Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol28.Issue1.2001:

Operations Research Letters 27 (2000) 235–241
www.elsevier.com/locate/dsw

Analysis of the M= D= 1-type queue based on an integer-valued
rst-order autoregressive process (
Soohan Ahna; ∗ , Gyemin Leeb , Jongwoo Jeona
a Department
b Department

of Statistics, Seoul National University, Seoul 151-742, South Korea
of Statistics, Gyeongsang National University, 660-701, South Korea

Received 1 November 1999; received in revised form 1 April 2000

Abstract
In this paper, we propose a queueing model based on an integer-valued rst-order autoregressive(INAR(1)) process. We
derive the queue length distribution and its asymptotic decay rate of the proposed model. Also, our numerical study shows that
the new model can be considered as an alternative approach to the well-known MMPP=D=1 queue in terms of performance
c 2000 Elsevier Science B.V. All rights reserved.
and amount of computational work.
Keywords: INAR(1) process; MMPP=D=1 queue; Asymptotic decay rate

1. Introduction
Integrated service communication systems usually
have very complicated input streams which are the
superposition of a number of several tracs. A typical
example is a statistical multiplexer, whose input consists of a superposition of packetized voice sources
together with data trac [7]. These input streams have
an important distinction that the number of packet
arrivals in adjacent time intervals can be highly
correlated, which signicantly aects queueing performance of the system. Thus, a great interest has
recently risen in the modeling of these complicated
( This research was supported by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.
∗ Corresponding author. Fax: +82-2-873-1146.
E-mail address: shahn@stats.snu.ac.kr (S. Ahn).

input streams and in the analysis of the resulting
queueing model.
Within this framework, various input processes
have been studied. A particularly interesting process
is the Markov modulated Poisson process(MMPP),

which is an extended Poisson process. It possesses
important properties which make it suitable for modeling complicated arrival processes. By using the
MMPP as an arrival process, various computer and
communication systems have been analyzed. However, the amount of computational work to analyze the
queueing model based on the MMPP, explodes as the
number of the aggregated input streams increases [3].
Hence, we are interested in an integer-valued rst
order autoregressive (INAR(1)) process introduced by
Al-Osh and Alzaid [1]. This process not only enables
us to appropriately model the correlation of trac
streams but also is analytically tractable. In this paper,

c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 4 7 - X

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S. Ahn et al. / Operations Research Letters 27 (2000) 235–241

we shall provide a computational algorithm for the
queue length distribution of the INAR(1)=D=1 queue,
and derive its asymptotic decay rate using the results
of Chang [2], Glynn and Whitt [4].
This paper is organized as follows. In Section 2,
we review the procedure of computing the stationary
queue length distribution of the MMPP=D=1 queue for
the comparison to the INAR(1)=D=1 queue. In Section
3.1, we review the INAR(1) process and its properties.
Section 3.2 provides a recursive algorithm for computing the queue length distribution of the INAR(1)=D=1
queue. In Section 4, we obtain the asymptotic decay rate of the queue length distribution. Finally, we
present several numerical studies in order to compare the INAR(1)=D=1 queue with the MMPP=D=1
queue.

2. Review: MMPP/D/1 queue
In this section, we introduce, based on Lucantoni
[8], how to compute the stationary queue length distribution of the MMPP=D=1 queue where the deterministic service time is 1 and the number of the states of
the phase of the MMPP arrival process is m.
Let Q(n) and J (n), respectively, denote the number in the system and the phase of the MMPP arrival
process immediately after the nth departure from the

queue. Then, (Q(n); J (n)) is a Markov chain and has a
transition probability matrix P ∗ of the following form.


B0 B1 B2 B3 · · ·
 A0 A1 A2 A3 · · · 



(1)
P∗ = 
 0 A0 A1 A2 · · ·  :
 0
0 A0 A1 · · · 
··· ··· ··· ··· ···

Note that An ’s and Bn ’s are m×m square matrices. Let
also x=(x0 ; x1 ; : : :) be the stationary probability vector
of P ∗ , where xi ; i¿0, are m-vectors and xP ∗ = x.
For computing the stationary probability vector x

of P ∗ , the stochastic matrix G which satises the following nonlinear matrix equation (2), is needed.
G=

∞
X

formula
xn =

x0 B̃n +

n−1
X

xk Ãn+1−k

k=1

!

