On bayesian design in finite source queues∗
M. Eugenia Castellanos1 , Javier Morales2 , Asunción M. Mayoral2 , Roland Fried3 ,
and Carmen Armero4
1
2
3
4
Universidad Rey Juan Carlos, Madrid, Spain maria.castellanos@urjc.es
Universidad Miguel Hernández, Elche, Spain j.morales@umh.es,
asun.mayoral@umh.es
Universidad Carlos III, Madrid, Spain rfried@est-econ.uc3m.es
Universitat de València, Spain Carmen.Armero@uv.es
Summary. We develop a Bayesian analysis of queueing systems in applications of
the machine interference problem, like job-shop type systems, telecommunication
traffic, semiconductor manufacturing or transport. Bayesian guarantees of system
performance can be given using the predictive or the posterior distribution. While
these distributions can be obtained by standard Monte Carlo integration in case
of an M/M/c//r queueing system with exponential operational and service times,
more refined Monte Carlo sampling (MCMC) strategies are needed in general.
Key words: machine interference problem, queuing design, Bayesian performance
criteria, MCMC methods
1 Introduction
Queueing models allow to analyze problems which involve congestion. Once the
characteristics of the system have been identified, we can derive conclusions about
its temporal evolution and steady-state behavior, sometimes analytically, mostly by
simulation [K75], [K76]; [GH98]; [LK00]; [M03]. Statistics becomes important in the
presence of uncertainty about the stochastic model. Bayesian inference is well suited
for analyzing queues because we can evaluate posterior and predictive distributions
for system performance measures taking different sources of uncertainties into account. Moreover, the Bayesian approach provides an optimal decision framework for
design purposes since incorporation of losses and costs is straightforward [AB99].
In an M/M/c//r queueing system, we analyze a set of r machines, which are
subject to failures and maintained by c repair crews. The Bayesian framework allows
us to derive performance criteria concerning the availability of the machines, without
needing to specify values for unknown model parameters. The queuing system can
thus be designed choosing a suitable number of repair crews such that a certain
acceptable system performance is achieved, with guarantees optionally in terms of
∗
Research under grants MTM2004-03290; GV05/018; TSI2004-06801-C04-01;
MTM2004-02934.
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the predictive or the posterior distribution. We discuss such criteria and illustrate
them on simulated data.
Section 2 presents the basic model and criteria measuring system performance.
Section 3 develops Bayesian design strategies guaranteeing reliable performance and
exemplifies our proposals. Section 4 presents some extensions of the basic model
needing more refined computational techniques.
2 Problem formulation
2.1 M/M/c//r queueing system
We start with a very simplistic model for illustration. Assume that a company uses r
identical machines for production. The machines break down at random time points
and must be repaired (maintained). The number c of repair crews needs to be chosen
sufficiently large to guarantee a satisfactory availability of the machines. A high
availability can be achieved by a huge choice of c, but for the prize of large costs. We
assume a unique type of maintenance service, discarding different kinds of damages,
inventory limits and transportation times. Section 4 presents some extensions of this
basic setting.
In the language of queueing systems, the machines act as users and the crews
as servers. When an user arrives at the system and finds an idle server, (s)he is
attended immediately. Otherwise the user needs to wait in a queue until a server
becomes available. Simple queueing systems are defined by the arrival pattern, the
service mechanism including the number of servers, the distribution of the service
times, the capacity of the waiting room, the size of the customer population and
the discipline, that is the order in which the users in the queue are selected for
service. The problem as stated before can be modelled as a queueing system with
the following characteristics:
• Arrivals to the system: Machines (users) get periodically into the system
and are declared “non-operative” when stopping for being served.
• Service mechanism: If any of the c servers is idle, an incoming user is attended
immediately; otherwise, (s)he needs to wait in the queue until a server gets idle.
• Finite population: The number r of users is finite.
• Discipline: The order of the queue is FIFO, meaning the user “first in the
system” (stopped to be maintained) is “first out” (starts maintenance).
• Steady state: We assume the system to work under stationarity.
If we assume that the operational times To and the maintenance times Tm vary
independently according to exponential distributions with means 1/λ and 1/µ respectively, the problem becomes an M/M/c//r queueing system. When assuming
To or Tm to follow another distribution not being an exponential, e.g. a gamma, the
first (second) of the M’s is replaced by a G.
