Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol26.Issue1.2000:
Operations Research Letters 26 (2000) 43–47
www.elsevier.com/locate/orms
On the waiting times in queues with dependency between
interarrival and service times
Alfred Muller ∗
Institut fur Wirtschaftstheorie und Operations Research, Universitat Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany
Received 1 November 1998; received in revised form 1 July 1999
Abstract
In this paper a queueing system with partial correlation is considered. We assume that the amount of work (service time)
brought in by a customer and the subsequent interarrival time are dependent. We show that in this case stronger dependence
between interarrival and service times leads to decreasing waiting times in the increasing convex ordering sense. This
c 2000 Elsevier Science B.V. All rights reserved.
generalizes a result of Chao (Oper. Res. Lett. 17 (1995) 47–51).
Keywords: Queues with partial correlation; Monotonic dependence; Increasing convex order
1. Introduction
Most queueing models considered in the literature
assume independence between the service times and
the interarrival times. In practice, however, they will
be dependent. A typical example of such a situation is
the case, when the arrival of a customer with a long
service time discourages the next arrival. Therefore,
the aim of this paper is to investigate the e ect of
dependencies between the service times and the subsequent interarrival times on the waiting times of the
customers.
There are many articles on state-dependent queues,
but most of them assume that the service and=or interarrival times depend on the queue length. There are
only a few papers, which assume a direct dependency
between arrival and service patterns. Hadidi [4,5] conFax: +49-721-608-6057.
E-mail address: mueller@wior.uni-karlsruhe.de (A. Muller)
∗
siders a single-server queue, where the joint distribution of the interarrival and service times are from the
class of the so-called Wicksell–Kibble bivariate exponential distributions. For this case he derives a recursion for the Laplace transform of the waiting time
distribution. Mitchell and Paulson [8] present some
simulation studies that indicate that the waiting times
decrease monotonically with the correlation between
service and arrival times. Chao [3] has shown this
theoretically for the case of the class of bivariate exponential distributions of Marshall–Olkin type. It is the
purpose of this paper to extend his result to singleserver queues with arbitrary joint distributions of the
service times and the subsequent interarrival times. Our
main result will be, that stronger dependence between
the arrival and service patterns (in the sense of being more positive quadrant dependent) leads to shorter
waiting times in the increasing convex ordering sense.
Our paper is organized as follows. In the next
section we will collect the most important de nitions
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 ( 9 9 ) 0 0 0 6 0 - 7
44
A. Muller / Operations Research Letters 26 (2000) 43–47
and facts about stochastic order relations. They will
then be used in Section 3 to proof the main result.
Finally, we will give some examples in Section 4.
order relation, which is well known as lower orthant
ordering, see e.g. [10] for more details and references.
Next, we want to characterize the class of
functions f : R2 → R, for which X 6c Y implies
Ef(X )6Ef(Y ). To do so we need the notion of
supermodularity, which we will introduce now.
2. Stochastic orders and dependence
The most important notion for positive dependence
of bivariate distributions is the so-called positive quadrant dependence (PQD). We say that a bivariate random vector X = (X1 ; X2 ) is PQD, if
P(X1 ¿a1 ; X2 ¿a2 )¿P(X1 ¿a1 )P(X2 ¿a2 )
∀a1 ; a2 ∈ R;
De nition 2. A function f : R2 → R is said to be
supermodular, if
f(x1 + ; x2 + ) − f(x1 + ; x2 )
¿f(x1 ; x2 + ) − f(x1 ; x2 )
(2)
or equivalently, if
for all x1 ; x2 ∈ R and all ; ¿ 0.
P(X1 6a1 ; X2 6a2 )¿P(X1 6a1 )P(X2 6a2 )
∀a1 ; a2 ∈ R:
If f is twice di erentiable, then it is easy to see that
f is supermodular if and only if
This positive dependence concept compares a bivariate distribution with a bivariate random vector of independent random variables with the same marginals.
This can naturally be generalized to a dependence ordering, that compares the dependence structure of arbitrary bivariate distributions.