(I − Ã1 )−1 ;

(3)

where,
Ãn = An + An+1 G + An+2 G 2 + : : : ;
B̃n = Bn + Bn+1 G + Bn+2 G 2 + : : : :

(4)

Lucantoni [8] also provides the procedure of obtaining
the vector x0 . The stationary queue length distribution
(0); (1); : : : can be easily obtained by
(n) = e′ xn ;
where e is the column vector of 1’s.
From Eqs. (2) – (4) and Lucantoni [8], the computation of the stationary queue length distribution
of the MMPP=D=1 queue, requires approximations of
An ; Bn ; Ãn ; B̃n and G and truncations of the innite matrix sequence {Ai } and {Bi } depending on a certain
accuracy. Hence, there are two problems in computation, i.e., the rst is that the computational work explodes as the number of states of the phase of the
MMPP increases [3], and the second is that the speed

of convergence becomes seriously slow under heavy
trac conditions [6].

3. Analysis of INAR(1)/D/1 queue
3.1. INAR(1) process
In this section, we introduce the denition of the
INAR(1) process and describe its properties based on
Al-Osh and Alzaid [1]. We shall rst present the denition of the operator ◦ based on Steutel and Van Harn
[9]. Let X be a non-negative integer-valued random
variable and then for any  ∈ [0; 1] the operator ◦ is
dened by
◦X =

X
X

Zi

(5)

i=1

An G n :

(2)

n=0

Given the matrix G and the vector x0 , the vectors
xn ; n = 1; 2; : : : ; are recursively obtained from the

where Zi is independent of X and is a sequence of
i.i.d. random variables such that
P(Zi = 1) = 1 − P(Zi = 0) = :

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S. Ahn et al. / Operations Research Letters 27 (2000) 235–241

From the denition of the operator ◦, it is clear that

0 ◦ X = 0; 1 ◦ X = X; E( ◦ X ) = E(X ) and for any
 ∈ [0; 1];  ◦ ( ◦ X ) = () ◦ X .
The stationary INAR(1) process {An : n = 1; 2; : : :}
is dened by
An =  ◦ An−1 + n ;

(6)

where {n } is a sequence of i.i.d. Poisson random variables with mean  and n is independent of An−1 . From
now on, we shall call  and n correlation factor and
innovation factor, respectively.
The mean of An is simply E[An ] = E[An−1 ] + .
Since the process {An } is stationary, it should be that
E[An ] =


:
1−

The INAR(1) process dened in (6) can be interpreted as follows: the rst term  ◦ An−1 is the number of packets induced by the trac streams at time

n − 1 and the second term n is the number of packets
arriving from outside at time n. Since the INAR(1)
process includes the Poisson process as a special case,
the INAR(1)=D=1 queue can be considered as an extended M=D=1 queue.
From the denition of An and Qn , we can easily
see that (An ; Qn ) is a two-dimensional discrete-time
Markov chain, whose state space is {(i; j): 06i6j +
1; 06j ¡ ∞}. Let us denote
i; j = lim P(An = i; Qn = j)

For any non-negative integer k, the covariance at lag
k;
(k), is given by
(k) = Cov(An−k ; An ) = k
(0)

(ii) An is the INAR(1) process.
(iii) Qn = (Qn−1 + An − 1)+ , where (x)+ = max{0; x}.

(7)

where
(0) = Var(An ) = =(1 − ). Eq. (7) shows that
the autocorrelation decays exponentially with lag k
and has the same form as the Yule–Walker equation
in the AR(1) process.
Al-Osh and Alzaid [1] describe four techniques for
estimating  and  for a given realization a0 ; : : : ; an .
These are Yule–Walker estimation, conditional least
squares estimation, maximum likelihood estimation
conditional on the initial observation, and the unconditional maximum likelihood estimation.

n→∞

for all possible states and dene
ai; j = P(An = j|An−1 = i)
min(i; j)  
X
i
k (1 − )i−k pj−k ;
=
k

where pk = P(1 = k); k = 0; 1; : : : : Then we obtain the following balance equation for 06i6j +
1; 06j ¡ ∞,
i; j = (0; 0 a0; 0 + 1; 0 a1; 0 )1(i=j=0)
j−i+2

+
3.2. Queue-length distribution of the INAR(1)=D=1
queue
In this section, based on the INAR(1) process, we
propose a new queueing system and derive the queue
length distribution of the proposed queueing model.
Let {n ; n¿0} be a sequence of times when services occur. We assume that the number of arrivals
which can be served at each time n is 1. And we
shall dene An and Qn as number of packets arrived at
the system during the (n−1 ; n ] and the queue length
in the system at time +
n , respectively. Then, a new
queueing model, called the INAR(1)=D=1 queue, can
be dened as follows.
(i) 1 − 0 ; 2 − 1 ; : : : are constant 1 and 0 = 0.