A steady-state solution of the queueing system always exists as the population
of users is finite and the size of the queue cannot increase to infinity. The steady
state solution of the number of non-operating users in the system, Nno , in case of
the M/M/c//r queuing system depends on the parameters λ and µ and the number
of crews c [GH98]:
On bayesian design in finite source queues
8
n
>
< (r −r!n)!n! µλ p0 ,
n = 0, 1, . . . , c − 1
n
P (Nno = n|λ, µ) =
λ
r!
>
p0 , n = c, c + 1, . . . , r.
:
n−c
µ
(r − n)!c
P (Nno = 0|λ, µ) =
n=0
(1)
c!
where p0 denotes P (Nno = 0|λ, µ), evaluated from:
" c−1
X
1383
r!
(r − n)!n!
n
λ
µ
+
r
X
n=c
r!
(r − n)!cn−c c!
n #−1
λ
µ
The distribution of the time Tq a machine needs to wait for maintenance is also
known in this model, namely [GH98]:
8 c−1
X (r − n) P (Nno = n | λ, µ)
>
>
>
, if t = 0
>
>
>
r − E(Nno | λ, µ)
< n=0
r−1
P (Tq ≤ t | λ, µ) = X
(r − n) P (Nno = n | λ, µ)
>
Fga (t | n − c + 1, cµ)
>
>
>
r − E(Nno | λ, µ)
>
n=c
>
:
+ P (T = 0 | λ, µ)
if t > 0
(2)
q
Rt
and 0 if t < 0, where Fga (t | a, b) = 0 Ga(s | a, b)ds, t > 0 stands for the
distribution function of a Gamma with expectation a/b and variance a/b2 .
2.2 Reliable maintenance
The number Nno of non-operating users in the system and the time Tq that a machine
has to wait for maintenance are classical congestion measures in queueing systems.
A popular summary measure of system performance in machine repair models is the
operational availability:
Ao =
E(To )
,
E(To ) + E(Tno )
where Tno = Tm + Tq is the non-operational time and To is again the operational
time between consecutive maintenances [KGE98], [RKK00]. It can be seen as a first
order approximation of the expected availability E(A), where
A=
To
.
To + Tno
A queuing system can be designed by determining the minimal c meeting some
demand on the expected availability, or by finding the value of c from which on
the increase of the availability levels out. However, these criteria cannot be applied
directly since the distributions of To and Tno depend on the unknown failure and
service rates λ and µ. Additional statistical reasoning is needed therefore, analyzing
some data to get information on λ and µ and include it into the design. We propose
Bayesian modelling for finding feasible values of c, guaranteeing satisfactory availability with high security, and avoiding a compromise in form of assuming values
for λ and µ. A related design problem in terms of the operational capacity, i.e. the
number of machines working at any time point, has been addressed in [MCMFA05].
For this measure it is also reasonable to consider the probability of fulfillment in
addition to the average fulfillment considered here.
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M. Eugenia Castellanos et al.
3 Bayesian maintenance design
In a Bayesian formulation of a queueing model with parameter uncertainty, the
background knowledge is subsumed into a prior distribution for the unknown λ
and µ. Updating this prior by combination with the likelihood obtained from the
measured data results in the posterior distribution. A predictive analysis integrates
all information available on λ and µ contained in the posterior.
3.1 Bayesian model specification
A Bayesian model for a queueing system is given by two levels:
1. Data level. The data consists of no life (or failure) times {to1 , . . . , tono } and nm
maintenance (or service) times {tm1 , . . . , tmnm }, regarded as independent realizations of random variables To and Tm with distributions Fo and Fm , respectively. In an M/M/c//r queuing system, Fo and Fm are exponential distributions
characterized by their means 1/λ and 1/µ.
2. Prior level. In the absence of prior information, a non-informative Jeffrey’s prior
is assumed with independency between λ and µ, p(λ, µ) = p(λ) · p(µ) ∝
(1/λ) · (1/µ). Otherwise, the natural choice for the prior is a product of
Gamma distributions Ga(·|ko , so ) and Ga(·|km , sm ), with parameters (ko , so )
and (km , sm ) expressing the information available on λ and µ. Jeffrey’s prior
corresponds to degenerate Gamma distributions.