@2
f(x)¿0
@x1 @x2
De nition 1. Let X ; X ′ be two bivariate random vectors with the same marginals. Then we say that X ′ is
more PQD than X (written X 6c X ′ ), if
P(X1 6a1 ; X2 6a2 )6P(X1′ 6a1 ; X2′ 6a2 )
∀a1 ; a2 ∈ R;
(1)
or equivalently, if
P(X1 ¿ a1 ; X2 ¿ a2 ) 6 P(X1′ ¿ a1 ; X2′ ¿ a2 )
∀a1 ; a2 ∈ R:
Remark. (1) This de nition can be found, e.g. in [6],
where this order relation is also called concordance.
He also shows that this order relations exhibits all
desirable properties of a dependence order. Especially,
X 6c X ′ implies Cov(X1 ; X2 )6Cov(X1′ ; X2′ ), and this
order relation is invariant under scale transformations.
In fact, it is the weakest order relation with these two
properties, since it can be shown that X 6c X ′ holds,
if and only if Cov(f(X1 ); g(X2 ))6Cov(f(X1′ ); g(X2′ ))
for all non-decreasing functions f; g : R → R.
(2) If we only require (1) without assuming that
X and X ′ have the same marginals, then we get an
for all x ∈ R2 :
For more details about supermodularity and the corresponding supermodular stochastic order in arbitrary
dimensions see e.g. [2,7] or [11]. Now, we can state
the following well-known result. It can be found e.g.
in [12].
Theorem 3. For bivariate random vectors X ; X ′ the
following conditions are equivalent:
(a) X 6c X ′ ;
(b) Ef(X )6Ef(X ′ ) for all supermodular functions
f : R2 → R; such that the expectation exists.
For univariate random variables X; Y we say as
usual that they are ordered in increasing convex order, written X 6icx Y , if Ef(X )6Ef(Y ) holds for all
increasing convex functions f, such that the expectation exists. Our main tool in the next section will now
be the following theorem.
Theorem 4. Let X = (X1 ; X2 ) and Y = (Y1 ; Y2 ) be
two bivariate random vectors. Then X 6c Y implies
Y1 − Y2 6icx X1 − X2 .
Proof. We will show even more, namely that Ef(X1 −
X2 )¿Ef(Y1 − Y2 ) for any convex (not necessarily
increasing) function f : R → R. For any convex function f we de ne the bivariate function g(x1 ; x2 ) :=
45
A. Muller / Operations Research Letters 26 (2000) 43–47
−f(x1 − x2 ). Then
g(x1 + ; x2 + ) − g(x1 + ; x2 )
− g(x1 ; x2 + ) + g(x1 ; x2 )
= − f(x1 + − x2 − ) + f(x1 + − x2 )
+ f(x1 − x2 − ) − f(x1 − x2 )
=f(x1 + + z + ) − f(x1 + + z)
− f(x1 + z + ) + f(x1 + z)
us assume that Wn′ 6icx Wn . We de ne n := Sn − Tn
and ′n := Sn′ − Tn′ , respectively. According to our
assumptions n and Wn (resp. ′n and Wn′ ) are independent. Theorem 4 implies that ′n 6icx n . Since
it is well known that increasing convex order is
closed with respect to convolution (see e.g. [9]),
this implies Wn′ + ′n 6icx Wn + n . Now let f be
an increasing convex function. Then the function
x → f(x+ ) is also increasing convex, and hence
we have Ef((Wn′ + ′n )+ )6Ef((Wn + n )+ ), i.e.
′
6icx Wn+1 .
Wn+1
¿0:
(In the second equality we used the substitution z:=
− x2 − .) Hence g is supermodular, and thus we can
deduce from Theorem 3 that X 6c Y implies Eg(X ) =
−Ef(X1 − X2 )6 − Ef(Y1 − Y2 ) = Eg(Y ). Since this
holds for any convex function f, we thus have shown
that Y1 − Y2 6icx X1 − X2 holds.
