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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 generalizes a result of Chao (Oper. Res. Lett. 17 (1995) 47–51). c 2000 Elsevier Science B.V. All rights reserved.
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 eect of dependencies between the service times and the sub-sequent 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 in-terarrival 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]
con-∗Fax: +49-721-608-6057.
E-mail address:[email protected] (A. Muller)
siders a single-server queue, where the joint distribu-tion of the interarrival and service times are from the class of the so-called Wicksell–Kibble bivariate ex-ponential distributions. For this case he derives a re-cursion 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 expo-nential distributions of Marshall–Olkin type. It is the purpose of this paper to extend his result to single-server 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 be-ing 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 denitions
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.
2. Stochastic orders and dependence
The most important notion for positive dependence of bivariate distributions is the so-calledpositive quad-rant dependence(PQD). We say that a bivariate ran-dom vectorX= (X1; X2) is PQD, if
This positive dependence concept compares a bivari-ate distribution with a bivaribivari-ate random vector of in-dependent random variables with the same marginals. This can naturally be generalized to a dependence or-dering, that compares the dependence structure of ar-bitrary bivariate distributions.
Denition 1. LetX;X′be two bivariate random
vec-tors with the same marginals. Then we say thatX′ is
more PQD thanX (writtenX6cX′), if
Remark. (1) This denition can be found, e.g. in [6], where this order relation is also calledconcordance. He also shows that this order relations exhibits all desirable properties of a dependence order. Especially,
X6c X′implies Cov(X
1; 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 thatX6cX′ holds,
if and only if Cov(f(X1); g(X2))6Cov(f(X1′); g(X2′))
for all non-decreasing functionsf; g:R→R. (2) If we only require (1) without assuming that
X andX′ have the same marginals, then we get an
order relation, which is well known aslower 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 X6cY implies Ef(X)6Ef(Y). To do so we need the notion of supermodularity, which we will introduce now.
Denition 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)
for allx1; x2∈Rand all; ¿0.
Iffis twice dierentiable, then it is easy to see that
fis supermodular if and only if
@2 @x1@x2
f(x)¿0 for allx∈R2:
For more details about supermodularity and the cor-responding 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 vectorsX;X′the
following conditions are equivalent: (a) X6cX′;
(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 or-der, writtenX6icxY, ifEf(X)6Ef(Y) holds for all increasing convex functionsf, such that the expecta-tion exists. Our main tool in the next secexpecta-tion will now be the following theorem.
Theorem 4. Let X = (X1; X2) and Y = (Y1; Y2) be two bivariate random vectors.Then X6cY implies Y1−Y26icxX1−X2.
Proof. We will show even more, namely thatEf(X1−
X2)¿Ef(Y1 −Y2) for any convex (not necessarily
−f(x1−x2). Then
(In the second equality we used the substitution z:=
−x2−.) Hencegis supermodular, and thus we can
deduce from Theorem 3 thatX6cYimpliesEg(X) = −Ef(X1−X2)6−Ef(Y1−Y2) =Eg(Y). Since this
holds for any convex functionf, we thus have shown thatY1−Y26icxX1−X2 holds.
3. Main result
We consider a single server queueing system. The interarrival time between customer nand 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
indepen-dent and iindepen-dentically distributed, i.e. there is only de-pendence 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 byWn the waiting time of customern.
Ac-cording to Lindley’s equation we haveWn+1= (Wn+
Sn−Tn)+, wherex+:=max{x;0}andW1= 0.
Now assume that we have another queue with inter-arrival and service times (S′
n; Tn′) with the same
prop-erties, and letW′
n be the corresponding waiting times.
Then we get the following main result.
Theorem 5. If(Sn; Tn)6c(Sn′; Tn′);thenWn′6icxWnfor
alln∈N.
Proof. We proceed by induction. Forn= 1 the asser-tion is clearly true, sinceW1=W1′= 0. Therefore let
pendent. Theorem 4 implies that ′
n6icxn. Since
it is well known that increasing convex order is closed with respect to convolution (see e.g. [9]), this implies W′
n +′n6icxWn +n. Now let f be
an increasing convex function. Then the function
x → f(x+) is also increasing convex, and hence
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 of dependency betweenS andT.
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(−) andX3∼exp(), and
dene
S:= min{X1; X3} and T:= min{X2; X3}:
Then a straightforward calculation shows that the vec-tor (S; T) indeed has the survival function described above. This construction has a natural interpretation as a shock model. For more details about this distri-bution we refer to Barlow and Proschan [1].
Chao [3] claims that the denition of the Marshall– Olkin distribution can be extended to negative de-pendence by allowing ¡0 in the denition of the survival function given in (3). This is not true, how-ever. For ¡0 Eq. (3) doesnotdene a proper sur-vival function. This can be seen as follows: Dene
non-negative function. But family of distributions is increasing in the6c-sense for increasing. This follows immediately from the fact that
@
@F(s; t) = F(s; t)·(s+t−max{s; t})¿0:
Hence in this model the waiting time of any customer is a decreasing function ofin 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 ofX andY. In fact, in this case Y is almost sure an increasing function ofX, namelyY =F−1
2 (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 Sn6stTn, (a condition which is fullled in
most of the typical models of stable queues!), then the Frechet bound has the property that P(Sn6Tn) = 1,
and hence this lower bound forWnis zero, i.e. in this
case the queue is always empty! This is trivial, since
P(Sn6Tn) = 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
or-dered with respect to6c, if and only if they have the same marginals and Cov(X1; X2)6Cov(X1′; X2′). Since
6cis invariant under scale transformations, this can immediately be extended to the bivariate log-normal vectorsY= (eX1;eX2) andY′= (eX′
1;eX2′).
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 withSnandTnhaving the same distributions, but being
independent. A very easy to check sucient condition for a random vector (X1; X2) to be PQD is, thatX2is
stochastically increasing inX1, i.e.P(X2¿ t|X1=s)
is increasing ins for allt. 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 timesof 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 the service timeSn=s, the next customer entering the
system arrives after Tn=PNi=1s Yi time units, where
Y1; Y2; : : :are independent and identically distributed
according to F, and Ns is geometrically distributed
with parameterps. Ifs→psis increasing, thens→
Nsis 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
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