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Asymptotics in the MAP/G/1 Queue with Critical LoadJeongsim Kim a & Bara Kim ba Department of Mathematics Education , Chungbuk National University , Chungbuk , Koreab Department of Mathematics and Telecommunication Mathematics Research Center , KoreaUniversity , Seoul , KoreaPublished online: 21 Dec 2009.

To cite this article: Jeongsim Kim & Bara Kim (2009) Asymptotics in the MAP/G/1 Queue with Critical Load, StochasticAnalysis and Applications, 28:1, 157-168, DOI: 10.1080/07362990903415866

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Stochastic Analysis and Applications, 28: 157–168, 2010Copyright © Taylor & Francis Group, LLCISSN 0736-2994 print/1532-9356 onlineDOI: 10.1080/07362990903415866

Asymptotics in theMAP/G/1 Queuewith Critical Load

JEONGSIM KIM1 AND BARA KIM2

1Department of Mathematics Education,Chungbuk National University, Chungbuk, Korea2Department of Mathematics and TelecommunicationMathematics Research Center, Korea University, Seoul, Korea

When the offered load � is 1, we investigate the asymptotic behavior of thestationary measure for the MAP/G/1 queue and the asymptotic behavior of the lossprobability for the finite buffer MAP/G/1/K + 1 queue. Unlike Baiocchi [StochasticModels 10(1994):867–893], we assume neither the time reversibility of the MAPnor the exponential moment condition for the service time distribution. Our resultgeneralizes the result of Baiocchi for the critical case � = 1 and solves the problemconjectured by Kim et al. [Operations Research Letters 36(2008):127–132].

Keywords Loss probability; Markovian arrival process; Stationary measure;Stationary probability vector; Wiener-Hopf theory.

Mathematics Subject Classification Primary 60K25; Secondary 60J10.

1. Introduction

The Markovian arrival process (MAP) is governed by an underlying r state Markovprocess having transition rate cij , 1 ≤ i �= j ≤ r, from state i to j without an arrivaland having transition rate dij , 1 ≤ i� j ≤ r, from state i to j with an arrival. Wedefine cii = −�

∑j �=i cij +

∑rj=1 dij�, 1≤ i ≤ r. Let C and D be r × r matrices whose

�i� j�-components are cij and dij , respectively. Assume that C +D is irreducible. Let� denote the stationary probability vector of C +D, that is, � is a solution of

��C +D� = 0� �1 = 1� (1)

Received March 5, 2009; Accepted June 22, 2009This research was supported by a Korea University Grant and the MIC (Ministry

of Information and Communication), Korea, under the ITRC (Information TechnologyResearch Center) support program supervised by the IITA (Institute of InformationTechnology Assessment).

Address correspondence to Bara Kim, Department of Mathematics andTelecommunication Mathematics Research Center, Korea University, 1, Anam-dong,Seongbuk-gu, Seoul 136-701, Korea; E-mail: bara@korea.ac.kr

157

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158 Kim and Kim

The so-called fundamental arrival rate of the MAP is given by �= �D1. Here andsubsequently, 1 denotes a column vector with all its components equal to one.

We consider both the infinite buffer MAP/G/1 and the finite bufferMAP/G/1/K + 1 queues where the arrival process is MAP with fundamental arrivalrate �. The service times, denoted by random variable V , are independent of eachother and have an identical distribution function H�·�. The offered load � is definedas � = ��V .

In this article, we concentrate on the case where the offered load � is 1. Theaim of this article is two-fold. First, we find the following asymptotic behavior ofthe stationary measure � = ��0� �1� �2� � � � � for the MAP/G/1 queue:

limn→� �n = c��

where c is positive when ��V 2� < � and 0 when ��V 2� = �. Second, we obtainthe following asymptotic behavior of the loss probability P

�K+1�Loss for the finite buffer

MAP/G/1/K + 1 queue:

limK→�

KP�K+1�Loss = �′′�1−�

2� (2)

where ��z� denotes the Perron–Frobenius eigenvalue of A�z� = ∫ �0 e�C+Dz�tdH�t�, 0 ≤

z ≤ 1.This article is inspired by Baiocchi [2] and Kim et al. [6]. Baiocchi derived (2)

under the conditions that the MAP is time reversible and that the service time Vhas a finite exponential moment ��e�V � < � for some � > 0. Kim et al. conjecturedthat (2) still holds without the time reversibility of the MAP.