3. Main result
We consider a single server queueing system. The
interarrival time between customer n and n + 1 will
be denoted by Tn , and Sn shall be the service time
of customer n. The bivariate random vector (Sn ; Tn )
may have an arbitrary bivariate distribution. We only
assume that the vectors (Sn ; Tn ); n ∈ N are independent and identically distributed, i.e. there is only dependence between the service time of a customer and
the interarrival time of the following customer, but no
dependence between interarrival and service times of
other customers. This means especially that the arrival
process is a renewal process.
Denote by Wn the waiting time of customer n. According to Lindley’s equation we have Wn+1 = (Wn +
Sn − Tn )+ , where x+ :=max{x; 0} and W1 = 0.
Now assume that we have another queue with interarrival and service times (Sn′ ; Tn′ ) with the same properties, and let Wn′ be the corresponding waiting times.
Then we get the following main result.
Theorem 5. If (Sn ; Tn )6c (Sn′ ; Tn′ ); then Wn′ 6icx Wn for
all n ∈ N.
Proof. We proceed by induction. For n = 1 the assertion is clearly true, since W1 = W1′ = 0. Therefore let
4. Examples
1. Chao [3] considered the case of the bivariate
exponential distribution of Marshall–Olkin type. His
main result is a special case of Theorem 5. In fact, let
(S; T ) have such a bivariate exponential distribution
with S ∼ exp() and T ∼ exp(). Then the survival
function is given by
t) = P(S ¿ s; T ¿ t)
F(s;
= exp(−( −
)s − ( −
)t −
max{s; t});
(3)
where 06
6min{; } is a parameter for the degree
of dependency between S and T .
It is easy to construct random variables S; T with
this distribution from three independent exponentially
distributed random variables X1 ; X2 ; X3 . In fact, let
X1 ∼ exp( −
); X2 ∼ exp( −
) and X3 ∼ exp(
), and
de ne
S:= min{X1 ; X3 } and
T := min{X2 ; X3 }:
Then a straightforward calculation shows that the vector (S; T ) indeed has the survival function described
above. This construction has a natural interpretation
as a shock model. For more details about this distribution we refer to Barlow and Proschan [1].
Chao [3] claims that the de nition of the Marshall–
Olkin distribution can be extended to negative dependence by allowing
¡ 0 in the de nition of the
survival function given in (3). This is not true, however. For
¡ 0 Eq. (3) does not de ne a proper survival function. This can be seen as follows: De ne
g(x) := P(S6x; T 6x). Then g obviously must be a
46
A. Muller / Operations Research Letters 26 (2000) 43–47
non-negative function. But
0) − F(0;
x) − F(x;
0) + F(x;
x)
g(x) = F(0;
= 1 − e−x − e−x + e−(+−
)x ;
and thus
g′ (x) = e−x + e−x − ( + −
)e−(+−
)x :
Hence limx→0 g′ (x) =
. Since g(0) = 0 this would
imply that g is negative for small values of x, if
¡ 0.
Hence (3) only makes sense for
¿0.
Now, it is easy to see that for xed and this
family of distributions is increasing in the 6c -sense
for increasing
. This follows immediately from the
fact that
@
t) · (s + t − max{s; t})¿0:
F(s; t) = F(s;
@
Hence in this model the waiting time of any customer
is a decreasing function of
in the increasing convex
ordering sense. In Chao [3] this has been shown by a
tedious direct calculation.
2. According to Lorentz’s inequality (see e.g. [12])
there is an upper bound in the set of all bivariate
distributions with xed marginals F1 ; F2 , namely the
so-called upper Frechet bound with the distribution
function
F(x; y) := min{F1 (x); F2 (y)}:
This means that we have a full coupling of X and Y .
In fact, in this case Y is almost sure an increasing
function of X , namely Y = F2−1 (F1 (X )) almost sure.
