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Bayesian design in queues: An application to aeronautic maintenance
Journal of Statistical Planning and Inference 137 (2007) 3058 – 3067 www.elsevier.com/locate/jspi
Bayesian design in queues: An application to aeronautic maintenance夡 Javier Moralesa , M. Eugenia Castellanosb , Asunción M. Mayorala , Roland Friedc , Carmen Armerod,∗ a Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain b Departamento de Estadística e Investigación Operativa, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain c Departamento de Estadística, Universidad Carlos III de Madrid, 28903 Getafe, Madrid, Spain d Department d’Estadística i Investigacio Operativa, Universitat de València, 46100 Burjassot, València, Spain
Received 11 October 2005; accepted 30 June 2006 Available online 15 March 2007
Abstract We exploit Bayesian criteria for designing M/M/c//r queueing systems with spares. For illustration of our approach we use a real problem from aeronautic maintenance, where the numbers of repair crews and spare planes must be sufficiently large to meet the necessary operational capacity. Bayesian guarantees for this to happen can be given using predictive or posterior distributions. © 2007 Elsevier B.V. All rights reserved. MSC: 62F15; 62K05 Keywords: M/M/c//r queueing system with spares; Operative performance criteria; Predictive distribution; Posterior distribution
1. Introduction Maintenance scheduling and machine repair times are crucial in all industrial and business branches which depend on the correct performance of a set of machines. Inventory problems require quick replacement of broken down machines to guarantee continuous functioning. Optimal planning of resources becomes essential then to assure some operational performance level. We develop a Bayesian analysis of a queueing system with a set of r machines which are subject to failures. The machines are maintained by c repair crews and can be replaced by s spare machines when broken down. This problem can be modeled as an M/M/c//r queueing system with s spares. This is a finite source queueing model and is commonly referred to as machine interference problem (MIP from now on), a particular case of closed queueing networks (Medhi, 2003) in the literature. Some interesting papers in this context are Iravani et al. (2000) in manufacturing, Almási and Sztrik (1998) in telecommunication, Erkip (1993) in transport and Schoemig (1999) in semiconductor manufacturing. The Bayesian framework allows us to derive natural performance criteria in terms of the number of working machines. 夡
Research under Grants MTM2004-03290; GV05/018; TSI2004-06801-C04-01; MTM2004-02934.
∗ Corresponding author. Tel.: +34963544309; fax: +34963543238.
E-mail address: [email protected] (C. Armero). 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2006.06.046
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The queuing system can be designed by choosing suitable numbers of repair crews and spares such that a certain acceptable global performance is achieved, with guarantees optionally assessed by predictive or posterior distributions. We focus the exposition on maintenance problems in aeronautics (Kreimer and Mehrez, 1998; Ghobbar and Friend, 2002; Jain et al., 2004; Kilpi and Vepsäläinen, 2004; Sleptchenko et al., 2005; de Smidt-Destombes et al., 2005; Marseguerra et al., 2005), but the underlying ideas are much more general. An airline needs a sufficient number of available planes ready to operate at any time. Non-predictable failures force stops at the hangars for repair but the existence of spare planes reduces non-operativity. For easier comprehension of our proposals we only consider a single type of failure: generalization to several types of failures is just an issue of model complexity but does not affect the appropriateness of the proposed criteria. This paper constitutes a first approach to queues within the MIP from a Bayesian perspective. Most statistical papers on queueing systems are in the frequentist framework, but there has been an increasing interest in Bayesian methods recently: M/M/c queues are discussed by Armero and Bayarri (1996), M/G/c with phase type service are examined by Ausín et al. (2004), bulk queues are considered by Armero and Conesa (2000), and non-parametric Bayesian models in discrete-time queues are studied by Conti (1999). See Armero and Bayarri (1999) for a review of basic analyses and an exhaustive list of Bayesian references. The paper is organized as follows. Section 2 presents a real problem from aeronautic maintenance and states the general model. Section 3 develops different Bayesian criteria for identification of reliable queueing designs. Section 4 illustrates our proposals on a real problem studied by Rodrigues et al. (2000). Finally, Section 5 provides some conclusions and proposes extensions for a full decision-theoretical approach to queueing design based on utilities.
2. An aeronautic maintenance problem and general setting 2.1. A queueing problem from aeronautic maintenance We examine a simplified version of a real aeronautic problem regarding the A-4 fleet maintenance of the Argentine Air Force and the Brazilian Navy. Rodrigues et al. (2000) studied the impact of consolidating aviation component spare inventory management and reducing transportation cycle times. Major repair work had to be sent to the manufacturers in the United States and the resulting long repair cycle times adversely affected operational readiness. We investigate the problem of quantifying system performance for an inventory/repair facility created in situ to repair and provide spare parts to planes from both Forces. With some small simplifications to clarify our proposals, it is straightforward to identify the following queueing system: • Arrivals: Planes, or precisely their mechanical components, fail randomly over time. The failure times are assumed to follow an exponential distribution. Remember that for simplicity we consider a single type of failure and thus a single type of component for inventory. • Service mechanism: Service means repairing a broken part. The servers are the c repair crews. If one of the repair crews is idle, a broken part is repaired immediately; otherwise, it needs to wait in a queue until a crew gets idle. The repair times are assumed to be exponentially distributed. • Finite population: A certain number r of operating planes is always required. The Argentine Air Force had 30 aircraft and the Brazilian Navy 20, so r = 50. Moreover, there are s spare parts. Whenever one of the r operating planes fails, the broken part is replaced by a spare if any is available. In this case we assume that the plane becomes operational again immediately, i.e. that removal and installation times of broken/spare parts are negligible. The spare parts can be regarded as spare planes, or briefly spares, and consequently, the potential number of users is r + s. • Discipline: The order of the waiting queue is FIFO, meaning the part “first in the system” (broken down) is “first out” (starts maintenance). If we assume that the operational times To and the maintenance times Tm are independent and follow exponential distributions with mean 1/ and 1/, respectively, the problem becomes an M/M/c//r queueing system with s spares.
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2.2. Steady-state solution The steady-state solution of a queue (if it exists) is usually expressed via the number of users in the system. In our situation, this corresponds to the number of broken planes Nno and always exists since the population is finite. For given numbers of maintenance crews c and spares s, the stationary distribution of Nno depends on and and reads as follows (Gross and Harris, 1998): • If c s,
• If c > s,
⎧ r n n ⎪ n = 0, 1, . . . , c − 1, ⎪ ⎪ n! p0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ r n n P (Nno = n|, ) = p0 , n = c, c + 1, . . . , s − 1, ⎪ cn−c c! ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ r s r! ⎪ ⎩ p0 , n = s, s + 1, . . . , s + r. (r − n + s)!cn−c c!