The posterior distribution of λ and µ, obtained from combining the prior with the
likelihood, in case of an M/M/c//r queuing system is given by:
p(λ, µ|data) = Ga(λ|no , tno ) · Ga(µ|nm , tnm ),
Pno
Pnm
(3)
with tno = i=1 toi and tnm = j=1 tmj . In the general case of a G/G/c//r queuing
system, there is not necessarily an analytic expression of the posterior distribution,
so that MCMC techniques are needed for approximation.
All system performance measures like the availability A inherit a posterior
distribution from (4). This means that we do not just have an estimate of, say,
P (A ≥ a|λ, µ) for each a ∈ [0, 1], but a full posterior probability distribution for
it, p(P (A ≥ a | λ, µ) | data), specifying the full information and uncertainty in
the model. Estimations, as well as the confidence in them, can be expressed via
percentiles, standard deviations, or probabilities of ranges.
The predictive steady state distribution of A is obtained from the expectations
of the parameter-dependent steady state solutions P (A ≥ a|λ, µ) with respect to
(w.r.t.) the posterior (4),
Z
P (A ≥ a|data) =
P (A ≥ a|λ, µ) p(λ, µ|data) d(λ, µ) , for a ∈ [0, 1].
(4)
Although there will not always exist an analytic formula of the posterior distribution,
Bayesian inference can be performed using stochastic simulation. We generate M
values {λi , µi }M
i=1 from the posterior (4), and then generate values from To |λi and
Tno |λi , µi . In view of the definition of Tno , we need to simulate from Tm and from
Tq using (2). With the simulated values {(to,i , tno,i )}M
i=1 we get draws from the
predicted availability, ai = to,i /(to,i + tno,i ), and can approximate the predicted
average availability.
On bayesian design in finite source queues
1385
3.2 Bayesian performance criteria for reliable maintenance
In a similar line as [MCMFA05], we can reformulate the demand on the availability A
from Section 2.2 in terms of the mean of the predictive distribution P (A ≥ a|data),
for a given level of availability a ∈ [0, 1], or the posterior distribution of the mean
E(A|λ, µ), p(E(A|λ, µ)|data), as follows:
1. Achieving a predicted mean availability larger than a:
E(A|data) ≥ a.
(5)
The predicted mean availability can easily be approximated averaging values
from the distribution of A given the data, using the above algorithm.
2. Achieving a sufficiently large posterior probability β ∈ [0, 1] that the mean availability
P post [E(A|λ, µ) ≥ a] ≥ β.
(6)
This posterior probability can be approximated by Monte Carlo as follows:
P [E(A|λ, µ) ≥ a|data] ≈
#{i, E(A|λi , µi ) ≥ a}
,
M
where in this expression we additionally need an estimate of the expectation in
the numerator, which can be obtained P
from a sample {aki }K
k=1 for each λi , µi ,
k
approximating the expectation by ai = K
k=1 ai /K.
To determine suitable choices of c we select one of these goals and evaluate the corresponding measure for different values of c. In the predictive goal (5), we average
over all possible λ and µ weighted by their posterior distribution, while in the posterior goal (6) we require that the values of λ and µ for which we achieve reliable
performance have high probability. Predictive criteria are easily comprehensive as
they are based on expected system performance; they are perfectly suitable e.g. if
we design many queuing systems and average performance guarantees are sufficient.
Posterior criteria imply that the selected goal is achieved in most of the systems
designed.
3.3 An example
[DVV04] consider a radar consisting of r = 64 identical pieces, which are subject
to break-down and need to be maintained for the radar to work. Following their
assumptions we use λ = 0.00008 (failures/h) and µ = 0.006 (repairs/h), but these
values are usually not known exactly. We generate 50 exponential draws for both To
and Tm using these values of λ and µ. The goal is a mean availability of 90%.