Thus, we can derive a lower bound for the waiting
times from the Frechet bound. Note however, that if
we have Sn 6st Tn , (a condition which is ful lled in
most of the typical models of stable queues!), then the
Frechet bound has the property that P(Sn 6Tn ) = 1,
and hence this lower bound for Wn is zero, i.e. in this
case the queue is always empty! This is trivial, since
P(Sn 6Tn ) = 1 means that the next arrival occurs after
the end of the service of the present customer with
probability one.
3. According to Slepian’s inequality (see e.g. [13,
p. 8 ]), bivariate normal vectors X and X ′ are ordered with respect to 6c , if and only if they have the
same marginals and Cov(X1 ; X2 )6Cov(X1′ ; X2′ ). Since
6c is invariant under scale transformations, this can
immediately be extended to the bivariate log-normal
′
′
vectors Y = (e X1 ; e X2 ) and Y ′ = (e X1 ; e X2 ).
4. The most important application of Theorem 5
is, that we can nd an upper bound for the waiting
times of any queue with (Sn ; Tn ) PQD by considering
the waiting times of the corresponding GI=GI=1-queue
with Sn and Tn having the same distributions, but being
independent. A very easy to check sucient condition
for a random vector (X1 ; X2 ) to be PQD is, that X2 is
stochastically increasing in X1 , i.e. P(X2 ¿ t|X1 = s)
is increasing in s for all t. In our setting this means
that the distribution of the next interarrival time is
stochastically increasing in the service time of the last
customer. This is a very natural assumption. Let us
consider a typical example, where we have such a
situation. Consider a GI=GI=1-queue with impatient
customers. We assume, that an arriving customer can
observe the service time s of his predecessor, and that
he does not enter the queue with a probability ps ,
where ps is an increasing function of s. Let F be
the distribution of the interarrival times. Then, given
next customer entering the
the service time Sn = s, the P
Ns
Yi time units, where
system arrives after Tn = i=1
Y1 ; Y2 ; : : : are independent and identically distributed
according to F, and Ns is geometrically distributed
with parameter ps . If s → ps is increasing, then s →
Ns is stochastically increasing, and thus it follows from
Shaked and Shanthikumar [10, Theorem 1. A.4] that
Tn is stochastically increasing in Sn , and hence the
vector (Sn ; Tn ) is PQD in this case.
References
[1] R.E. Barlow, F. Proschan, Statistical Theory of Reliability
and Life Testing: Probability Models, Holt, Rinehart and
Winston, New York, 1975.
[2] N. Bauerle, Inequalities for stochastic models via
supermodular orderings, Commun. Statist. — Stochastic
Models 13 (1997) 181–201.
[3] X. Chao, Monotone e ect of dependency between interarrival
and service times in a simple queueing system, Oper. Res.
Lett. 17 (1995) 47–51.
[4] N. Hadidi, Queues with partial correlation, SIAM J. Appl.
Math. 40 (1981) 467– 475.
[5] N. Hadidi, Further results on queues with partial correlation,
Oper. Res. 33 (1985) 203–209.
[6] H. Joe, Multivariate Models and Dependence Concepts,
Chapman and Hall, London, 1997.
[7] A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization
and its Applications, Academic Press, New York, 1979.
[8] C.R. Mitchell, A.S. Paulson, M=M=1-queues with interdependent arrival and service processes, Naval Res. Logistics
Quart. 40 (1979) 467– 475.
A. Muller / Operations Research Letters 26 (2000) 43–47
[9] A. Muller, Stochastic orders generated by integrals: a uni ed
study, Adv. Appl. Probab. 29 (1997) 414 – 428.
[10] M. Shaked, J.G. Shanthikumar, Stochastic Orders and their
Applications, Academic Press, London, 1994.
[11] M. Shaked, J.G. Shanthikumar, Supermodular stochastic
orders and positive dependence of random vectors,
J. Multivariate Anal. 61 (1997) 86 –101.
47
[12] A.H. Tchen, Inequalities for distributions with given
marginals, Ann. Probab. 8 (1980) 814 –827.