(1)
⎧ r n n ⎪ n = 0, 1, . . . , s − 1, p0 , ⎪ ⎪ ⎪ n! ⎪ ⎪ ⎪ ⎪ n ⎨ r s r! P (Nno = n|, ) = p0 , n = s, s + 1, . . . , c − 1, ⎪ (r − n + s)!n! ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ r s r! ⎪ ⎩ p0 , n = c, c + 1, . . . , s + r. n−c (r − n + s)!c c!
(2)
In both cases, p0 = P (Nno = 0|, ) can be computed from the condition s+r n=0 P (Nno = n|, ) = 1. The stationary distributions of other common performance measures related to Nno (as queueing size, number of busy repair crews, occupancy rates, etc.) can be easily derived from that for Nno . In particular, the variable No describing the number of operational components can be expressed as No = r + s − Nno . All these variables are independent of the repair discipline, which only affects the individual waiting times. 2.3. Performance criteria based on operative requirements Designing the queuing system will basically consist in determining the number of maintenance crews c and spares s necessary to meet some demand on the operational readiness. We define several criteria which guarantee a sufficiently large number of operative components No and thus an operational fleet. A full decision-theoretic approach to queuing design based on costs or losses would be the most attractive approach. We will discuss this subject later on. We assume that the main goal is to guarantee that r planes are ready to fly at any time point. This can be expressed in terms of No as: • Goal 1: Guarantee an average number of operative components at least equal to the required ready-to-fly r, E(No ) r. • Goal 2: Assure a high probability of having at least r operative components available, P (No r) , for a sufficiently large ∈ [0, 1]. Goal 1 assures that the mean number of operative planes, averaging over time, will be adequate for the required working fleet. Goal 2 establishes guarantees about the number of planes available at any time point. Goal 2 is more stringent than Goal 1 for large as exigency concerns the most frequent behavior, whereas Goal 1 just refers to the mean behavior, which may not be the usual. Thus, verifying Goal 2 provides more confidence on the required capacity.
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However, these criteria cannot be applied directly since the distribution of No , and thus its expectation and probabilities, depend on the unknown parameters and describing the failure and service rates, respectively. We need to estimate these parameters and apply statistical reasoning to deduce information on No . This is the objective of the next section. 3. Bayesian maintenance design 3.1. Posterior and predictive distribution We start by formulating the Bayesian queueing model in terms of the likelihood function and a prior distribution. The available data consists of no life times {to1 , . . . , tono } and nm maintenance (repair) times {tm1 , . . . , tmnm }, which are independent realizations of exponentially distributed random variables To and Tm with mean and , respectively. An objective analysis is proposed to avoid influences of the prior. Assuming independence and no a priori knowledge on the parameters, a non-informative Jeffreys’s prior is used: p(, ) = p()p() ∝ (1/)(1/).
(3)
The resulting posterior distribution, which contains all the available information on the parameters, is given by p(, |data) = Ga(|no , tno ) Ga(|nm , tnm ), where tno =
no
i=1 toi , tnm 2
=
nm
j =1 tmj
(4)
and Ga(|, ) denotes the density at of a Gamma distribution with mean /
and variance / . Every congestion measure mentioned in Section 2.2 inherits a posterior distribution from p(, |data) in (4) since it depends on and . This means that we have a full posterior probability density for E(No |, ) and P (No r|, ), denoted by p(E(No |, )|data) and p(P (No r|, )|data), respectively. Another powerful tool offered by the Bayesian approach is a predictive analysis. The predictive steady-state distribution of the number of non-operating planes Nno incorporates all information about and . It is given by the expectations of the conditional probabilities P (No = n|, ) with respect to the posterior p(, |data): P (No = n|data) = E post [P (No = n|, )],
n = 0, 1, . . . , r + s.
(5)
Whatever analysis based on posterior or predictive distributions can be performed using stochastic simulation with a random sample from p(, |data) in (4) and Monte Carlo integration. 3.2. Bayesian performance criteria The performance criteria proposed in Section 2.3 for choosing c and s can be reformulated in terms of the predictive distribution of No , P (No = n|data), or the posterior distribution p(P (No = n|, )|data) as follows: • Predictive Criterion 1 (PredC1): The predictive expected number of operational components is greater than r, the required operative fleet: E(No |data) r,
(6)
where E(No |data) is the expectation of the predictive in (5), or • Predictive Criterion 2 (PredC2): The predictive probability of having at least r operational components available is large enough, P (No r|data) , where ∈ [0, 1] is sufficiently large.
(7)
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Again the second criterion is more demanding and allows to control the level of exigency about the system performance via the probability . The first criterion, though providing a sufficiently large expected fleet size, can result in unreliable designs leading to many situations with too few operational planes. E(No |data) and P (No r|data) are also the respective means of the posteriors p(E(No |, )|data) and p(P (No r|, )|data). Both posteriors contain more information than their mean and they give rise to alternative versions of the initial Goals 1 and 2: • Posterior Criterion 1 (PostC1): Guarantee at a high level ∗ ∈ [0, 1] that the average number of operational components exceeds the required r, P post [E(No |, ) r] ∗ .
(8)
• Posterior Criterion 2 (PostC2): Guarantee at a high level ∗ ∈ [0, 1] that the probability of obtaining the required number r of operational components is high enough, P post [P (No r|, )] ∗ .
(9)
In the posterior versions of Goals 1 and 2, the probability P post (·) is assessed from the posterior distribution of p(E(No |, )|data) and p(P (No r|, )|data), respectively. In the predictive criteria, we average over all possible values of (, ) weighted by the posterior distribution, while in the posterior goals we insist on the values of and for which (c, s) is suitable to have high probability, so that reliability is guaranteed in most of the cases. Both types of criteria seem reasonable. Predictive goals refer to mean behaviors, and so they are more intuitive and easily comprehensible. Posterior criteria provide more confidence in satisfying the operational requirements as they quantify their reliability given the observed data. Consequently, posterior criteria will generally lead to larger values of c and s than their predictive versions if sufficiently high probabilities are required. As commented before, Goal 2 is, whatever the version, more demanding than Goal 1 since it is based on a probable occurrence instead of the mean behavior. In this respect, Criterion 2 would be preferable to Criterion 1 when searching for a high reliability. If the only objective is to reach the highest guarantees for assuring the readiness of the fleet, PostC2 is clearly the most suitable criterion. PredC2 is a bit less demanding but easier to understand. Less stringent requirements can be expressed in terms of mean performance, with criterion PostC1 providing more reliable system configurations than its predictive version PredC1. 4. An example from air-force maintenance As introduced in Section 2 we work on a simplification of the real aeronautic problem described by Rodrigues et al. (2000). The number of A-4 planes is r = 50, which also represents the required capacity. The initial problem was focussed on choosing the number of spares necessary to assure readiness of the fleet. Rodrigues et al. (2000) assume the life and the maintenance times to be exponentially distributed and estimate the respective rates as 180 and 30 days, but they do not provide data. They only work with these point estimates and do not introduce their variability into the analysis. We have constructed a sample of convenience with 250 repair and 250 life times by simulating from 1 1 exponential distributions with parameters = 180 and = 30 . From the non-informative prior in (3) and the artificial data we obtain the posterior distribution of (, ), p(, |data) = Ga(|250, 45 950) Ga(|250, 7358).