(8)

k=0

X

(9)

k; j−i+1 ak; i ;

k=0

where 1(·) is an indicator function. Let Q denote the
stationary queue length random variable and
(j) ≡ P(Q = j) =

j+1
X

i; j :

i=0

We also dene for r; s ¿ 0
X
X
i; j r i s j
ai; j s j ;
(r; s) =
ai (s) =
j

i; j

and (s) = E[s1 ]. Then, we obtain
ai (s) = (1 −  + s)i (s)

(10)

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S. Ahn et al. / Operations Research Letters 27 (2000) 235–241

of the INAR(1)=D=1 queue. Let

and


(r; s) = 1 −



1
B + s−1 (1 −  + rs; s) (rs);
s

xn ≡ (0; n ; 1; n ; : : : ; n+1; n )′ ;
(11)

where B=0; 0 a0; 0 +1; 0 a1; 0 . From the above equation,
we derive the following theorem which is necessary
for computing the stationary queue-length distribution
of the INAR(1)=D=1 queue. Let the trac intensity
 ≡ E[A1 ] = =(1 − ).
Theorem 1. Under the condition  ¡ 1;
B = 1 − :

(12)

This theorem says that when the queue is empty
after the service is completed, the probability that there
is no arrival before the next service, equals 1 – trac
intensity.

xn(1) ≡ (0; n ; 1; n )′ ;
xn(2)

n = 0; 1; 2; : : : ;

n = 1; 2; : : : ;
′

≡ (2; n ; : : : ; n+1; n ) ;

(13)

n = 1; 2; : : : :

Note that (n) = e′ xn ; n¿0: Dene n × n(n + 3)=2
matrices An ; n = 1; 2; : : : ; by
 ∗

′
an−1; n
0n′
0n−1
· · · · · · · · · 02′
′
∗
′
 0n+1
an−2;
02′ 
n 0n−1 · · · · · · · · ·



..  ;
An ≡  ...
···
··· ··· ··· ···
. 
 ′

′
′
′
∗
 0n+1
0n
0n−1 · · · 04 a1; n 02′ 
′
′
0n+1
0n′
0n−1
· · · · · · 03′ a0;∗ n

(14)

where 0n is the n-dimensional column vector of 0’s
and a∗m; n ; 06m6n−1; n¿1; are (m+2)-dimensional
row vectors dened by

Proof. From Eq. (11), we have
s(r; s) = (s − 1)B + (1 −  + rs; s) (rs):
Dierentiating both sides of the above equation with
respect to s, we obtain
(r; s) + s

@
@
(r; s) = B + (1 −  + rs; s) (rs)
@s
@s
@
(rs):
+(1 −  + rs; s)
@s

It is easily shown that
@
(rs)|(r=1; s=1) = ;
@s

s

@
(r; s)|(r=1; s=1) = E[Q]
@s

∗
am;
n ≡ (a0; n−m+1 ; a1; n−m+1 ; : : : ; am+1; n−m+1 ):

We also dene (n + 2) × (n + 1) matrices Bn ; n =
0; 1; 2; : : : ; as follows


b0; 1
;
(16)
B0 ≡
b1; 1



b0; 1 b0; 2 · · · b0; n+1
Bn ≡  b1; 1 b1; 2 · · · b1; n+1  ;
0n
In
b0; 1 = C(1 − a1; 1 );

@
(1 −  + rs; s)|(r=1; s=1) = E[A1 ] + E[Q]:
@s

b0; k = −C(ak; 0 (1 − a1; 1 ) + ak; 1 a1; 0 );

1 = B + E[1 ] + E[A1 ]:

n = 1; 2; 3; : : : ;

where, In is the n × n identity matrix and

and

Thus, we can obtain the following equation:

(15)

b1; 1 = Ca0; 1 ;

b1; k = −C(ak; 0 a0; 1 − ak; 1 a0; 0 );
C = (a0; 0 − a0; 0 a1; 1 + a1; 0 a0; 1 )

k = 2; 3; : : : ;

k = 2; 3; : : : ;
−1

:

Since E[A1 ] = =(1 − ) in Section 3.1, we derive Eq.
(12). So the theorem is completely proved.