[13] Y.L. Tong, Probability Inequalities in Multivariate
Distributions, Academic Press, New York, 1980.
www.elsevier.com/locate/orms
On the waiting times in queues with dependency between
interarrival and service times
Alfred Muller ∗
Institut fur Wirtschaftstheorie und Operations Research, Universitat Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany
Received 1 November 1998; received in revised form 1 July 1999
Abstract
In this paper a queueing system with partial correlation is considered. We assume that the amount of work (service time)
brought in by a customer and the subsequent interarrival time are dependent. We show that in this case stronger dependence
between interarrival and service times leads to decreasing waiting times in the increasing convex ordering sense. This
c 2000 Elsevier Science B.V. All rights reserved.
generalizes a result of Chao (Oper. Res. Lett. 17 (1995) 47–51).
Keywords: Queues with partial correlation; Monotonic dependence; Increasing convex order
1. Introduction
Most queueing models considered in the literature
assume independence between the service times and
the interarrival times. In practice, however, they will
be dependent. A typical example of such a situation is
the case, when the arrival of a customer with a long
service time discourages the next arrival. Therefore,
the aim of this paper is to investigate the e ect of
dependencies between the service times and the subsequent interarrival times on the waiting times of the
customers.
There are many articles on state-dependent queues,
but most of them assume that the service and=or interarrival times depend on the queue length. There are
only a few papers, which assume a direct dependency
between arrival and service patterns. Hadidi [4,5] conFax: +49-721-608-6057.
E-mail address: mueller@wior.uni-karlsruhe.de (A. Muller)
∗
siders a single-server queue, where the joint distribution of the interarrival and service times are from the
class of the so-called Wicksell–Kibble bivariate exponential distributions. For this case he derives a recursion for the Laplace transform of the waiting time
distribution. Mitchell and Paulson [8] present some
simulation studies that indicate that the waiting times
decrease monotonically with the correlation between
service and arrival times. Chao [3] has shown this
theoretically for the case of the class of bivariate exponential distributions of Marshall–Olkin type. It is the
purpose of this paper to extend his result to singleserver queues with arbitrary joint distributions of the
service times and the subsequent interarrival times. Our
main result will be, that stronger dependence between
the arrival and service patterns (in the sense of being more positive quadrant dependent) leads to shorter
waiting times in the increasing convex ordering sense.
Our paper is organized as follows. In the next
section we will collect the most important de nitions
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 ( 9 9 ) 0 0 0 6 0 - 7
44
A. Muller / Operations Research Letters 26 (2000) 43–47
and facts about stochastic order relations. They will
then be used in Section 3 to proof the main result.
Finally, we will give some examples in Section 4.
order relation, which is well known as lower orthant
ordering, see e.g. [10] for more details and references.
Next, we want to characterize the class of
functions f : R2 → R, for which X 6c Y implies
Ef(X )6Ef(Y ). To do so we need the notion of
supermodularity, which we will introduce now.
2. Stochastic orders and dependence
The most important notion for positive dependence
of bivariate distributions is the so-called positive quadrant dependence (PQD). We say that a bivariate random vector X = (X1 ; X2 ) is PQD, if
P(X1 ¿a1 ; X2 ¿a2 )¿P(X1 ¿a1 )P(X2 ¿a2 )
∀a1 ; a2 ∈ R;
De nition 2. A function f : R2 → R is said to be
supermodular, if
f(x1 + ; x2 + ) − f(x1 + ; x2 )
¿f(x1 ; x2 + ) − f(x1 ; x2 )
(2)
or equivalently, if
for all x1 ; x2 ∈ R and all ; ¿ 0.
P(X1 6a1 ; X2 6a2 )¿P(X1 6a1 )P(X2 6a2 )
∀a1 ; a2 ∈ R:
If f is twice di erentiable, then it is easy to see that
f is supermodular if and only if
This positive dependence concept compares a bivariate distribution with a bivariate random vector of independent random variables with the same marginals.
This can naturally be generalized to a dependence ordering, that compares the dependence structure of arbitrary bivariate distributions.
