follows from the standard monotone class argument. Let

n2 1

Xn = -1 k k/n<X<(k+l)/n}

+ n l{X>n}l k=0

Then, X,n X as n --* o and, by the discussion above, for any E > 0, there exists an xo > 0 such that

I E[Xn o Or(x) 1A] - E[Xn I L(0) = 0] P(A)I < E for all x > xo.

Hence, letting n tend to infinity, we obtain (5.7) by the monotone convergence theorem and the

P-integrability of X.



884 N. MIYOSHI

Next, we prove (5.8). Let XI = X A 1 for 1 > 0. By the definition of Palm probability PON in (3.1), for any e > 0,

E [XI o Or(x) 1A] = 7E E[(XI o Or(x) 1A) o 8T, l{T,>-,}] n<O

and, thus,

1 1 0

< Eo[X1 o Or(x) 1A] -- E[(X o r(x) 1A) o OTo I{TO>-}] P(N((-8, 0]) > 2).

(5.12) Note that Or(x) o 0To

= 0r(x+To)

for x > -To, P-almost surely. Therefore, since 0-'A {To > -E}) E , applying (5.7) to the above,

lim E[(XI o Or(x) 1A) o OTo l{1O>-l)] = lim E[Xi o Or(x+To) lO -An{To>-E) x 00oo x---+ oo T

= E[Xi I L(0) = 0] P(TolA n {To > -e}). (5.13)

Here, since e is arbitrary, take e 4 0. Then P(ToIA An {To > -e})/(Xe) -+ Po(A) in (5.13) by Ryll-Nardzewski's local definition of PO and also P(N((-E, 0])

> 2)/E -+ 0 in the last

expression of (5.12) by Korolyuk's estimate (see [5, Section 1.5]), that is, we obtain (5.8) for

XI. Finally, since XI f X as 1 -- o, the monotone convergence theorem and P-integrability of X complete the proof.

In the proof above, we have proved the mixing property of the shift operators {Oi(x))ER, in

(5.9). As other fundamental properties of {Oi(x)}xEIR+,

we can easily show that the conditional distribution P(- I L(0) = 0) is invariant in Oi(x) for any x > 0 and that the P-ergodicity of

{Ot itEr implies that

1 fX lim - IA o Oi(y) dy = P(A I L(O) = 0) P-a.s., Ae •; x -- oo X

these results are presented in Appendix A.

Lemma 5.3. Suppose that Fe is subexponential. Then, under Assumptions 4.1 and 4.2, for any Fo-measurable, nonnegative and P-integrable random variable X,

1 f 0 E[X l{L(O)=0}] lim 1 0

(X, z) dz = - [X 1 )=

o Na, (5.14)

x-- Fe(x) F

x(1 - P)

where cD(X, z) is given in (5.3).

Proof We first prove the lemma in the case of X = 1, that is, noting that D(1, z) = Fo(z),

1 [0

f0* (5.15) lim

00 Fo(z) dz = -a.s.(5.15)

x00 FeW(x) i

Under Assumption 4.1(i), for any e > 0, there exists Po -almost surely a •0>_

0 (depending on the realization) such that

Fo (x) F(x)-fo* < for all x >?o0, PN-a.s. F(x)




Thus, for any x > ?o,

Fo(z) dz - - F(z) dz - F(z) dz Fe (x ) I Fe(x) J F(Z) Fe(x) x

1 - Fo fo) F (z) dz

Fe (x)W x F(z)

< o P?-a.s., IX

where we use the fact that Fe(x) = pt fx -F (z) dz. Hence, (5.15) holds. Now, we prove the lemma in the case where X is Fo-measurable, nonnegative and P-

integrable. Applying Fubini's theorem twice, we have S

(X, z) dz = EO [X o Or(w-z) I So = w, Fo-] dz dFo(w) f x N

= EO[X o E0(y) I So = W, Fo•]

dy dFo(w)

x00 --

E?[X o 0,(y)