This article asserts that (2) holds without the time reversibility for the MAPand the exponential moment condition for the service time. This implies that theconjecture in Kim et al. [6] is true.

For finite buffer queues, the loss behavior has been one of the important topicsof research in telecommunication systems and should be estimated accurately forbuffer dimensioning. Many researchers have analyzed asymptotic loss probabilityfor various queueing systems with finite buffer. (For more details on related works,see, e.g., [2–5, 9, 13].)

2. Asymptotics for the Stationary Measure of the MAP/G/1 Queue

We consider the MAP/G/1 queue where the arrival process is MAP withrepresentation �C�D� and the service times V are independent and identicallydistributed. We assume � = 1. The MAP/G/1 queue is described by a Markovchain �Xn� Jn� on the state space �i� j� � i = 0� 1� � � � � j = 1� � � � � r, where Xn

and Jn denote the number of customers in the system and the phase of theMAP, respectively, immediately after the nth customer’s departure. The transitionprobability matrix P of the Markov chain �Xn� Jn� is given by

P =

B0 B1 B2 B3 · · ·A0 A1 A2 A3 · · ·

A0 A1 A2 · · ·� � �

� � �� � �

� (3)

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Asymptotics in the MAP/G/1 Queue 159

where An is the r × r matrix whose �i� j�-component is the conditional probability ofa departure occurring with the arrival process in phase j, while during that servicethere were n arrivals, given that the previous departure left at least one customer inthe system and the arrival process in phase i. Further, Bn is the r × r matrix whose�i� j�-component is the conditional probability of a departure occurring with thearrival process in phase j, leaving n customers in the system, given that the previousdeparture left the system empty and the arrival process in phase i. It follows fromthese definitions that Bn = �−C−1�DAn, n ≥ 0.

We define the generating function A�z� of the matrix sequence An � n ≥ 0 by

A�z� ≡�∑n=0

Anzn� 0 ≤ z ≤ 1�

which is expressed as follows (see [8]):

A�z� =∫ �

0e�C+Dz�tdH�t��

From the assumption that C +D is irreducible, it follows that A�z� is irreducible for0 < z ≤ 1. The Perron–Frobenius eigenvalue of A�z�, 0<z ≤ 1 is denoted by ��z�.We note that the stationary probability vector � defined by (1) is also the stationaryprobability vector of the irreducible stochastic matrix A ≡ A�1�.

By Theorem A.2.3 of the Appendix in [11], we have �′�1−� = �A′�1−�1 = 1.Further, it can be shown that �′′�1−� > 0 and

��V 2� < � ⇔ A′′�1−� < � ⇔ �′′�1−� < �� (4)

The following lemma is also due to Theorem A.2.3 of the Appendix in [11].

Lemma 1 ([11]). We have

�′′�1−� = �A′′�1−�1+ 2�A′�1−��I − A+ 1��−1�A′�1−�− I�1�

We remark that both sides of the above equation are finite if ��V 2� < � andinfinite if ��V 2� = �.

Since � = 1, the Markov chain �Xn� Jn� is null recurrent and there isa stationary measure that is unique up to a multiplicative constant. Let � =��0� �1� �2� � � � � with �i = ��i1� � � � � �ir�, i ≥ 0, be a stationary measure of the Markovchain �Xn� Jn�.

The following is the first main result of this article. The proof is deferred to theend of this section.

Theorem 1. We have

limn→� �n = c��

where

c = 2�′′�1−�

�0�−C�−11�V

�

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160 Kim and Kim

Remark. By (4), the coefficient c is positive when��V 2�<� and 0 when��V 2�=�.

The stationary measure � of �Xn� Jn� is closely related to a Markovadditive process �Xn� Jn� � n ≥ 0 on � � � �−1� 0� 1� � � � × 1� � � � � r with transitionprobability matrix

P ≡

· · · −2 −1 0 1 2 3 · · ·���

−101���

� � �� � �

� � �� � �

A0 A1 A2 A3 · · ·A0 A1 A2 A3 · · ·

A0 A1 A2 A3 · · ·� � �

� � �� � �

� � �

�

(5)

The stationary measure � = ��0� �1� �2� � � � � satisfies the following lemma.

Lemma 2. We have

�n = �0�−C�−1DR−�n− 1�� n = 2� 3� � � � � (6)

where R−�n� is the r × r matrix whose �i� j�-component is

�R−�n��ij = �( s−−1∑

k=0

��Xk�Jk�=�n�j�

∣∣ �X0� J0� = �0� i�)

with s− ≡ infn ≥ 1 � Xn < 0.