(10)
We have computed the posterior and predictive performance measures defining the criteria in Section 3.2. They have been evaluated on a range of c = 6, . . . , 10 repair crews and s = 9, . . . , 44 spares, with the upper limit for the spares chosen as in Rodrigues et al. (2000). The eligible combinations (c, s) clearly depend on the operational requirements in terms of the fleet size and the operational level imposed.
J. Morales et al. / Journal of Statistical Planning and Inference 137 (2007) 3058 – 3067
1
80
0.9
0.9
75
0.8
0.8
70
70
0.7
65
65
60
60
55
55
50
50
0.3
0.3
45
45
0.2
0.2
40
40
0.1
0.1
35
35
0
c= 6 c= 7 c= 8 c= 9 c= 10
80 75
9
14
20
26
32
38
Probability
85
85
E (No)
3063
44
1
0.7 c= 6 c= 7 c= 8 c= 9 c= 10
0.6 0.5 0.4
0.5 0.4
0 9
Spare components
0.6
14
20
26
32 38
44
Spare components
Fig. 1. Predictive expected operational fleet (left) and predicted probability of the required 50 planes (right) as a function of the number of crews c and spares s.
Table 1 Predictive expectations, E(No |data) (E), and probabilities P (No 50|data) (P) underlying PredC1 and PredC2 for selected combinations of crews and spares c
s
E
P
8
15 33
50.02 58.98
0.55 0.70
9
11 13 17 23
50.58 52.08 55.18 59.99
0.64 0.72 0.81 0.88
10
9 10 12 14
50.16 51.01 52.75 54.53
0.63 0.71 0.80 0.87
Fig. 1 displays the predictive expected number of operational components E(No |data) and the predicted probability of having 50 working planes, P (No 50|data). The measures are presented in dependence on the number of spares and the number of repair crews. They have been computed by Monte Carlo integration from the posterior p(, |data) in (10). The horizontal lines mark the demand on the fleet size, 50 aircraft, and the required probability = 0.8. Remembering that the required operational fleet is 50 aircraft, we observe that with c = 6, 7 repair crews we need many spares to guarantee the predictive Criterion 1 (PredC1), that is E(No |data) 50 planes. For c 8 we achieve PredC1 easily. While for c = 8 crews at least 15 spares are needed, s = 11 and 9 are already sufficient for c = 9 and 10, respectively. With regard to the second predictive criterion (PredC2), if we choose = 0.8 as the minimum probability for P (No 50|data), c = 6, 7 and now also c = 8 crews are practically ruled out since only c 9 assures PredC2 for the numbers of spares considered here. In case of c = 9 or 10 repair crews, s = 17 and 12 are the minimum numbers of spares compatible with PredC2, respectively. Comparison of both graphics allows to appreciate the more demanding nature of PredC2 as compared to PredC1, especially when requiring a high probability in the former. This becomes even more obvious from Table 1: Goal PredC1 is achieved in all (c, s) combinations examined there, but not PredC2.
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1
0.9
0.9
0.8
0.8
0.8
0.7
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c= 6 c= 7 c= 8 c= 9 c= 10
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0.9
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1
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1 0.9 0.8
c= 6 c= 7 c= 8 c= 9 c= 10
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0.1 0
0
0 9
14
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26
32
38
44
0 9
Spare components
14
20
26
32 38
44
Spare components
Fig. 2. Posterior probability that the expected number of operative components exceeds the required 50 aircraft (left) and posterior probability of assuring at 80% level the required operational fleet (right). In both graphics the horizontal line corresponds to the fixed level ∗ = 0.8.
Table 2 Performance measures P post [E(No |, ) 50] (E) and P post [P (No 50|, ) 0.8] (P) for posterior criteria PostC1 and PostC2 c
s
E
P
8
15 33
0.54 0.81
0.12 0.46
9
11 13 17 23
0.67 0.83 0.94 0.97
0.15 0.35 0.63 0.80
10
9 10 12 14
0.60 0.81 0.95 0.99
0.06 0.22 0.60 0.80
Fig. 2 depicts the posterior guarantees involved in the criteria PostC1 and PostC2, namely P post [E(No |, )50] and P post [P (No 50|, )], with = 0.8, respectively. As expected, PostC2 is even harder to achieve than PostC1, whatever probability level ∗ is considered. In fact, for ∗ = 0.8, PostC1 is compatible with c 8, while PostC2 is only achieved for c 9 repair crews. The posterior probabilities involved in both posterior criteria are actually quite low for values c 8. When ∗ = 0.8 and c = 9, s = 13 and 23 spares are necessary to verify PostC1 and PostC2, respectively, while for c = 10 we need s = 10 and 14, respectively. Table 2 shows the probabilities involved in the posterior criteria for the same combinations of crews and spares as in Table 1. Note the strong level of exigency imposed through the probabilities and/or ∗ , particularly the latter. As complementary material for better understanding of the posterior criteria, Fig. 3 presents the posterior densities of E(No |, ) and P (No 50|, ). The displays are for c = 9 repair crews and s ∈ {11, 13, 17, 23} spares. The vertical lines mark the required 50 aircraft and the probability ∗ = 0.8. The degree of certainty in verifying the capacity demands imposed by PostC1 and PostC2 can be observed as the right tail area starting at 50 in the case of PostC1 and at 0.8 for PostC2. When the number of spares increases, the densities move to the right and thus the posterior probabilities in PostC1 and PostC2 also increase.
J. Morales et al. / Journal of Statistical Planning and Inference 137 (2007) 3058 – 3067
c=9 s=11 s=13 s=17 s=23
0.20
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c=9 s=11 s=13 s=17 s=23
6
Density
Density
0.15
0.10
4
2
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0.00
0 40
50
60
70
Expected operative components
80
0.0
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0.4
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Probability of required operativity
Fig. 3. Posterior distribution of E(No |, ) (left) and of P (No 50|, ) (right).