I So = w, Fo•0] dFo(w) dy. (5.16) x+y

Here, under Assumption 4.2, we have by Lemma 5.2 that, for any e > 0, there exists Po -almost surely an

i0o > 0 (depending on the realization) such that

I E?[X o 8,(y) I So = w, TO-] - E[X I L(0) = 0]1 < e for all y > io, Po-a.s. (5.17)

Then, using such an r0o,

we obtain from (5.16) that

D (X, z) dz < sup Eo [X o 0,(y) I So = w, Fo-]Fo(x + y) dy x 0 w>x

+ (E[X L(0) = 0] + e) Fo(x + y) dy

_ Fo(x) sup EON[X o 0,(y) I So = w, Fo-] dy

Jo w>xN

+ (E[X I L(0) = 0] + e) Fo(x + y)dy PoN-a.s.

(5.18) 010 N

For the first term on the right-hand side of (5.18), we have by Lemma 2.2 and Assumption 4.1 that

Fo(x) F(x) Fo(x) lim = lim = 0 Po-a.s. x--00 Fe(x) x-+00 Fe(x) F(x)

On the other hand, for the second term on the right-hand side of (5.18), we have from (5.15) that

1 1 f* lim Fo(x + y) dy = lim Fo(y) dy =-Po-a.s. x-*o Fe(x) fo x -+ FeW(x)

Thus, since e > 0 is arbitrary, 1 f0

E[X _

{L(0)=0}] lim sup D(X, z) dz <E[X 1L(O)= fo* P-a.s.,

x--o Fe(x)Jx - (1 - p) where we use the fact that P(L(0) = 0) = 1 - p.



886 N. MIYOSHI

Similarly, from (5.16) and (5.17), we obtain that

4(X, z) dz > (E[X I L(O) = 0] - 8) Fo(x + y) dy Po?-a.s.

Here, by Lemma 2.1(i) and (5.15),

1 DO Fe (x + 11o) 1 0

lim Fo(x + y) dy - lim (? _

Fo(y) dy x-00 Fe (x) o 0 Fe (x) Fe(x + o) +Fo() dy

_ fo poPa.s., A

N

and so

lim inf -1 0

D(X, z) dz E[X

1{L(O)=o)l fo* po -a.s., x +oo 0Fe(x)Jx (1

- p)

completing the proof.

We now are in a position to verify Lemma 4.1.

5.1. Proof of Lemma 4.1(i)

We prove (4.3) by induction on k. When k = 1, subtracting (5.2) from (5.4) in Lemma 5.1, we obtain that

E[X - X(')(x)] = E[ (O) j (X, z) dz 1IL(O)=0 ) = [ Eo [ q(X, z) dz1(L(o-)=ol, fx N-f-x

where the second equality follows from Papangelou's formula. Let X1 = X A I for I > 0. Then, since F(Xi, z) < lFo(z),

I f lKf 00 - 1 SD(XI, z) dz

I{L(0-)=ol F(z) dz

=

--Kf P?-a.s.,

Fe(x) W-FeW(x)

where Kf = sup,,o Fo(z)/F(z). Therefore, applying the dominated convergence theorem under Assumption 4.1 (ii) and then applying (5.14) from Lemma 5.3,

lim E[XI.x(1)(x)] =- E[fo

I{L(O-)=0)] E[Xl I{L(O)=O}]. --00 Fe(x) 1 - p

Since XI f X as 1 -+ 0o, the monotone convergence theorem and the P-integrability of X

imply (4.3) with k = 1. Now, suppose that (4.3) holds for some k > 1. Again, let X1 = X Al for 1 > 0. Substituting

(5.2) and (5.4) into

E[XI - X(k+l)(x)]

= E[XI . X(k+1) (oo) - E[ (O) / (Xi, z) dz x (k) x) +

E[(O) /0(X d (x)] + E X(0) D(Xl, z)dz x(k)(x)

-- E[Xi - X(k+l)(x)],




we obtain that

EI[XI - X(k+) (x)] = E[•(O)

j c (X, z) dz -(k) (x)