Proof. The stationary measure �nj � n = 0� 1� 2� � � � � j = 1� � � � � r satisfies

�nj =r∑

i=1

�0i�( −1∑

k=0

��Xk�Jk�=�n�j�

∣∣ �X0� J0� = �0� i�)�

n = 0� 1� 2� � � � � j = 1� � � � � r� (7)

where ≡ infn ≥ 1 � Xn = 0. For n = 1� 2� � � � , and j = 1� � � � � r,

�( −1∑

k=0

��Xk�Jk�=�n�j�

∣∣ �X0� J0� = �0� i�)

=�∑l=1

r∑i′=1

�(�X1� J1� = �l� i′�

∣∣ �X0� J0� = �0� i�)

×�( −1∑

k=1

��Xk�Jk�=�n�j�

∣∣ �X1� J1� = �l� i′�)

=�∑l=1

r∑i′=1

�Bl�ii′�( −1∑

k=1

��Xk�Jk�=�n�j�

∣∣ �X1� J1� = �l� i′�)

= �−C�−1D�∑l=1

r∑i′=1

�Al�ii′�( −1∑

k=1

��Xk�Jk�=�n�j�

∣∣ �X1� J1� = �l� i′�)� (8)

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Asymptotics in the MAP/G/1 Queue 161

Now, for n = 2� 3� � � � , and j = 1� � � � � r,

�( s−−1∑

k=0

��Xk�Jk�=�n−1�j�

∣∣ �X0� J0� = �0� i�)

=�∑l=1

r∑i′=1

���X1� J1� = �l− 1� i′�∣∣ �X0� J0� = �0� i��

×�( s−−1∑

k=1

��Xk�Jk�=�n−1�j�

∣∣ �X1� J1� = �l− 1� i′�)

=�∑l=1

r∑i′=1

�Al�ii′�( s−−1∑

k=1

��Xk�Jk�=�n−1�j�

∣∣ �X1� J1� = �l− 1� i′�)� (9)

We define w− ≡ infn ≥ 1 � Xn ≤ 0. Then, for n = 2� 3� � � � , j = 1� � � � � r, l =1� 2� � � � , and i′ = 1� � � � � r,

�( s−−1∑

k=1

��Xk�Jk�=�n−1�j�

∣∣ �X1� J1� = �l− 1� i′�)

= �( w−−1∑

k=1

��Xk�Jk�=�n�j�

∣∣ �X1� J1� = �l� i′�)

= �( −1∑

k=1

��Xk�Jk�=�n�j�

∣∣ �X1� J1� = �l� i′�)� (10)

Substituting (10) into (9), we have for n = 2� 3� � � � � j = 1� � � � � r,

�( s−−1∑

k=0

��Xk�Jk�=�n−1�j�

∣∣ �X0� J0� = �0� i�)

=�∑l=1

r∑i′=1

�Al�ii′�( −1∑

k=1

��Xk�Jk�=�n�j�

∣∣ �X1� J1� = �l� i′�)� (11)

Combining (8) with (11), we have for n = 2� 3� � � � � j = 1� � � � � r,

�( −1∑

k=0

��Xk�Jk�=�n�j�

∣∣ �X0� J0� = �0� i�)

= �−C�−1D�( s−−1∑

k=0

��Xk�Jk�=�n−1�j�

∣∣ �X0� J0� = �0� i�)� (12)

Finally, substituting (12) into (7) yields the desired assertion (6). �

The following lemma is proved by using a result of Wiener-Hopf theory.

Lemma 3. The matrix R−��� ≡ limn→� R−�n� exists.

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162 Kim and Kim

Proof. For the Markov additive process �Xn� Jn� � n = 0� 1� � � � , we consider thetime-reversed Markov additive process �X∗

n� J∗n � � n = 0� 1� � � � , which is a Markov

process on � � � �−1� 0� 1� � � � × 1� � � � � r with transition probability matrix

P∗ =

· · · −2 −1 0 1 2 3 · · ·���

−101���

� � �� � �

� � �� � �

A∗0 A∗

1 A∗2 A∗

3 · · ·A∗

0 A∗1 A∗

2 A∗3 · · ·

A∗0 A∗

1 A∗2 A∗

3 · · ·� � �

� � �� � �

� � �

�

where A∗k = �diag����−1A

k diag���, k = 0� 1� 2� � � � . Let

∗w+ = infn ≥ 1 � X∗n ≥ 0�

For n = 0� 1� 2� � � � , G∗+�n� is defined as the r × r matrix whose �i� j�-component is