Note that An ’s and Bn ’s are known matrices depending
only on ai; j ’s dened in Eq. (8).
Computation of x0 :
From Eq. (9) and Theorem 1, we obtain

Now, we shall obtain a recursive algorithm which
provides the exact stationary queue length distribution

1 −  = 0; 0 a0; 0 + 1; 0 a1; 0 ;

1; 0 = 0; 0 a0; 1 + 1; 0 a1; 1 :

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S. Ahn et al. / Operations Research Letters 27 (2000) 235–241

Solving this equation, it can be easily seen that
x0 = (0; 0 ; 1; 0 )′ = (1 − ) × B0 :

(17)

Computation of x1 :
From Eq. (9), it can be easily seen that
x1(2) = 2; 1 = A1 x0 :

(18)

Also, Eq. (9) and Theorem 1 yield the following equation:
0; 0 = 1 −  + p0; 1 a0; 0 + 1; 1 a1; 0 + 2; 1 a2; 0 ;
1; 1 = 0; 1 a0; 1 + 1; 1 a1; 1 + 2; 1 a2; 1 :
Solving this equation, we obtain
x1(1) = (0; 1 ; 1; 1 )′

 

b0; 1 b0; 2
b0; 1
(0; 0 ; x1(2) )′ :
+
= (1 − ) ×
b1; 1 b1; 2
b1; 1
(19)
Hence,
x1 = B1 (1 − ; 0)′ + B1



1 02′
0 A1



0; 0
x0



:

(20)

Computation of xn ; n¿2:
Assume that we know xi ; 06i6n − 1. At rst, we
consider xn(2) . Note that Eq. (9) yields
n+1−m; n =

m+1
X

06m6n − 1; n¿2:

k; m ak; n−m+1 ;

k=0

(21)

Hence, it can be easily seen that
′
′
; xn−2
; : : : ; x0′ )′ :
xn(2) = An (xn−1

n+1
X

k; n ak; 1 ;

k=2

a0; 0 0; n + a1; 0 1; n = 0; n−1 −

n+1
X

(23)

k; n ak; 0 :

k=2

Solving Eq. (23), it follows that
"
0; n = C 0; n−1 (1 − a1; 1 )
−

n+1
X
k=2

1; n = C 0; n−1 a0; 1 −

n+1
X

#

k; n {ak; 0 a0; 1 − ak; 1 a0; 0 } :

k=2

(24)

This equation yields



0; n−1
b0; 1 b0; 2 · · · b0; n+1
(1)
:
xn =
b1; 1 b1; 2 · · · b1; n+1
xn(2)
From Eqs. (22) and (25), we can obtain


0; n−1
 x


′
1 0n(n+3)=2
 n−1 
xn = Bn
 ..  :
0n
An
 . 

(25)

(26)

x0

Thus, we can recursively obtain the exact stationary
queue length distribution of the INAR(1)=D=1 queue.
From the result of this section and properties of
the INAR(1) process, the INAR(1)=D=1 queue has
some advantages that the INAR(1) process can model
the correlation of trac streams as the MMPP, that
the stationary queue length distribution can be computed exactly and that the computational work of obtaining the stationary queue length distribution of the
INAR(1)=D=1 queue is very small, and does not increase as the trac intensity becomes high. Furthermore, we shall show by numerical studies in Section
5, that the INAR(1)=D=1 queue can be a good substitute for the MMPP=D=1 queue.