+ E[•(O) fj (X, z) dz dx(k)(y) . (5.19) [ O--y

Consider the first term of the right-hand side of (5.19). Summing (5.4) in Lemma 5.1 over k = 1, 2, ..., we obtain that

E [X(O) f (c(X, z) dz = E[XI 1{L(0)>}] < oo,

and thus, by the induction hypothesis,

lim IE (0) j D(Xi, z) dz-y(k)(x) X-'oo Fe(x) L

k-1 00

S- EO[f LL

ol{L(o-)=i}] E[(O) J (X1, z) dz

l{L(O)=k-l-i} k-i

P Eo[f* 1{L(o-)=i}] E[XI 1{L(O)=k-i}], (5.20) 1-p i=O

where we use (5.4) in the second equality. Next, for the second term of the right-hand side in (5.19), we show that

lim r E 1(0) 00

(X, z)dzdx(k)(Y)] x ->0

Fe (Wx ) - y

-= EoN[fo* l{L(O-)=k}] E[Xi I{L(O)=O}]. (5.21) 1-p

By Egoroff's theorem (see e.g. [1, p. 94] or [7, p. 187]) and Lemma 5.3, we have that, for any e > 0, there exists an event U, such that Po?(U,) > 1 - e and the convergence of (5.14) is uniform on Ue, that is, there exists a constant xo > 0 such that

1 00 (X, z) dz - E[X1 1{L(O)=O}] f*< for all x > xo on U8, (5.22)

Fe (x)Jx d (1 - p)

where clearly U, e •o-

since D(Xi, z) and fo* are both Fo_-measurable. Applying Papan- gelou's formula to the second term of the right-hand side of (5.19) and then decomposing by U8,

E [X(0) J D(Xi, z) dz dx(k) (y)] - hEo4 Q((Xi, z) dz dx(k) (y) 1iU Ex-y N

x-y X z y

+ E0N P (Xi, z) dz dx (k) (y) 1U .

x-y

(5.23)



888 N. MIYOSHI

For the second term of the right-hand side of (5.23), since ( (Xi, z) < IFo(z),

E?[ fSy

(Xl, z) dz dx (k)(y) lUc] 5: E[ Kf Fe(x - y) dx(k)(y) 1uc

where Kf = supz>oFo(z)/F(z). Here, suppose that EO?[Kf l{L(0-)=k} 1uc] > 0 for a

moment. Then EO?[Kfx(k) (x) 1u]/ E?[Kf l{L(0-)=k} 1U] is a proper distribution function and, applying Papangelou's formula,

Eo [Kf -(k)(x) 1Ug] E[X(O)KfX(k)(x) 1uc] Fe (x) Fe(x)

which has a limit as x -- oo by the induction hypothesis. Thus, from (2.3) in Lemma 2.1(iii) it follows that

lim E Kf Fe(x - y) dx (k)(y) 1U = E[K 1L(O-)=k U

x-+ O Fe(x) o

Therefore, whether EOy[Kf l{L(0-)=k} luc] > 0 or not,

lim sup Eo Di (Xl, z) dz dx (k) (y) 1l < - E?[Kf l{L(O-)=k} 1U] x--+o Fe(x) LJx-y

< lp Eo[Kf 1uc] < o. (5.24)

For the first term of the right-hand side of (5.23), it follows from (5.22) that, for any x > xo,

E[ Eo (Xl, z) dzdx(k) () i,)U [ /~-y

< E[Xt 1

L(O)=O}] Eo x -xo

Fe(x - y) dx (k)(Y) A(1 --p)

N[fo rxY

+ XE -Xo

Fe(x - y) dx(k) (y)] + Eo (Xl, z) dz dx (k) (y)

pE[XI 1{L(o)=)}] EO[ f Fx(x - y) dx(k)(y) +

EX.E[ Fe(x - y) dx(k)(y)