@2
f(x)¿0
@x1 @x2
De nition 1. Let X ; X ′ be two bivariate random vectors with the same marginals. Then we say that X ′ is
more PQD than X (written X 6c X ′ ), if
P(X1 6a1 ; X2 6a2 )6P(X1′ 6a1 ; X2′ 6a2 )
∀a1 ; a2 ∈ R;
(1)
or equivalently, if
P(X1 ¿ a1 ; X2 ¿ a2 ) 6 P(X1′ ¿ a1 ; X2′ ¿ a2 )
∀a1 ; a2 ∈ R:
Remark. (1) This de nition can be found, e.g. in [6],
where this order relation is also called concordance.
He also shows that this order relations exhibits all
desirable properties of a dependence order. Especially,
X 6c X ′ implies Cov(X1 ; X2 )6Cov(X1′ ; X2′ ), and this
order relation is invariant under scale transformations.
In fact, it is the weakest order relation with these two
properties, since it can be shown that X 6c X ′ holds,
if and only if Cov(f(X1 ); g(X2 ))6Cov(f(X1′ ); g(X2′ ))
for all non-decreasing functions f; g : R → R.
(2) If we only require (1) without assuming that
X and X ′ have the same marginals, then we get an
for all x ∈ R2 :
For more details about supermodularity and the corresponding supermodular stochastic order in arbitrary
dimensions see e.g. [2,7] or [11]. Now, we can state
the following well-known result. It can be found e.g.
in [12].
Theorem 3. For bivariate random vectors X ; X ′ the
following conditions are equivalent:
(a) X 6c X ′ ;
(b) Ef(X )6Ef(X ′ ) for all supermodular functions
f : R2 → R; such that the expectation exists.
For univariate random variables X; Y we say as
usual that they are ordered in increasing convex order, written X 6icx Y , if Ef(X )6Ef(Y ) holds for all
increasing convex functions f, such that the expectation exists. Our main tool in the next section will now
be the following theorem.
Theorem 4. Let X = (X1 ; X2 ) and Y = (Y1 ; Y2 ) be
two bivariate random vectors. Then X 6c Y implies
Y1 − Y2 6icx X1 − X2 .
Proof. We will show even more, namely that Ef(X1 −
X2 )¿Ef(Y1 − Y2 ) for any convex (not necessarily
increasing) function f : R → R. For any convex function f we de ne the bivariate function g(x1 ; x2 ) :=
45
A. Muller / Operations Research Letters 26 (2000) 43–47
−f(x1 − x2 ). Then
g(x1 + ; x2 + ) − g(x1 + ; x2 )
− g(x1 ; x2 + ) + g(x1 ; x2 )
= − f(x1 + − x2 − ) + f(x1 + − x2 )
+ f(x1 − x2 − ) − f(x1 − x2 )
=f(x1 + + z + ) − f(x1 + + z)
− f(x1 + z + ) + f(x1 + z)
us assume that Wn′ 6icx Wn . We de ne n := Sn − Tn
and ′n := Sn′ − Tn′ , respectively. According to our
assumptions n and Wn (resp. ′n and Wn′ ) are independent. Theorem 4 implies that ′n 6icx n . Since
it is well known that increasing convex order is
closed with respect to convolution (see e.g. [9]),
this implies Wn′ + ′n 6icx Wn + n . Now let f be
an increasing convex function. Then the function
x → f(x+ ) is also increasing convex, and hence
we have Ef((Wn′ + ′n )+ )6Ef((Wn + n )+ ), i.e.
′
6icx Wn+1 .
Wn+1
¿0:
(In the second equality we used the substitution z:=
− x2 − .) Hence g is supermodular, and thus we can
deduce from Theorem 3 that X 6c Y implies Eg(X ) =
−Ef(X1 − X2 )6 − Ef(Y1 − Y2 ) = Eg(Y ). Since this
holds for any convex function f, we thus have shown
that Y1 − Y2 6icx X1 − X2 holds.
