![](bg2.png)
systems are notoriously difficult to analyse as the
k-limited service discipline does not satisfy the so-
called branching property; see (Resing, 1993). In our
case, we have the added difficulty of an additional
preparation phase before service. Traditional approx-
imation methods seem to be of little help. For exam-
ple, heavy traffic approximations for polling systems
seem to be mainly suitable for the study of charac-
teristics of the customers, as is typical in the polling
literature, rather than the server, as is our case here;
cf. (van der Mei, 2007). Here, we assume that there
are always waiting customers in front of each station.
Therefore, the analysis of the model is parallel to the
study of the server of a polling-type system, in which
each of the queues is overloaded. As such, heavy traf-
fic diffusion approximations cannot be utilised for this
system, since we consider a system that is overloaded,
rather than critically loaded, while fluid approxima-
tions are equally not straightforward due to the addi-
tional preparation phase.
This model is a layered network in which a server,
while executing a service, may request a higher-layer
service and wait for it to be completed. Layered queu-
ing networks occur naturally in all kinds of informa-
tion and e-commerce systems, grid systems, and real-
time systems such as telecom switches; see (Franks
et al., 2009) and references therein for an overview.
Layered queues are characterised by simultaneous or
separate phases where entities are no longer classified
in the traditional roles of “servers” and “customers”,
but may have also a dual role of being either a server
to other entities (of lower layers) or a customer to
higher-layer entities. Think of a peer-to-peer network,
where users are both customers when downloading a
file, but also servers to users who download their files.
For our system, one may view the preparation time
of a customer as a first phase of service. The ser-
vice station (lower layer) acts in this case as a server.
However, the second phase of service (the actual op-
eration) does not necessarily follow immediately. The
service station might have to ‘wait’ for the server to
finish working on other stations. At this stage, the
service stations act as customers waiting to be served
by the higher layer, the server. Thus, we see that
each service station acts both as a ‘server’ (preparing
the customer) and as a ‘customer’ (waiting until the
server completes his tasks in the previous stations).
This model leads to a Lindley-type equation,
which for two stations leads to the equation (in its
steady-state form) as W
D
=
B − A − W
+
. Here, B
denotes the preparation time, A denotes service time
and W is the waiting time of the server. The dif-
ference from the original (Lindley, 1952) equation
is the minus sign in front of W at the right-hand
side of the equation, which in Lindley’s equation is
a plus. Lindley’s equation describes the waiting time
of a customer in a single-server queue. It is one of
the fundamental and best-studied equations in queu-
ing theory. For a detailed study on Lindley’s equa-
tion we refer to (Asmussen, 2003; Cohen, 1982) and
the references therein. The implications of this “mi-
nor” difference in sign are rather far-reaching, since
even for two stations, in the particular case we study
in this paper, Lindley’s equation has a simple solu-
tion, while for our equation it is probably not pos-
sible to derive an explicit expression without making
some additional assumptions. In the applied probabil-
ity literature, there has been considerable interest in
the class of Markov chains described by the recursion
W
n+1
= g(W
n
,X
n
). An important result is the duality
theory by (Asmussen and Sigman, 1996), relating the
steady-state distribution to a ruin probability associ-
ated with a risk process. See also (Borovkov, 1998)
and (Kalashnikov, 2002). However, duality does not
hold in our case, as our function is non-increasing in
its main argument. This fact produces some surpris-
ing results when analysing the equation.
We study the waiting time of the server for this
model. The waiting time satisfies the Lindley-type re-
cursion (2), which surprisingly emerges when study-
ing maximum weight independent sets in sparse ran-
dom graphs. Specifically, consider an n-node sparse
random (potentially regular) graph and let the nodes
of the graph be equipped with nonnegative weights,
independently generated according to some common
distribution. Rather than only the size of the max-
imum independent set, consider also the maximum
weight of an independent set. (Gamarnik et al., 2006)
show that for certain weight distributions, a limiting
result can be proven both for the maximum indepen-
dent set and the maximum weight independent set.
What is crucial in this computation is recursion (2);
cf. (Gamarnik et al., 2006, Eq. (3)). This recursion
provides another surprising link between queuing the-
ory and random graphs.
At a glance, other than the analytical results, the
major insights we gain for this system are summarised
as follows. First, we observe that variability in prepa-
ration times has a greater influence on the system than
that of service times. In the healthcare setting, one
could summarise it as follows: it pays more to have
a reliable nurse than a reliable specialist. See Fig-
ure 1 for an illustration. Second, a small variability of
preparation times actually improves the performance
of the server, in the sense that he waits less frequently;
cf. Figure 2. However, it also decreases the through-
put. Thus, the system’s designer may wish to consider
how to balance these conflicting goals. Next, when
Cyclic-typePollingModelswithPreparationTimes
213