�G∗+�n��ij = �

(�X∗

∗w+ � J ∗ ∗w+ � = �n� j�

∣∣ �X∗0 � J

∗0 � = �0� i�

)�

By Proposition 2.13 on page 315 of [1],

R−�n� = �diag����−1

( �∑k=0

�G∗+�

∗k�n�)

diag���� n = 0� 1� 2� � � � � (13)

where

�G∗+�

∗0�n� ={I if n = 0

O if n = 1� 2� � � � �

�G∗+�

∗k�n� =n∑

l=0

�G∗+�

∗�k−1��n− l�G∗+�l�� n = 0� 1� � � � � k = 1� 2� � � � �

We observe that �A0�ii > 0 for all i = 1� � � � � r, and∑�

n=1 An is a positive matrix.From this observation it can be shown that G∗

+�n� is a positive matrix for everyn = 0� 1� 2� � � � . By applying the matrix analogue of the Blackwell renewal theorem(see Theorem 1 in [12]), we can obtain the limit of

∑�k=0�G

∗+�

∗k�n� in the right-handside of (13) as follows:

limn→�

�∑k=0

�G∗+�

∗k�n� = 1g∗ ∑�

n=1 nG∗+�n�1

1g∗�

where g∗ is the stationary distribution of∑�

n=0 G∗+�n�. Therefore (13) implies that

limn→�R−�n� = �diag����−1

((g∗

�∑n=1

nG∗+�n�1

)−1

1g∗)

diag���

=(g∗

�∑n=1

nG∗+�n�1

)−1

�diag����−1�g∗���

which completes the proof. �

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Asymptotics in the MAP/G/1 Queue 163

According to Lemmas 2 and 3, the next corollary is immediate.

Corollary 1. The limit limn→� �n exists.

Now we prove Theorem 1.

Proof of Theorem 1. According to Corollary 1, the limit limn→� �n exists. The limitcan be obtained by Abelian theorem for power series:

limn→� �n = lim

z→1−�1− z���z�� (14)

where ��z� ≡ ∑�n=0 �nz

n. Now we calculate the right-hand side of (14). The systemof equations, �P = �, leads to

��z� = �0�−C�−1�C +Dz�(z�zI − A�z��−1 − I

)�

on z ∈ � � �z� ≤ 1� det�zI − A�z�� �= 0. For 0 < z ≤ 1, let ��z� and ��z� be the leftand right Perron–Frobenius eigenvectors of A�z�, respectively, scaled by ��z���z� =��z�1 = 1. We note that the Perron–Frobenius eigenvalue ��z� of A�z� is of algebraicmultiplicity 1; ��z� is continuous in z, 0 < z ≤ 1; and ��1� = 1. Hence, it can beshown that there is �> 0 such that �zI − A�z��−1 − 1

z−��z���z���z� is bounded in z ∈

�1− �� 1�. Therefore,

limz→1−

�1− z���z� = limz→1−

�0�−C�−1�C +Dz�z�1− z�

z− ��z���z���z�

= limz→1−

�0�−C�−1

(1

1− z�C +D�−D

)z�1− z�2

z− ��z�

× (1− �1− z��′�1−�+ o�1− z�

)(� + o�1�

)= 2

�′′�1−��0�−C�−1

(�C +D��′�1−�+D1

)�� (15)

Let ��z�, 0 < z ≤ 1, be the eigenvalue of C +Dz with the largest real part. Wenote ��1� = 0. Further, because ��z� = ��e��z�V �, 0 < z ≤ 1, and �′�1−� = 1, wehave �′�1−� = ��V�−1. Therefore the relation �C +Dz���z� = ��z���z�, 0 < z ≤ 1,leads to

�C +D��′�1−�+D1 = ��V�−11� (16)

Substituting (16) into (15) and using (14), we complete the proof. �

3. Asymptotics for the Loss Probability in the MAP/G/1/K + 1 Queue

We consider the MAP/G/1/K + 1 queue with � = 1. The second main theoremdescribes the asymptotic behavior of the loss probability P

�K+1�Loss for the

MAP/G/1/K + 1 queue as K tends to infinity:

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164 Kim and Kim

Theorem 2. The loss probability P�K+1�Loss for the MAP/G/1/K + 1 queue satisfies

limK→�

KP�K+1�Loss = �′′�1−�

2� (17)

Remark. By (4), the right-hand side of (17) is finite when ��V 2� < � and infinitewhen ��V 2� = �.