3. Main result
We consider a single server queueing system. The
interarrival time between customer n and n + 1 will
be denoted by Tn , and Sn shall be the service time
of customer n. The bivariate random vector (Sn ; Tn )
may have an arbitrary bivariate distribution. We only
assume that the vectors (Sn ; Tn ); n ∈ N are independent and identically distributed, i.e. there is only dependence between the service time of a customer and
the interarrival time of the following customer, but no
dependence between interarrival and service times of
other customers. This means especially that the arrival
process is a renewal process.
Denote by Wn the waiting time of customer n. According to Lindley’s equation we have Wn+1 = (Wn +
Sn − Tn )+ , where x+ :=max{x; 0} and W1 = 0.
Now assume that we have another queue with interarrival and service times (Sn′ ; Tn′ ) with the same properties, and let Wn′ be the corresponding waiting times.
Then we get the following main result.
Theorem 5. If (Sn ; Tn )6c (Sn′ ; Tn′ ); then Wn′ 6icx Wn for
all n ∈ N.
Proof. We proceed by induction. For n = 1 the assertion is clearly true, since W1 = W1′ = 0. Therefore let
4. Examples
1. Chao [3] considered the case of the bivariate
exponential distribution of Marshall–Olkin type. His
main result is a special case of Theorem 5. In fact, let
(S; T ) have such a bivariate exponential distribution
with S ∼ exp() and T ∼ exp(). Then the survival
function is given by
t) = P(S ¿ s; T ¿ t)
F(s;
= exp(−( −
)s − ( −
)t −
max{s; t});
(3)
where 06
6min{; } is a parameter for the degree
of dependency between S and T .
It is easy to construct random variables S; T with
this distribution from three independent exponentially
distributed random variables X1 ; X2 ; X3 . In fact, let
X1 ∼ exp( −
); X2 ∼ exp( −
) and X3 ∼ exp(
), and
de ne
S:= min{X1 ; X3 } and
T := min{X2 ; X3 }:
Then a straightforward calculation shows that the vector (S; T ) indeed has the survival function described
above. This construction has a natural interpretation
as a shock model. For more details about this distribution we refer to Barlow and Proschan [1].
Chao [3] claims that the de nition of the Marshall–
Olkin distribution can be extended to negative dependence by allowing
¡ 0 in the de nition of the
survival function given in (3). This is not true, however. For
¡ 0 Eq. (3) does not de ne a proper survival function. This can be seen as follows: De ne
g(x) := P(S6x; T 6x). Then g obviously must be a
46
A. Muller / Operations Research Letters 26 (2000) 43–47
non-negative function. But
0) − F(0;
x) − F(x;
0) + F(x;
x)
g(x) = F(0;
= 1 − e−x − e−x + e−(+−
)x ;
and thus
g′ (x) = e−x + e−x − ( + −
)e−(+−
)x :
Hence limx→0 g′ (x) =
. Since g(0) = 0 this would
imply that g is negative for small values of x, if
¡ 0.
Hence (3) only makes sense for
¿0.
Now, it is easy to see that for xed and this
family of distributions is increasing in the 6c -sense
for increasing
. This follows immediately from the
fact that
@
t) · (s + t − max{s; t})¿0:
F(s; t) = F(s;
@
Hence in this model the waiting time of any customer
is a decreasing function of
in the increasing convex
ordering sense. In Chao [3] this has been shown by a
tedious direct calculation.
2. According to Lorentz’s inequality (see e.g. [12])
there is an upper bound in the set of all bivariate
distributions with xed marginals F1 ; F2 , namely the
so-called upper Frechet bound with the distribution
function
F(x; y) := min{F1 (x); F2 (y)}:
This means that we have a full coupling of X and Y .
In fact, in this case Y is almost sure an increasing
function of X , namely Y = F2−1 (F1 (X )) almost sure.