1-,p N

+ E[ X(0) c(XI, z) dz (k)(x - x0) - E [(O) f (Q (Xi, z) dz -(k)(x) ,

(5.25)

where we use Papangelou's formula again in the second inequality. The last two terms on the right-hand side of (5.25) divided by Fe(x) cancel each other as x * o00 by the induction

hypothesis and Lemma 2.1(i). For the first and second terms on the right-hand side of (5.25), similar to the above discussion, we find from (2.3) in Lemma 2.1(iii) that

lim Eo fE Fe(x - y) dx(k)(y) = EO [f,* l{L(0-)=k}], x-o00 Fe(x)

lim Eoj Fe(x - y) dx (k)(y) = po(L(0-) = k). x --o Fe(x)Wo




Therefore,

lim sup - E% Il (X1, z) dz dx (k) (y) lu x- Fe(x) L x-y

< PE[Xl 1{L(0)=0}]

E?[fo* l{L(0-)=k}] + eX P?(L(0-)

= k). (5.26) 1-p

Summing (5.26) and (5.24) and noting that Eo[Kf] < 00 and Po (UC) < e, since e is arbitrary, we obtain that

1 of

lim sup E X (O) () ((Xt,

z) dz dx (k)() x-+c.o Fe(x) x- y

S Eo[f o* l{L(O-)=k}] E[Xi I{L(O)=O}]. -1-p

On the other hand, from (5.22) and (5.23) it follows that, for any x > xo,

E[X(0) fj -c(X, z) dz dx (k)(y)] (5.27) [Ox--y

X Eo { Dj (X1, z) dz dx (k)) (lUe x x E -yOO p E[Xl l{L(O)= EOf If

-x

Fe(x - y) dx (k) (y) 1]U

1-p N

S x

E?o Fe(x - y) dx (k) (Y)

SFe(x)E[XI 1{L(O)=O]E

Eo [fx(k)(x - xo) lu,] - eE Fe(x - y) dx (k (y) . S-p 0

(5.28)

Similar to the above, the second term on the right-hand side of (5.28) divided by Fe (x) converges to eX Po(L(0-) = k) < 00 as x --+ co. For the first term, clearly EO?[f oX(k)(x - xo) 1] -

Eo[fo* l{L(O-)=k} 1U,] as x -- oo, and also EO [fo l{L(O-)=k} lU8] -+ EO[fo* l{L(O-)=k}] as

e 4. 0. Therefore,

lim inf I

E (O) (Xl, z) dz dx (k) (Y) x -- 0Fe(x) f x- y

> p EEJ[fo* l{L(O-)=k}] E[Xl I{L(O)=O}]. -1-p

Hence, (5.21) holds. Finally, summing up (5.20) and (5.21), and then applying the standard monotone class argument as 1 -+ 00, (4.3) holds for k + 1.

5.2. Proof of Lemma 4.1(ii) The proof follows similar lines to that of the corresponding result for the distribution of the

sum of independent random variables (see e.g. [3, Lemma 4.2(b)]). First, since L(0) < 00 P-almost surely under p < 1, we can see from (3.5) that there exists an io > 0 such that

E[II(0) I L(0) = i] Eo[So I L(0-) = i] < 1 for all i > io. Therefore, we can take y < 1



890 N. MIYOSHI

such that y > E[L(0) I L(0) = i] Eo?[So I L(0-) = i] for all i > io. Thus, letting

S-

supo<i<io E[X(O) I L(0) = i]EO[So I L(0-) = i] (> 1) if io > 0,

1 if io = 0,

we obtain from (3.5) that, for k = 0, 1, 2, ...

P(L(0) = k) < (1 - p) ) k.

