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A correlated M
=
G
=
1-type queue with randomized server repair
and maintenance modes
David Perry
a, M.J.M. Posner
b;∗aDepartment of Statistics, The University of Haifa, Haifa, 31905, Israel
bDepartment of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, ONT,
Canada M5S 3G8
Received 1 June 1998; received in revised form 1 August 1999
Abstract
We consider a single machine, subject to breakdown, that produces items to inventory, continuously and uniformly. While the machine is working (an ON period), there is a deterministic net ow into the buer. There are also two dierent types of OFF periods; one is a repair operation initiated after a breakdown, and the other is a preventive maintenance operation. The length of an OFF period depends on the length of the preceding ON period. During OFF periods there is a uniform outow from the inventory buer. The buer content process can then be described as a uid model. The main tool employed in determining its steady-state law exploits an equivalence relationship between the buer content process and the virtual waiting time process of a special single-server queue in which the interarrival times and the service requirements depend on each other. We present an appropriate cost function and provide an optimization scheme illustrated by some numerical results. c2000 Elsevier Science B.V. All rights reserved.
Keywords:Production=inventory; Correlated uid queues; Repair; Maintenance; On=o unreliable machine; Level crossings
1. Introduction
In this paper, we study a special type of manufacturing problem incorporating machine reliability and maintenance. Items are produced continuously and uniformly by a single machine that is subject to breakdown. During a machine ON time there is a deterministic net ow into the inventory buer at rate ¿0, where
is the production rate minus the demand rate. If a failure now occurs before some time ˜a has elapsed, then a machine repair operation will start; this is an OFF time of type 1. However, if a failure does not occur before time ˜a, then the controller stops production to initiate a preventive maintenance (PM) action; this is an OFF
∗Corresponding author. Fax: +1-416-978-3453. E-mail address: [email protected] (M.J.M. Posner)
time of type 2. During an OFF time of either type 1 or type 2, there is a deterministic demand generating a linear outow from the inventory buer at rate , whenever the buer level is positive. However, negative inventory is not allowed, so that during OFF times there will be neither inow nor outow whenever the system is empty.
In the context of such a model framework, we attempt to maximize the net revenue accumulation rate with respect to the decision variable ˜a, namely, how long we should permit the machine to operate before initiating PM. The net revenue function will incorporate revenue from production, holding costs, penalty cost for unsatised demands, and repair and maintenance costs. Each component is determined in accordance with the steady-state distribution of the buer contents.
Our model can be interpreted either as a certain queueing model with dependence between an interarrival time and the preceding service request, or as a uid=production model with a two-state random environment. The two models are directly related in the sense that the steady-state law of the buer contents in the uid interpretation can be expressed in terms of the workload in the queueing interpretation. As a stochastic model, the uid interpretation may be more natural than the queueing one; however, the queueing interpretation enables us to locate the problem in the general setting of queueing models and to use well-established tools and results from queueing theory for the solution. The analysis is based on the fact (cf. [13]) that the conditional steady-state buer content distribution, given that it is positive, is independent of the inow rate. Thus, by setting=∞, we generate a sample path in which the ON periods are deleted and are represented by upward jumps, while the OFF periods are then glued together. The resulting process can be interpreted as the workload process of a certain single-server queue in which customers arrive with a service request at a single server. Service requests of successive customers are independent and identically distributed random variables having a general distribution (corresponding to the general failure time distribution of the ON time). Upon arrival, the service request is registered. If the service request is less than a=a˜, then, the next interarrival time corresponds to an OFF period of type 1; otherwise, the service time becomes exactly equal toa (is cut o at a), and the next interarrival time corresponds to an OFF time of type 2.
At rst glance it is not easy to see that the uid production=inventory model and the queueing model are equivalent (in the sense that they have the same conditional steady-state laws). Kella and Whitt [13] were the rst to prove it for a generalized version of the GI=G=1 queue (in which the interarrival times and service times are not necessarily independent); however, they did not focus on the equivalence between the queueing system and the production=inventory model.
The uid production=inventory model is perhaps better motivated from the operational modeling point of view. However, the queueing interpretation, which is of independent interest due to the correlation feature, is better motivated from the stochastic analysis point of view. This interdependence feature will lead to the development of a new and original adaptation of level-crossing theory [8], which will augment the applications already presented by Perry and Posner [16–18]. For related works in production and manufacturing using uid models see [14,15,7,21,19]. A detailed survey on replacement policies and preventive maintenance models can be found in [1,20].
depends on the length of the preceding interarrival interval; here the dependence structure corresponds to a situation where work arrives at a gate which is opened at exponentially distributed intervals.
The paper is organized as follows. In Section 2 we describe the dynamics of the production=inventory model and develop the structure of the associated queueing model. In Section 3, we introduce the relationship between the buer content process and the queueing process. In Section 4, we analyze the stationary law of the queueing system. In Section 5 we present the cost function and provide an optimization scheme, leading to some numerical results.
2. Model dynamics
Let {Z(t):t¿0} be the buer content level process. Then, the sample paths of Z(t) are continuous and piecewise linear withZ(t)¿0. In order to derive the dynamics governing the evolution ofZ(t) we rst dene an articial net input contents process Y(t). The process Y(t) has the same properties as Z(t) except that negative inventory is allowed; that is, during ON times the inow accumulates at rate ¿0 and during OFF times the outow is released at rate ¿0.