Thus, we can derive a lower bound for the waiting
times from the Frechet bound. Note however, that if
we have Sn 6st Tn , (a condition which is ful lled in
most of the typical models of stable queues!), then the
Frechet bound has the property that P(Sn 6Tn ) = 1,
and hence this lower bound for Wn is zero, i.e. in this
case the queue is always empty! This is trivial, since
P(Sn 6Tn ) = 1 means that the next arrival occurs after
the end of the service of the present customer with
probability one.
3. According to Slepian’s inequality (see e.g. [13,
p. 8 ]), bivariate normal vectors X and X ′ are ordered with respect to 6c , if and only if they have the
same marginals and Cov(X1 ; X2 )6Cov(X1′ ; X2′ ). Since
6c is invariant under scale transformations, this can
immediately be extended to the bivariate log-normal
′
′
vectors Y = (e X1 ; e X2 ) and Y ′ = (e X1 ; e X2 ).
4. The most important application of Theorem 5
is, that we can nd an upper bound for the waiting
times of any queue with (Sn ; Tn ) PQD by considering
the waiting times of the corresponding GI=GI=1-queue
with Sn and Tn having the same distributions, but being
independent. A very easy to check sucient condition
for a random vector (X1 ; X2 ) to be PQD is, that X2 is
stochastically increasing in X1 , i.e. P(X2 ¿ t|X1 = s)
is increasing in s for all t. In our setting this means
that the distribution of the next interarrival time is
stochastically increasing in the service time of the last
customer. This is a very natural assumption. Let us
consider a typical example, where we have such a
situation. Consider a GI=GI=1-queue with impatient
customers. We assume, that an arriving customer can
observe the service time s of his predecessor, and that
he does not enter the queue with a probability ps ,
where ps is an increasing function of s. Let F be
the distribution of the interarrival times. Then, given
next customer entering the
the service time Sn = s, the P
Ns
Yi time units, where
system arrives after Tn = i=1
Y1 ; Y2 ; : : : are independent and identically distributed
according to F, and Ns is geometrically distributed
with parameter ps . If s → ps is increasing, then s →
Ns is stochastically increasing, and thus it follows from
Shaked and Shanthikumar [10, Theorem 1. A.4] that
Tn is stochastically increasing in Sn , and hence the
vector (Sn ; Tn ) is PQD in this case.
References
[1] R.E. Barlow, F. Proschan, Statistical Theory of Reliability
and Life Testing: Probability Models, Holt, Rinehart and
Winston, New York, 1975.
[2] N. Bauerle, Inequalities for stochastic models via
supermodular orderings, Commun. Statist. — Stochastic
Models 13 (1997) 181–201.
[3] X. Chao, Monotone e ect of dependency between interarrival
and service times in a simple queueing system, Oper. Res.
Lett. 17 (1995) 47–51.
[4] N. Hadidi, Queues with partial correlation, SIAM J. Appl.
Math. 40 (1981) 467– 475.
[5] N. Hadidi, Further results on queues with partial correlation,
Oper. Res. 33 (1985) 203–209.
[6] H. Joe, Multivariate Models and Dependence Concepts,
Chapman and Hall, London, 1997.
[7] A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization
and its Applications, Academic Press, New York, 1979.
[8] C.R. Mitchell, A.S. Paulson, M=M=1-queues with interdependent arrival and service processes, Naval Res. Logistics
Quart. 40 (1979) 467– 475.
A. Muller / Operations Research Letters 26 (2000) 43–47
[9] A. Muller, Stochastic orders generated by integrals: a uni ed
study, Adv. Appl. Probab. 29 (1997) 414 – 428.
[10] M. Shaked, J.G. Shanthikumar, Stochastic Orders and their
Applications, Academic Press, London, 1994.
[11] M. Shaked, J.G. Shanthikumar, Supermodular stochastic
orders and positive dependence of random vectors,
J. Multivariate Anal. 61 (1997) 86 –101.
47
[12] A.H. Tchen, Inequalities for distributions with given
marginals, Ann. Probab. 8 (1980) 814 –827.
[13] Y.L. Tong, Probability Inequalities in Multivariate
Distributions, Academic Press, New York, 1980.