(5.29)

Now, let C = supi>0,P(L(O)=i)>o E[X(O)fo I L(0) = i]. Then, from (4.3),

k-1 P(L(0) = k, V(0) > x) 1 L lim = E[k(0) fo*

1{L(o)=i)] P(L(0) = k - 1 - i) Fe(x) WM(1 - p) i=O

k-i

)C LP(L(0) = i)P(L(0) = k - 1 - i). (1 -

i=O

Therefore, we obtain from (5.29) that, for any e > 0, there exists an xo > 0 such that

P(L(0) = k, V(0) > x) C(1 -p) Y2ioy (k +) Fe (x) y faly

<C(1- p) ,2io

(I + E)kyk for all x > xo,

- s/y (

where we use the fact that e(k + e) < (1 + E)k for any k > 1 and e E (0, 1]. On the other hand, for x < xo, since Fe(x) > Fe(xo), we obtain from (5.29) that

P(L(0) = k, V(0) > x) P(L(0) = k) 1 -p y' k

Fe (x) - Fe(x) - Fe(xo) (7Y 1

Finally, setting

K = (1-p)( ylii max - Y ii Fe (O

y) IY( Y Fe (x o)

Lemma 4.1(ii) holds.

6. Concluding discussion

In this paper, we have offered a general set of conditions for the proposition that the

stationary workload, actual waiting time and sojourn time distributions in stationary single- server queues have the same subexponential asymptotics when the equilibrium residual service time distribution is subexponential. Our proof relies heavily on the connection between the Palm probability and the stochastic intensity. Assuming the existence of stochastic intensity is weak in the sense that the MAS has one, but it is not so trivial and rules out some important cases even for renewal arrivals. Since it is not required in [116] and [4] for example, the existence of stochastic intensity should not be necessary. Namely, there should be another way that does not rely on the stochastic intensity but has enough generality with dependent service times in non-Markovian settings.




Appendix A. Fundamental properties of (Oi(x) )x>0

In this appendix, for the mapping i(x), x E +, defined in (5.10), we prove the following fundamental properties of {Oi(x) }x>0.

Lemma A.1. (i) The conditional probability P(. I L(0) = 0) is invariant of Oi(x) for any x O0, that is, P(O[I)A I L(0) = 0) = P(A I L(0) = 0)forany A e F.

(ii) If {Or }tER is ergodic in P, then

lim - 1A o Oi(y) dy = P(A I L(0) = 0) P-a.s., x - 00 Xo

Proof (i) The proof is similar to that of the invariance property of the Palm probability PO with respect to T,, ne Z+ (see e.g. [5, Section 1.2.2]). Let

ftL(s)=Ods.

I (t) -- 1{L(s)=0} ds.

Then E[I(1)] = P(L(0) = 0) = 1 - p, and the conditional probability P(- I L(0) = 0) is

given as the Palm probability PI for random measure dI (t), that is,

P(A I L(0) = 0) = l E 1A o lOi(y)dy =-PI(A), AE•F. 1-p -0

Thus,

PI(O-1 A) = 1 E 1A o Oi(x+y) dy

[I (1) L. o x I (1)+x

1 E 1A o Oi(y) dy - A o Oi(y) dy + 1A o

Oi(y) dy . 1 - p 00If(1)

Here, note that

1A o Oi(y) dy =

1( A o Oi(y) dy o 01 P-a.s., I(1) J

and the invariance property of P with respect to Ot implies that PI (Oji-) A) = PI (A).

(ii) By the definition of i (x), we have

1 x i(x) 1 1i(x)

f 1A o i(y)dy = i(X) 1An{L(O)=O} ooOs ds. x 0 x i (x) fo

From the P-ergodicity of {Ot h R and the stability condition p < 1, i (x) -+ co as x -> co P-almost surely, and, thus, the ergodic theorem implies that

1 fi(x) lim i( lAn{L(0)=O} o 0s ds = P(A n {L(0) = 0}) P-a.s., x oo i(x) f

x lim = P(L(0) = 0) = 1 - p P-a.s.,

x-+oo i(x)

which completes the proof.



892 N. MIYOSHI

Acknowledgements

The author wishes to thank Tetsuya Takine and Karl Sigman for valuable discussions. Thanks are also due to the anonymous referee for some helpful comments and suggestions.

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On the Subexponential Properties in Stationary Single-Server Queues: A Palm-Martingale Approach