5. Conclusions and extensions We have presented a Bayesian approach for assessing efficiency in an M/M/c//r queueing system with s spares. Our proposal has been formulated in terms of aeronautic maintenance, but it can also be applied to similar problems arising in areas such as transport, telecommunication, production planning, semiconductor manufacturing, etc. In fact, we started working from the machine interference problem, which is commonly used to model utilities in industry. We have put forward several Bayesian performance criteria for the system design, each of them guaranteeing a required operational capacity in a different sense. The decision on which criterion to use will depend on the special characteristics of the system and the particular objectives of the operator. Predictive criteria imply a satisfactory expected performance. Posterior criteria are more demanding and require to obtain the desired capacity with large confidence. We note that the predictive approach is conceptually easier and more comprehensible but posterior criteria can be more appropriate if the reliability requirements are very strong. A predictive or posterior analysis for the proper choice of crews and spares could also be based on other performance measures such as the operational availability and the cycle time. The cycle time is defined as To + Tno , where To is the operational time of a component and Tno the non-operational time that includes the waiting time in queue and the repair time. It is an important congestion measure in transport and manufacturing (Schoemig, 1999). The operational availability (Kang et al., 1998) is defined as the quotient: Ao =
E(To ) , E(To ) + E(Tno )
and can be seen as an approximation of the average proportion of operational time. Because we do not consider reliability improvement of the aircraft working, operational times To are not influenced by the number of crews and spares. Within our framework of searching for suitable choices of c and s, working on either the operational availability or the expected cycle time, E(To ) + E(Tno ), would both be equivalent to considering the mean time out of order, E(Tno ). A Bayesian decision approach would provide its full benefits if we were able to specify costs, not only for spares and repair crews, but also penalties when the operational capacity requirements are not met. In the simplest case, fixed costs per time unit could be assumed for each missing aircraft. Instead of minimizing costs for spares and crews under restrictions on the system performance, we could apply a straightforward predictive approach and
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minimize the expected total costs. However, costs due to a lack of reliability are likely to be non-linear, and sometimes the consequences can hardly be quantified at all. Nevertheless, a full decision framework is the most suitable approach to the design of queuing systems and we defer it to future work. In order to discuss the concepts underlying our proposals we have concentrated on a simple basic situation. Many possible extensions could be incorporated in a natural way into more complex Bayesian queueing models suiting real problems. We just want to mention a few of them: • The distributional assumptions for maintenance and operational times can be relaxed. Instead of exponential, we can consider broader classes of distributions such as the Gamma or the Erlang. Probabilistic results are more complex then and consequently, also Bayesian ones. A sensitivity analysis with regard to different distributional assumptions is currently under investigation. • The repair service can be decomposed into different phases (inspection, diagnosis, revision, . . .), with individual durations. • Heterogeneous users and types of failures can be considered, as well as inventory limits when some pieces must be replaced. • Maintenance units operating from relatively fixed locations can be considered jointly to take advantage of pooling effects. These generalizations and many other modifications can be incorporated easily into a Bayesian model. Bayesian methods connect well with queueing systems: in addition to the straightforward incorporation of different sources of information, it is worth mentioning the natural derivation of criteria that really meet the desires for reducing costs or guaranteeing reliability. Acknowledgments The authors would like to thank two referees and the editor for their fruitful comments and helpful suggestions. References Almási, B., Sztrik, J., 1998. A queueing model for a non-reliable multi-terminal system with polling scheduling. J. Math. Sci. 92, 3974–3981. Armero, C., Bayarri, M.J., 1996. Bayesian questions and answers in queues. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (Eds.), Bayesian Statistics, vol. 5. Oxford University Press, Oxford, pp. 3–23. Armero, C., Bayarri, M.J., 1999. Dealing with uncertainties in queues and networks of queues: a Bayesian approach. In: Ghosh, S. (Ed.), Multivariate, Design and Sampling. Marcel Dekker, New York, pp. 57–608. Armero, C., Conesa, D., 2000. Prediction in Markovian bulk arrival queues. Queueing Systems 34, 327–350. Ausín, M., Wiper, M., Lillo, R., 2004. Bayesian estimation for the M/G/1 queue using a phase type approximation. J. Statist. Plann. Inference 118, 83–101. Conti, P.L., 1999. Large sample Bayesian analysis for Geo/G/1 discrete-time queueing models. Ann. Statist. 27, 6, 1785–1807. de Smidt-Destombes, K.S., van der Heijden, M.C., van Harten, A., 2005. On the interaction between maintenance, spare part inventories and repair capacity for a k-out-of-N system with wear-out. European J. Oper. Res., to appear. Erkip, N., 1993. A model to find optimal fleet size. Trans. Oper. Res. 5, 93–102. Ghobbar, A.A., Friend, C.H., 2002. Sources of intermittent demand for aircraft spare parts within airline operations. J. Air Transport Management 8, 221–231. Gross, D., Harris, C.M., 1998. Fundamentals of Queueing Theory. third ed. Wiley, New York. Iravani, S.M., Duenyas, I., Olsen, T.L., 2000. A production/inventory system subject to failure with limited repair capacity. Oper. Res. Informs 48, 951–964. Jain, M., Rakhee, Maheshwari, S., 2004. N -policy for a machine repair system with spares and reneging. Appl. Math. Modelling 25, 513–531. Kang, K., Gue, K.R., Eaton, D.R., 1998. Cycle time reduction for naval aviation depots. In: Medeiros, D.J., Watson, E.F., Carson, J.S., Manivannan, M.S. (Eds.), Proceedings of the 1998 Winter Simulation Conference, pp. 907–912. Kilpi, J., Vepsäläinen, A.P.J., 2004. Pooling of spare components between airlines. J. Air Transport Management 10, 137–146. Kreimer, J., Mehrez, A., 1998. Computation of availability of a real-time system using queueing theory methodology. J. Oper. Res. Soc. 49, 1095–1100. Marseguerra, M., Zio, E., Podofillini, L., 2005. Multiobjective spare part allocation by means of genetic algorithms and Monte Carlo simulation. Reliability Eng. System Safety 87, 325–335. Medhi, J., 2003. Stochastic Models in Queueing Theory. second ed. Academic Press, New York.