Figure 1 depicts the predicted mean availability, E(A|data), and the posterior
probability of the mean availability being larger than 0.8 or 0.9, P (E(A) ≥ 0.8|data)
and P (E(A) ≥ 0.9|data), for different c = 1, 2, . . . , 10. Monte Carlo integration was performed generating M = 10.000 values {λi , µi }M
i=1 from the posterior
p(λ, µ|data) = Ga(λ|50, 598252.6)· Ga(µ|50, 16062.42).
We see that the predictive goal is satisfied for c ≥ 3, and for c ≥ 4 there is
almost no further increase of the predicted mean availability. The posterior goal is
more demanding than the predictive goal if we insist on a high posterior probability.
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M. Eugenia Castellanos et al.
Predicted mean availability
Posterior mean availability
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
Larger than 0.8
Larger than 0.9
0.1
0.1
0
0
1
2
3
4
5
6
7
repair crews
8
9
10
1
2
3
4
5
6
7
repair crews
Fig. 1. Predicted mean availability (left) and posterior probability of the mean
availability being greater than 0.8 or 0.9 (right), for several values of c.
We need c ≥ 4 to achieve the posterior goal of an expectation greater than 0.9 with
probability also at least 0.9. When requiring only an expectation greater than 0.8,
c ≥ 3 crews satisfy this goal with probability more than 0.9. A reasonable choice
would thus be c = 4 or c = 3.
We note that the situation in [DVV04] is more complicated. The radar still works
if a few pieces are out of order, and we omit spares. The next section reports further
extensions of the basic model illustrated here.
4 Extensions
All but the simplest modifications of the M/M/c//r queueing system lead to a
severe increase of statistical and computational demands. We briefly outline three
extensions along with possible solutions: the first one maintains the model structure
and the extension only consists in the distribution of the life and the service time. The
second one maintains the same basic queueing model, but the complexity increases
by considering many queuing systems working independently. The third extension
deals with a queueing network with many connected queues. There are many more
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On bayesian design in finite source queues
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interesting extensions of queuing systems: the distribution of the life times can vary
due to aging, there can be patrolling services, spare machines, ancillary operators,
etc.
4.1 G/G/c//r queuing systems
A natural question is to examine the sensitivity of the results w.r.t. the assumption
of exponential life and service times by studying other, more flexible distributions.
Introducing an additional source of uncertainty, we can consider the Erlang distribution, Er(a, b), a popular and flexible distribution in queues which is a particular
Gamma distribution (the first parameter a is an integer). If this model is assumed
for the operational and maintenance times, To ∼ Er(ao , bo ) and Tm ∼ Er(am , bm ),
not only Formulas (1) and (2) describing the steady-state solution and the distribution of the time in queue are no longer valid, but also we need a prior distribution
for the parameters (ao , bo , am , bm ). The resulting posterior distributions can usually
only be handled by MCMC techniques. Consequently, computing the predictive and
posterior criteria, E(A | data) and P [E(A | λ, µ) ≥ a | data] with more general
models needs more sophisticated techniques and computational efforts.
4.2 Independent but quasi-identical queueing systems
This situation is made up by q independent and quasi-identical M/M/ci //ri queueing systems, i = 1, . . . , q. As a typical example of this situation, we consider a
multinational company with several factories at different sites. The factories work
independently and the machines as well as the repair crews have the same characteristics at all sites. The company wants to set up a new factory and uses information
about the reliability of the machines and the quality of the repair teams in the
existing factories to design the new one. Consequently, we assume that the failure rates λ1 , . . . , λq and the maintenance rates µ1 , . . . , µq at the different sites are
random samples from common populations with distributions indexed by some hyperparameters. Bayesian hierarchical models are particularly well suited for dealing
with this multilevel scenario and MCMC methods are required to deduce the system
performance.
4.3 Queueing networks
We consider a system of machines that may break down because of different failures,
requiring different types of repairs. These different repairs share common steps and
follow a fixed protocol that, depending on the type of failure, sends a broken down
machine to a sequence of specialized services. Consequently, the machine waits in
queue for a first service, and after having been attended it waits in another queue
for the next service, and so on. The complexity of such models is due to the fact
that there is an arrival and a service parameter for each queue (node), and the
probabilistic results for the total system are expressed as multivariate distributions.
Sophisticated MCMC techniques and statistical tools are needed for complex queueing networks.
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