For the MAP/G/1/K + 1 queue with � = 1, the asymptotic behavior ofthe form (17) was first obtained by Baiocchi [2]. He assumed that the servicetime V has a finite exponential moment ��e�V �<� for some � > 0 and theMAP is time reversible in the sense that C = diag���C�diag����−1 and D =diag���D�diag����−1. Under these conditions, he derived (17). Recently, based onnumerical evidence, Kim et al. [6] conjectured that (17) still holds without thetime reversibility of the MAP. Theorem 2 asserts that (17) holds without the timereversibility for the MAP and the exponential moment condition for the servicetime. This implies that the conjecture in Kim et al. [6] is true. The remainder of thissection is mainly devoted to the proof of Theorem 2.

The MAP/G/1/K + 1 queue can be described by a Markov chain�X�K+1�

n � J �K+1�n � with transition probability matrix P�K+1� given by

P�K+1� =

B0 B1 B2 · · · BK−1 BK

A0 A1 A2 · · · AK−1 AK

O A0 A1 · · · AK−2 AK−1

���� � �

� � ����

���

O O O · · · A0 A1

� (18)

where Bn =∑�

i=n Bi and An =∑�

i=n Ai. The stationary distribution of�X�K+1�

n � J �K+1�n � is denoted by ��K+1� = ��

�K+1�0 � � � � � �

�K+1�K � with �

�K+1�i =

���K+1�i1 � �

�K+1�i2 � � � � � �

�K+1�ir �, 0 ≤ i ≤ K.

We now prove a series of lemmas.

Lemma 4. We have

limn→�

1 �n

�n = �� (19)

limminn�K−n→�

1

��K+1�n �

�K+1�n = �� (20)

where · denotes the l1-norm of a vector.

Proof. We prove only (19) because (20) is proved by similar arguments. We observethat the Markov additive process �Xn� Jn� � n ≥ 0 with transition probabilitymatrix (5) is null recurrent and that �� � � � �� �� �� � � � � is a stationary measure. Fix iand j in 1� � � � � r with i �= j. Letting

��n�i� = infl ≥ 1 � �Xl� Jl� = �n� i��

�n = infl ≥ 1 � Xl = n�

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Asymptotics in the MAP/G/1 Queue 165

we have

�nj

�ni

= �( ��n�i�−1∑

k=0

��Xk�Jk�=�n�j�

∣∣ �X0� J0� = �n� i�

)

≥ �( ��n�i�∧�0−1∑

k=0

��Xk�Jk�=�n�j�

∣∣ �X0� J0� = �n� i�

)

= �( ��0�i�∧�−n−1∑

k=0

��Xk�Jk�=�0�j�

∣∣ �X0� J0� = �0� i�)

→ �j

�ias n → ��

Therefore lim infn→��nj�ni

≥ �j�i. Interchanging the roles of i and j, lim infn→�

�ni�nj

≥ �i�j.

Thus we have

limn→�

�nj

�ni

= �j

�i�

which leads to (19). �

Lemma 5. For n ≥ 0,

limK→�

��K+1�K−n = 0�

Proof. We define

��K+1��k�i� = inf

{l ≥ 1 �

(X

�K+1�l � J

�K+1�l

) = �k� i�}�

��K+1�k = inf

{l ≥ 1 � X

�K+1�l = k

}�

Let �Xk� Jk� � k = 0� 1� � � � be a Markov process on � � � �−2�−1� 0× 1� � � � � rwith transition probability matrix

P ≡

· · · −3 −2 −1 0���

−2−10

� � �

� � �� � �

� � ����

A0 A1 A2 A3

A0 A1 A2

A0 A1

�

and

��k�i� = inf{l ≥ 1 �

(Xl� Jl

) = �k� i�}�

�k = inf{l ≥ 1 � Xl = k

}�

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166 Kim and Kim

Then we have

1

��K+1�K−n�i

= �(��K+1��K−n�i�

∣∣ (X�K+1�0 � J

�K+1�0

) = �K − n� i�)

≥ �(��K+1��K−n�i� ∧ �

�K+1�0

∣∣ (X�K+1�0 � J

�K+1�0

) = �K − n� i�)

= �(��−n�i� ∧ �−K

∣∣ (X0� J0) = �−n� i�

)�

which leads to

lim infK→�

1

��K+1�K−n�i

≥ �(��−n�i�

∣∣ (X0� J0) = �−n� i�

)� (21)