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Rodrigues, M.B., Karpowicz, M., Kang, K., 2000. A readiness analysis for the Argentine air force and the Brazilian navy A-4 fleet via consolidated logistics support. In: Joines, J.A., Barton, R.R., Kang, K., Fishwick, P.A. (Eds.), Proceedings of the 2000 Winter Simulation Conference, pp. 1068–1074. Schoemig, A.K., 1999. On the corrupting influence of variability in semiconductor manufacturing. In: Farrington, P.A., Nembhard, H.B., Sturrock, D.T., Evans, G.W. (Eds.), Proceedings of the 1999 Winter Simulation Conference, pp. 837–842. Sleptchenko, A., van der Heijden, M.C., van Harten, A., 2005. Using repair priorities to reduce stock investment in spare part networks. European J. Oper. Res. 163, 733–750.
Bayesian design in queues: An application to aeronautic maintenance夡 Javier Moralesa , M. Eugenia Castellanosb , Asunción M. Mayorala , Roland Friedc , Carmen Armerod,∗ a Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain b Departamento de Estadística e Investigación Operativa, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain c Departamento de Estadística, Universidad Carlos III de Madrid, 28903 Getafe, Madrid, Spain d Department d’Estadística i Investigacio Operativa, Universitat de València, 46100 Burjassot, València, Spain
Received 11 October 2005; accepted 30 June 2006 Available online 15 March 2007
Abstract We exploit Bayesian criteria for designing M/M/c//r queueing systems with spares. For illustration of our approach we use a real problem from aeronautic maintenance, where the numbers of repair crews and spare planes must be sufficiently large to meet the necessary operational capacity. Bayesian guarantees for this to happen can be given using predictive or posterior distributions. © 2007 Elsevier B.V. All rights reserved. MSC: 62F15; 62K05 Keywords: M/M/c//r queueing system with spares; Operative performance criteria; Predictive distribution; Posterior distribution
1. Introduction Maintenance scheduling and machine repair times are crucial in all industrial and business branches which depend on the correct performance of a set of machines. Inventory problems require quick replacement of broken down machines to guarantee continuous functioning. Optimal planning of resources becomes essential then to assure some operational performance level. We develop a Bayesian analysis of a queueing system with a set of r machines which are subject to failures. The machines are maintained by c repair crews and can be replaced by s spare machines when broken down. This problem can be modeled as an M/M/c//r queueing system with s spares. This is a finite source queueing model and is commonly referred to as machine interference problem (MIP from now on), a particular case of closed queueing networks (Medhi, 2003) in the literature. Some interesting papers in this context are Iravani et al. (2000) in manufacturing, Almási and Sztrik (1998) in telecommunication, Erkip (1993) in transport and Schoemig (1999) in semiconductor manufacturing. The Bayesian framework allows us to derive natural performance criteria in terms of the number of working machines. 夡
Research under Grants MTM2004-03290; GV05/018; TSI2004-06801-C04-01; MTM2004-02934.
∗ Corresponding author. Tel.: +34963544309; fax: +34963543238.
E-mail address: [email protected] (C. Armero). 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2006.06.046
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The queuing system can be designed by choosing suitable numbers of repair crews and spares such that a certain acceptable global performance is achieved, with guarantees optionally assessed by predictive or posterior distributions. We focus the exposition on maintenance problems in aeronautics (Kreimer and Mehrez, 1998; Ghobbar and Friend, 2002; Jain et al., 2004; Kilpi and Vepsäläinen, 2004; Sleptchenko et al., 2005; de Smidt-Destombes et al., 2005; Marseguerra et al., 2005), but the underlying ideas are much more general. An airline needs a sufficient number of available planes ready to operate at any time. Non-predictable failures force stops at the hangars for repair but the existence of spare planes reduces non-operativity. For easier comprehension of our proposals we only consider a single type of failure: generalization to several types of failures is just an issue of model complexity but does not affect the appropriateness of the proposed criteria. This paper constitutes a first approach to queues within the MIP from a Bayesian perspective. Most statistical papers on queueing systems are in the frequentist framework, but there has been an increasing interest in Bayesian methods recently: M/M/c queues are discussed by Armero and Bayarri (1996), M/G/c with phase type service are examined by Ausín et al. (2004), bulk queues are considered by Armero and Conesa (2000), and non-parametric Bayesian models in discrete-time queues are studied by Conti (1999). See Armero and Bayarri (1999) for a review of basic analyses and an exhaustive list of Bayesian references. The paper is organized as follows. Section 2 presents a real problem from aeronautic maintenance and states the general model. Section 3 develops different Bayesian criteria for identification of reliable queueing designs. Section 4 illustrates our proposals on a real problem studied by Rodrigues et al. (2000). Finally, Section 5 provides some conclusions and proposes extensions for a full decision-theoretical approach to queueing design based on utilities.
2. An aeronautic maintenance problem and general setting 2.1. A queueing problem from aeronautic maintenance We examine a simplified version of a real aeronautic problem regarding the A-4 fleet maintenance of the Argentine Air Force and the Brazilian Navy. Rodrigues et al. (2000) studied the impact of consolidating aviation component spare inventory management and reducing transportation cycle times. Major repair work had to be sent to the manufacturers in the United States and the resulting long repair cycle times adversely affected operational readiness. We investigate the problem of quantifying system performance for an inventory/repair facility created in situ to repair and provide spare parts to planes from both Forces. With some small simplifications to clarify our proposals, it is straightforward to identify the following queueing system: • Arrivals: Planes, or precisely their mechanical components, fail randomly over time. The failure times are assumed to follow an exponential distribution. Remember that for simplicity we consider a single type of failure and thus a single type of component for inventory. • Service mechanism: Service means repairing a broken part. The servers are the c repair crews. If one of the repair crews is idle, a broken part is repaired immediately; otherwise, it needs to wait in a queue until a crew gets idle. The repair times are assumed to be exponentially distributed. • Finite population: A certain number r of operating planes is always required. The Argentine Air Force had 30 aircraft and the Brazilian Navy 20, so r = 50. Moreover, there are s spare parts. Whenever one of the r operating planes fails, the broken part is replaced by a spare if any is available. In this case we assume that the plane becomes operational again immediately, i.e. that removal and installation times of broken/spare parts are negligible. The spare parts can be regarded as spare planes, or briefly spares, and consequently, the potential number of users is r + s. • Discipline: The order of the waiting queue is FIFO, meaning the part “first in the system” (broken down) is “first out” (starts maintenance). If we assume that the operational times To and the maintenance times Tm are independent and follow exponential distributions with mean 1/ and 1/, respectively, the problem becomes an M/M/c//r queueing system with s spares.