(22)

Now, we consider xn(1) . Using the equations of
0; n−1 and 1; n in (9), we can get the following
equations:
a0; 1 0; n − (1 − a1; 1 )1; n = −

"

#

k; n {ak; 0 (1 − a1; 1 ) + ak; 1 a1; 0 } ;

4. Asymptotic decay rate
In this section, we obtain the asymptotic decay
rate of the stationary queue length distribution of
the INAR(1)=D=1 queueing model. By the results of
Chang [2], and Glynn and Whitt [4], the asymptotic
decay rate can be obtained if the INAR(1) process
satises the large deviation principle(LDP).
Let {An }; n = 1; 2; : : : denote the INAR(1) process
dened in Section 2 and () denote the asymptotic
logarithmic moment generating function dened by
Pn
1
log E[e i=1 Ai ]:
(27)
() = lim
n→∞ n
Now, we will show the LDP for the INAR(1) process.

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S. Ahn et al. / Operations Research Letters 27 (2000) 235–241

Theorem 2. The asymptotic logarithmic moment
generating function ; can be represented by



 (e − 1)
1 − e
() =

∞

if  ¡ − log();

i=1

Ai

otherwise

] = E[(s − s + ss)An −1 s
..
.
n−1
Y
(si );
= 1 (sn )

Pn−2
i=1

Ai

] (s)

(29)
5. Numerical studies

Since (s) = e(s−1) , it follows that
Pn−1
(si ) = e( i=1 si −(n−1))

 
s−1
= exp  (n − 1)
1−a

n−1
a(1 − s)
1−a
:
+
1−a
1−a

(30)

Since the INAR(1) process has a Poisson marginal
distribution with mean =(1 − ), for  ¡ − log(),

n→∞

1
log P(Q ¿ q) = − ∗ ;
q

where  ∗ is the unique positive solution of the equation () = .



s−a
s−a
n−1
:
+a
s−
sn =
1−a
1−a

() = lim

From Chang [2], and Glynn and Whitt [4], the stationary queue-length distribution of the INAR(1)=D=1
has the following asymptotic decay rate:
lim

where sn = s − a + asn−1 ; n¿2; a = s; s1 = s. For
a = s ¡ 1; sn converges and can be represented by
the following equation:

i=1

where  = = − .

q→∞

i=1

n−1
Y

!
p
2x +  − 4x + 2
= x log
2x
p
4x + 2 + 

+ −

2

(28)

Proof. From the property of the INAR(1) process, we
can show that for s ¿ 0,
E[s

∗ (x) = sup{x − ()}
¿0

Pn
and the sequence of random variables { i=1 Ai =n}
satises the LDP with the rate function ∗ which is
the Legendre transformation of ; that is; ∗ (x) =
sup [x − ()].

Pn

with the following rate function

Pn
(e − 1)
1
log E[e i=1 Ai ] =
n
1 − e

and for  ¿ − log(), P
() = ∞. Hence, by the
n
Gartner–Ellis theorem, { i=1 Ai =n} satises the LDP

In this section, we present several numerical studies
in order to compare the stationary queue length distribution of the INAR(1)=D=1 queue with that of the
MMPP=D=1 queue.
For the numerical studies, we consider the
MMPP=D=1 queue whose arrival process is the superposition of several homogeneous 2-state MMPP’s
with the following parameters:




−0:09 0:09
0:18
0
R=
; =
:
0:05 −0:05
0
0:05
We compute the queue length distributions of the
MMPP=D=1 queues for 10 cases where the number of
superpositions of the 2-state MMPP’s varies from 1
to 10.
For the INAR(1)=D=1 queue, we determine the correlation factor  and the innovation factor  such that
both the MMPP=D=1 queue and the INAR(1)=D=1
queue have the same trac intensity and asymptotic
decay rate. To do so, we use results of Kesidis et al.
[5] and Section 4.
Fig. 1 shows that the two queue length distributions
are very similar irrespective of the number of superpositions. On the other hand, the INAR(1)=D=1 queue
requires less amount of computational work than the

S. Ahn et al. / Operations Research Letters 27 (2000) 235–241

241

Fig. 1. Comparison of log-tail queue-length distributions of MMPP=D=1 and INAR(1)=D=1.

MMPP=D=1 queue when the number of superpositions is large. Consequently, the INAR(1)=D=1 queue
can be considered as an alternative approach to the
MMPP=D=1 queue when a number of homogeneous
2-state MMPP’s are superposed.
Acknowledgement
The authors thank the referee for careful reading
of the manuscript and the indications leading to the
improved representations of the recursive algorithm in
Section 3.1.
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