Since �Xk� Jk� � k = 0� 1� � � � is null recurrent, the right-hand side of (21) is infinityand the proof is complete. �

Lemma 6. We have

limK→�

( K∑k=0

�k )��K+1�0 = �0�

Proof. It is well known that

�k = �lNlk for l ≥ 0 and k ≥ 0�

where Nlk is the r × r matrix whose �i� j�-component is given by

�Nlk�ij = �( �l−1∑

t=0

��Xt�Jt�=�k�j�

∣∣ �X0� J0� = �l� i�

)�

By the left-skip free property of (3) and (18), we have

��K+1�k = �

�K+1�l Nlk if 0 ≤ k ≤ l�

Let � > 0 be given. According to Lemma 4, there is m such that

�1− ��� ≤ 1 �K−n

�K−n ≤ �1+ ��� if K − n ≥ m�

and

�1− ��� ≤ 1

��K+1�n �

�K+1�n ≤ �1+ ��� if n ≥ m and K − n ≥ m�

If K ≥ 2m, then

��K+1�0∑K−m

k=0

∥∥��K+1�k

∥∥ = ��K+1�K−m NK−m�0∑K−m

k=0

∥∥��K+1�K−m NK−m�k

∥∥=

1∥∥∥��K+1�K−m

∥∥∥��K+1�K−m NK−m�0∑K−m

k=0

∥∥∥ 1∥∥∥��K+1�K−m

∥∥∥��K+1�K−m NK−m�k

∥∥∥

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Asymptotics in the MAP/G/1 Queue 167

≤ 1+ �

1− �

�NK−m�0∑K−mk=0 �NK−m�k

≤(1+ �

1− �

)2 1 �K−m �K−mNK−m�0∑K−m

k=0

∥∥ 1 �K−m �K−mNK−m�k

∥∥=

(1+ �

1− �

)2 �K−mNK−m�0∑K−mk=0

∥∥�K−mNK−m�k

∥∥=

(1+ �

1− �

)2�0∑K−m

k=0 �k �

Similarly, if K ≥ 2m, then

��K+1�0∑K−m

k=0

∥∥��K+1�k

∥∥ ≥(1− �

1+ �

)2�0∑K−m

k=0 �k �

Therefore, if K ≥ 2m, then(1− �

1+ �

)2

�0 ≤∑K−m

k=0 �k ∑K−mk=0

∥∥��K+1�k

∥∥��K+1�0 ≤

(1+ �

1− �

)2

�0� (22)

Theorem 1 and Lemma 5 imply that

limK→�

∑K−mk=0 �k ∑Kk=0 �k

= 1�

and

limK→�

K−m∑k=0

∥∥��K+1�k

∥∥ = limK→�

K∑k=0

∥∥��K+1�k

∥∥ = 1�

respectively. Therefore, (22) leads to, for i = 1� 2� � � � � r,(1− �

1+ �

)2

�0i ≤ lim infk→�

K∑k=0

�k ��K+1�0i

≤ lim supk→�

K∑k=0

�k ��K+1�0i ≤

(1+ �

1− �

)2

�0i�

Letting � → 0+ completes the proof. �

Now we prove Theorem 2.

Proof of Theorem 2. According to Theorem 1 and Lemma 6, we have

limK→�

K��K+1�0 = �′′�1−�

2�V

�0�−C�−11�0�

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168 Kim and Kim

By substituting the above into the following basic formula (see, for example,Equation (9) in [2]):

P�K+1�Loss = �

�K+1�0 �−C�−11

�V + ��K+1�0 �−C�−11

�

we complete the proof. �

References

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Asymptotic theory. Stochastic Models 10:867–893.3. Choi, B.D., Kim, B., and Wee, I.-S. 2000. Asymptotic behavior of loss probability in

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8. Lucantoni, D.M. 1991. New results on the single server queue with a batch Markovianarrival process. Communication Statistics and Stochasic Models 7:1–46.

9. Miyazawa, M., Sakuma, Y., and Yamaguchi, S. 2007. Asymptotic behaviors of the lossprobability for a finite buffer queue with QBD structure. Stochasic Models 23:79–95.

10. Neuts, M.F. 1981. Matrix-Geometric Solutions in Stochastic Models: An AlgorithmicApproach. The Johns Hopkins University Press, Baltimore.

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12. Sgibnev, M.S. 2006. The matrix analogue of the Blackwell renewal theorem on the realline. Sbornik: Mathematics 197:69–86.

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