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2.2. Steady-state solution The steady-state solution of a queue (if it exists) is usually expressed via the number of users in the system. In our situation, this corresponds to the number of broken planes Nno and always exists since the population is finite. For given numbers of maintenance crews c and spares s, the stationary distribution of Nno depends on and and reads as follows (Gross and Harris, 1998): • If c s,
• If c > s,
⎧ r n n ⎪ n = 0, 1, . . . , c − 1, ⎪ ⎪ n! p0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ r n n P (Nno = n|, ) = p0 , n = c, c + 1, . . . , s − 1, ⎪ cn−c c! ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ r s r! ⎪ ⎩ p0 , n = s, s + 1, . . . , s + r. (r − n + s)!cn−c c!
(1)
⎧ r n n ⎪ n = 0, 1, . . . , s − 1, p0 , ⎪ ⎪ ⎪ n! ⎪ ⎪ ⎪ ⎪ n ⎨ r s r! P (Nno = n|, ) = p0 , n = s, s + 1, . . . , c − 1, ⎪ (r − n + s)!n! ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ r s r! ⎪ ⎩ p0 , n = c, c + 1, . . . , s + r. n−c (r − n + s)!c c!
(2)
In both cases, p0 = P (Nno = 0|, ) can be computed from the condition s+r n=0 P (Nno = n|, ) = 1. The stationary distributions of other common performance measures related to Nno (as queueing size, number of busy repair crews, occupancy rates, etc.) can be easily derived from that for Nno . In particular, the variable No describing the number of operational components can be expressed as No = r + s − Nno . All these variables are independent of the repair discipline, which only affects the individual waiting times. 2.3. Performance criteria based on operative requirements Designing the queuing system will basically consist in determining the number of maintenance crews c and spares s necessary to meet some demand on the operational readiness. We define several criteria which guarantee a sufficiently large number of operative components No and thus an operational fleet. A full decision-theoretic approach to queuing design based on costs or losses would be the most attractive approach. We will discuss this subject later on. We assume that the main goal is to guarantee that r planes are ready to fly at any time point. This can be expressed in terms of No as: • Goal 1: Guarantee an average number of operative components at least equal to the required ready-to-fly r, E(No ) r. • Goal 2: Assure a high probability of having at least r operative components available, P (No r) , for a sufficiently large ∈ [0, 1]. Goal 1 assures that the mean number of operative planes, averaging over time, will be adequate for the required working fleet. Goal 2 establishes guarantees about the number of planes available at any time point. Goal 2 is more stringent than Goal 1 for large as exigency concerns the most frequent behavior, whereas Goal 1 just refers to the mean behavior, which may not be the usual. Thus, verifying Goal 2 provides more confidence on the required capacity.
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However, these criteria cannot be applied directly since the distribution of No , and thus its expectation and probabilities, depend on the unknown parameters and describing the failure and service rates, respectively. We need to estimate these parameters and apply statistical reasoning to deduce information on No . This is the objective of the next section. 3. Bayesian maintenance design 3.1. Posterior and predictive distribution We start by formulating the Bayesian queueing model in terms of the likelihood function and a prior distribution. The available data consists of no life times {to1 , . . . , tono } and nm maintenance (repair) times {tm1 , . . . , tmnm }, which are independent realizations of exponentially distributed random variables To and Tm with mean and , respectively. An objective analysis is proposed to avoid influences of the prior. Assuming independence and no a priori knowledge on the parameters, a non-informative Jeffreys’s prior is used: p(, ) = p()p() ∝ (1/)(1/).
(3)
The resulting posterior distribution, which contains all the available information on the parameters, is given by p(, |data) = Ga(|no , tno ) Ga(|nm , tnm ), where tno =
no
i=1 toi , tnm 2
=
nm
j =1 tmj
(4)
and Ga(|, ) denotes the density at of a Gamma distribution with mean /
and variance / . Every congestion measure mentioned in Section 2.2 inherits a posterior distribution from p(, |data) in (4) since it depends on and . This means that we have a full posterior probability density for E(No |, ) and P (No r|, ), denoted by p(E(No |, )|data) and p(P (No r|, )|data), respectively. Another powerful tool offered by the Bayesian approach is a predictive analysis. The predictive steady-state distribution of the number of non-operating planes Nno incorporates all information about and . It is given by the expectations of the conditional probabilities P (No = n|, ) with respect to the posterior p(, |data): P (No = n|data) = E post [P (No = n|, )],
n = 0, 1, . . . , r + s.
(5)
Whatever analysis based on posterior or predictive distributions can be performed using stochastic simulation with a random sample from p(, |data) in (4) and Monte Carlo integration. 3.2. Bayesian performance criteria The performance criteria proposed in Section 2.3 for choosing c and s can be reformulated in terms of the predictive distribution of No , P (No = n|data), or the posterior distribution p(P (No = n|, )|data) as follows: • Predictive Criterion 1 (PredC1): The predictive expected number of operational components is greater than r, the required operative fleet: E(No |data) r,
(6)
where E(No |data) is the expectation of the predictive in (5), or • Predictive Criterion 2 (PredC2): The predictive probability of having at least r operational components available is large enough, P (No r|data) , where ∈ [0, 1] is sufficiently large.
(7)
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Again the second criterion is more demanding and allows to control the level of exigency about the system performance via the probability . The first criterion, though providing a sufficiently large expected fleet size, can result in unreliable designs leading to many situations with too few operational planes. E(No |data) and P (No r|data) are also the respective means of the posteriors p(E(No |, )|data) and p(P (No r|, )|data). Both posteriors contain more information than their mean and they give rise to alternative versions of the initial Goals 1 and 2: • Posterior Criterion 1 (PostC1): Guarantee at a high level ∗ ∈ [0, 1] that the average number of operational components exceeds the required r, P post [E(No |, ) r] ∗ .
(8)
• Posterior Criterion 2 (PostC2): Guarantee at a high level ∗ ∈ [0, 1] that the probability of obtaining the required number r of operational components is high enough, P post [P (No r|, )] ∗ .
(9)
In the posterior versions of Goals 1 and 2, the probability P post (·) is assessed from the posterior distribution of p(E(No |, )|data) and p(P (No r|, )|data), respectively. In the predictive criteria, we average over all possible values of (, ) weighted by the posterior distribution, while in the posterior goals we insist on the values of and for which (c, s) is suitable to have high probability, so that reliability is guaranteed in most of the cases. Both types of criteria seem reasonable. Predictive goals refer to mean behaviors, and so they are more intuitive and easily comprehensible. Posterior criteria provide more confidence in satisfying the operational requirements as they quantify their reliability given the observed data. Consequently, posterior criteria will generally lead to larger values of c and s than their predictive versions if sufficiently high probabilities are required. As commented before, Goal 2 is, whatever the version, more demanding than Goal 1 since it is based on a probable occurrence instead of the mean behavior. In this respect, Criterion 2 would be preferable to Criterion 1 when searching for a high reliability. If the only objective is to reach the highest guarantees for assuring the readiness of the fleet, PostC2 is clearly the most suitable criterion. PredC2 is a bit less demanding but easier to understand. Less stringent requirements can be expressed in terms of mean performance, with criterion PostC1 providing more reliable system configurations than its predictive version PredC1. 4. An example from air-force maintenance As introduced in Section 2 we work on a simplification of the real aeronautic problem described by Rodrigues et al. (2000). The number of A-4 planes is r = 50, which also represents the required capacity. The initial problem was focussed on choosing the number of spares necessary to assure readiness of the fleet. Rodrigues et al. (2000) assume the life and the maintenance times to be exponentially distributed and estimate the respective rates as 180 and 30 days, but they do not provide data. They only work with these point estimates and do not introduce their variability into the analysis. We have constructed a sample of convenience with 250 repair and 250 life times by simulating from 1 1 exponential distributions with parameters = 180 and = 30 . From the non-informative prior in (3) and the artificial data we obtain the posterior distribution of (, ), p(, |data) = Ga(|250, 45 950) Ga(|250, 7358).
(10)
We have computed the posterior and predictive performance measures defining the criteria in Section 3.2. They have been evaluated on a range of c = 6, . . . , 10 repair crews and s = 9, . . . , 44 spares, with the upper limit for the spares chosen as in Rodrigues et al. (2000). The eligible combinations (c, s) clearly depend on the operational requirements in terms of the fleet size and the operational level imposed.
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1
80
0.9
0.9
75
0.8
0.8
70
70
0.7
65
65
60
60
55
55
50
50
0.3
0.3
45
45
0.2
0.2
40
40
0.1
0.1
35
35
0
c= 6 c= 7 c= 8 c= 9 c= 10
80 75
9
14
20
26
32
38
Probability
85
85
E (No)
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44
1
0.7 c= 6 c= 7 c= 8 c= 9 c= 10
0.6 0.5 0.4
0.5 0.4
0 9
Spare components
0.6
14
20
26
32 38
44
Spare components
Fig. 1. Predictive expected operational fleet (left) and predicted probability of the required 50 planes (right) as a function of the number of crews c and spares s.
Table 1 Predictive expectations, E(No |data) (E), and probabilities P (No 50|data) (P) underlying PredC1 and PredC2 for selected combinations of crews and spares c
s
E
P
8
15 33
50.02 58.98
0.55 0.70
9
11 13 17 23
50.58 52.08 55.18 59.99
0.64 0.72 0.81 0.88
10
9 10 12 14
50.16 51.01 52.75 54.53
0.63 0.71 0.80 0.87
Fig. 1 displays the predictive expected number of operational components E(No |data) and the predicted probability of having 50 working planes, P (No 50|data). The measures are presented in dependence on the number of spares and the number of repair crews. They have been computed by Monte Carlo integration from the posterior p(, |data) in (10). The horizontal lines mark the demand on the fleet size, 50 aircraft, and the required probability = 0.8. Remembering that the required operational fleet is 50 aircraft, we observe that with c = 6, 7 repair crews we need many spares to guarantee the predictive Criterion 1 (PredC1), that is E(No |data) 50 planes. For c 8 we achieve PredC1 easily. While for c = 8 crews at least 15 spares are needed, s = 11 and 9 are already sufficient for c = 9 and 10, respectively. With regard to the second predictive criterion (PredC2), if we choose = 0.8 as the minimum probability for P (No 50|data), c = 6, 7 and now also c = 8 crews are practically ruled out since only c 9 assures PredC2 for the numbers of spares considered here. In case of c = 9 or 10 repair crews, s = 17 and 12 are the minimum numbers of spares compatible with PredC2, respectively. Comparison of both graphics allows to appreciate the more demanding nature of PredC2 as compared to PredC1, especially when requiring a high probability in the former. This becomes even more obvious from Table 1: Goal PredC1 is achieved in all (c, s) combinations examined there, but not PredC2.
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1
0.9
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c= 6 c= 7 c= 8 c= 9 c= 10
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1
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0
0 9
14
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26
32
38
44
0 9
Spare components
14
20
26
32 38
44
Spare components
Fig. 2. Posterior probability that the expected number of operative components exceeds the required 50 aircraft (left) and posterior probability of assuring at 80% level the required operational fleet (right). In both graphics the horizontal line corresponds to the fixed level ∗ = 0.8.
Table 2 Performance measures P post [E(No |, ) 50] (E) and P post [P (No 50|, ) 0.8] (P) for posterior criteria PostC1 and PostC2 c
s
E
P
8
15 33
0.54 0.81
0.12 0.46
9
11 13 17 23
0.67 0.83 0.94 0.97
0.15 0.35 0.63 0.80
10
9 10 12 14
0.60 0.81 0.95 0.99
0.06 0.22 0.60 0.80
Fig. 2 depicts the posterior guarantees involved in the criteria PostC1 and PostC2, namely P post [E(No |, )50] and P post [P (No 50|, )], with = 0.8, respectively. As expected, PostC2 is even harder to achieve than PostC1, whatever probability level ∗ is considered. In fact, for ∗ = 0.8, PostC1 is compatible with c 8, while PostC2 is only achieved for c 9 repair crews. The posterior probabilities involved in both posterior criteria are actually quite low for values c 8. When ∗ = 0.8 and c = 9, s = 13 and 23 spares are necessary to verify PostC1 and PostC2, respectively, while for c = 10 we need s = 10 and 14, respectively. Table 2 shows the probabilities involved in the posterior criteria for the same combinations of crews and spares as in Table 1. Note the strong level of exigency imposed through the probabilities and/or ∗ , particularly the latter. As complementary material for better understanding of the posterior criteria, Fig. 3 presents the posterior densities of E(No |, ) and P (No 50|, ). The displays are for c = 9 repair crews and s ∈ {11, 13, 17, 23} spares. The vertical lines mark the required 50 aircraft and the probability ∗ = 0.8. The degree of certainty in verifying the capacity demands imposed by PostC1 and PostC2 can be observed as the right tail area starting at 50 in the case of PostC1 and at 0.8 for PostC2. When the number of spares increases, the densities move to the right and thus the posterior probabilities in PostC1 and PostC2 also increase.
J. Morales et al. / Journal of Statistical Planning and Inference 137 (2007) 3058 – 3067
c=9 s=11 s=13 s=17 s=23
0.20
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c=9 s=11 s=13 s=17 s=23
6
Density
Density
0.15
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4
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0 40
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Fig. 3. Posterior distribution of E(No |, ) (left) and of P (No 50|, ) (right).
5. Conclusions and extensions We have presented a Bayesian approach for assessing efficiency in an M/M/c//r queueing system with s spares. Our proposal has been formulated in terms of aeronautic maintenance, but it can also be applied to similar problems arising in areas such as transport, telecommunication, production planning, semiconductor manufacturing, etc. In fact, we started working from the machine interference problem, which is commonly used to model utilities in industry. We have put forward several Bayesian performance criteria for the system design, each of them guaranteeing a required operational capacity in a different sense. The decision on which criterion to use will depend on the special characteristics of the system and the particular objectives of the operator. Predictive criteria imply a satisfactory expected performance. Posterior criteria are more demanding and require to obtain the desired capacity with large confidence. We note that the predictive approach is conceptually easier and more comprehensible but posterior criteria can be more appropriate if the reliability requirements are very strong. A predictive or posterior analysis for the proper choice of crews and spares could also be based on other performance measures such as the operational availability and the cycle time. The cycle time is defined as To + Tno , where To is the operational time of a component and Tno the non-operational time that includes the waiting time in queue and the repair time. It is an important congestion measure in transport and manufacturing (Schoemig, 1999). The operational availability (Kang et al., 1998) is defined as the quotient: Ao =
E(To ) , E(To ) + E(Tno )
and can be seen as an approximation of the average proportion of operational time. Because we do not consider reliability improvement of the aircraft working, operational times To are not influenced by the number of crews and spares. Within our framework of searching for suitable choices of c and s, working on either the operational availability or the expected cycle time, E(To ) + E(Tno ), would both be equivalent to considering the mean time out of order, E(Tno ). A Bayesian decision approach would provide its full benefits if we were able to specify costs, not only for spares and repair crews, but also penalties when the operational capacity requirements are not met. In the simplest case, fixed costs per time unit could be assumed for each missing aircraft. Instead of minimizing costs for spares and crews under restrictions on the system performance, we could apply a straightforward predictive approach and
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minimize the expected total costs. However, costs due to a lack of reliability are likely to be non-linear, and sometimes the consequences can hardly be quantified at all. Nevertheless, a full decision framework is the most suitable approach to the design of queuing systems and we defer it to future work. In order to discuss the concepts underlying our proposals we have concentrated on a simple basic situation. Many possible extensions could be incorporated in a natural way into more complex Bayesian queueing models suiting real problems. We just want to mention a few of them: • The distributional assumptions for maintenance and operational times can be relaxed. Instead of exponential, we can consider broader classes of distributions such as the Gamma or the Erlang. Probabilistic results are more complex then and consequently, also Bayesian ones. A sensitivity analysis with regard to different distributional assumptions is currently under investigation. • The repair service can be decomposed into different phases (inspection, diagnosis, revision, . . .), with individual durations. • Heterogeneous users and types of failures can be considered, as well as inventory limits when some pieces must be replaced. • Maintenance units operating from relatively fixed locations can be considered jointly to take advantage of pooling effects. These generalizations and many other modifications can be incorporated easily into a Bayesian model. Bayesian methods connect well with queueing systems: in addition to the straightforward incorporation of different sources of information, it is worth mentioning the natural derivation of criteria that really meet the desires for reducing costs or guaranteeing reliability. Acknowledgments The authors would like to thank two referees and the editor for their fruitful comments and helpful suggestions. References Almási, B., Sztrik, J., 1998. A queueing model for a non-reliable multi-terminal system with polling scheduling. J. Math. Sci. 92, 3974–3981. Armero, C., Bayarri, M.J., 1996. Bayesian questions and answers in queues. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (Eds.), Bayesian Statistics, vol. 5. Oxford University Press, Oxford, pp. 3–23. Armero, C., Bayarri, M.J., 1999. Dealing with uncertainties in queues and networks of queues: a Bayesian approach. In: Ghosh, S. (Ed.), Multivariate, Design and Sampling. Marcel Dekker, New York, pp. 57–608. Armero, C., Conesa, D., 2000. Prediction in Markovian bulk arrival queues. Queueing Systems 34, 327–350. Ausín, M., Wiper, M., Lillo, R., 2004. Bayesian estimation for the M/G/1 queue using a phase type approximation. J. Statist. Plann. Inference 118, 83–101. Conti, P.L., 1999. Large sample Bayesian analysis for Geo/G/1 discrete-time queueing models. Ann. Statist. 27, 6, 1785–1807. de Smidt-Destombes, K.S., van der Heijden, M.C., van Harten, A., 2005. On the interaction between maintenance, spare part inventories and repair capacity for a k-out-of-N system with wear-out. European J. Oper. Res., to appear. Erkip, N., 1993. A model to find optimal fleet size. Trans. Oper. Res. 5, 93–102. Ghobbar, A.A., Friend, C.H., 2002. Sources of intermittent demand for aircraft spare parts within airline operations. J. Air Transport Management 8, 221–231. Gross, D., Harris, C.M., 1998. Fundamentals of Queueing Theory. third ed. Wiley, New York. Iravani, S.M., Duenyas, I., Olsen, T.L., 2000. A production/inventory system subject to failure with limited repair capacity. Oper. Res. Informs 48, 951–964. Jain, M., Rakhee, Maheshwari, S., 2004. N -policy for a machine repair system with spares and reneging. Appl. Math. Modelling 25, 513–531. Kang, K., Gue, K.R., Eaton, D.R., 1998. Cycle time reduction for naval aviation depots. In: Medeiros, D.J., Watson, E.F., Carson, J.S., Manivannan, M.S. (Eds.), Proceedings of the 1998 Winter Simulation Conference, pp. 907–912. Kilpi, J., Vepsäläinen, A.P.J., 2004. Pooling of spare components between airlines. J. Air Transport Management 10, 137–146. Kreimer, J., Mehrez, A., 1998. Computation of availability of a real-time system using queueing theory methodology. J. Oper. Res. Soc. 49, 1095–1100. Marseguerra, M., Zio, E., Podofillini, L., 2005. Multiobjective spare part allocation by means of genetic algorithms and Monte Carlo simulation. Reliability Eng. System Safety 87, 325–335. Medhi, J., 2003. Stochastic Models in Queueing Theory. second ed. Academic Press, New York.
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