We now introduce the following notation:Tn+1=Tn+Un+1+Dn+1; n¿0; T0= 0, where Un andDn are ON times and OFF times, respectively. HereTn is the onset time of the nth ON period. The process {Y(t): t¿0} is then dened byY(0) = 0, and
Y(t) =
(
Y(Tn) +(t−Tn); Tn6t ¡ Tn+Un+1;
Y(Tn) +·Un+1−(t−Tn−Un+1); Tn+Un+16t ¡ Tn+1:
Finally, the buer content process {Z(t):t¿0} is obtained by applying a reection map to the process Y(t), so that Z(t) =Y(t)−min[0;inf06s6t Y(s)] (refer to Fig. 1).
We now construct the process {V(t):t¿0} by deleting the ON times and then gluing together the OFF times of theZ(t) process. Formally, we dene ˜Tn+1= ˜Tn+Dn+1; n=0;1; : : : ; with ˜T0=0, and ˜Y(t)= ˜Y( ˜T
− n)+
Un+1−(t−T˜n); T˜n6t ¡T˜n+1, with ˜Y(0) = 0. Then, V(t= ˜Y(t)−min[0;inf06s6tY˜(s)].
Note that Un and Dn are the same for both Z(t) and V(t). However, unlike in Z(t), the sample paths of V(t) are not continuous; V(t) is in fact a right-continuous jump process with left-hand limits. These are depicted in Fig. 1, where it is clear that V(t) is a queueing model analogue of Z(t) in which the jump sizes (services times) are bounded by a= ˜a·, and the subsequent exponential interarrival process (repair [with mean −01] or preventive maintenance time [with mean 1−1]) is governed by whether or not the preceding jump was less than a. Clearly, then, optimization with respect to ˜a or a are equivalent.
3. The relationship between Z(t) and V(t)
Assume that Z(t) is a regenerative process (so that V(t) is also a regenerative process), and dene BZ= inf{t ¿0:Z(t) = 0} andBV= inf{t ¿0:V(t) = 0}. Here,BZ andBV are the busy periods (busy cycles minus idle periods) associated with the processesZ(t) and V(t), respectively. Also, let B0=BZ−BV. In words, BV is the total amount of OFF time in a busy cycle during which the inventory level (buer content) is always positive, and B0 is the total amount of ON time during that busy cycle. Let N be the number of ON times
during a cycle.
By the denition of the model, we have
BV =
Z
BZ
N−1 X
n=0
Fig. 1. A typical sample function ofZ(t) and corresponding realization ofV(t).
and
B0= Z
BZ
N−1 X
n=0
1{Tn6t¡Tn+Un+1}dt:
Let FZ(x) = limt→∞P(Z(t)6x) and FV(x) = limt→∞P(V(t)6x), with Z=FZ(0) andV=FV(0). Then, ˜
FZ(x) = [FZ(x)−Z]=[1−Z] and ˜FV(x) = [FV(x)−V]=[1−V] are the steady-state conditional distributions of Z(t) andV(t) given that the respective systems are not empty.
We now want to show that ˜FZ(x) = ˜FV(x) for all x ¿0. By renewal theory,
˜
FZ(x) =
ER
BZ1{Z(t)6x}dt
EBZ
=E
R
BV1{Z(t)6x}dt+E R
B01{Z(t)6x}dt
EBZ
(3.1)
and
˜
FV(x) =
ER
BV1{Z(t)6x}dt
EBV
since both Z(t) and V(t) decrease at the same rate . It follows by a simple geometric argument that, with probability 1,
Z
B0
1{Z(t)6x}dt= Z
BV
1{Z(t)6x}dt; x ¿0: (3.3)
Setting x=∞in (3.3) we thereby obtain B0= (=)BV, and, sinceB0=BZ−BV, it follows that
EBZ=
1 +
EBV: (3.4)
Substituting (3.3) and (3.4) into (3.1) we see that the right-hand side of (3.1) is the same as the right-hand side of (3.2). Thus, ˜FZ(x) = ˜FV(x) (even if the interarrival times and the service times associated with the virtual waiting time process, V(t), are not independent). This equality relation was also obtained by Kella and Whitt [13] and later generalized by Kaspi et al. [12], but using a dierent approach.
4. The stationary law of Z(t) and V(t)
Let ˜G be the distribution of the ON time of a machine in the uid production–inventory model, and letG
be the distribution of the service requirement in the associated queueing model. Then, it is straightforward to see that ˜G(x=) =G(x); and in particular, 1−G˜( ˜a) = 1−G(a) is the probability that the machine ON time [service requirement] is truncated at ˜a[a].
The only situation for which explicit formulas can be obtained is the case where both repair times and preventive maintenance times are exponential. In this case, the associated queueing system is a special version of a single-server queue with Markovian arrivals. We therefore assume that the repair time is exponentially distributed (0) and preventive maintenance is exponentially distributed (1). Then, the associated queueing
system has the following properties. The service requirement has distribution G, and each arriving customer species his service requirement at his moment of arrival. If this service requirement is less thana, then the next interarrival time is exponentially distributed (0). Otherwise, if his service requirement is at leasta, then
it is truncated to a, and the next interarrival time is exponentially distributed (1). The ˜Tn’s are the arrival times of customers in that queueing system. If for some n; T˜n+1−T˜n is exponentially distributed (0), then,
for all ˜Tn6t ¡T˜n+1, we say that the status of the system at time t is {0}. Alternatively, if ˜Tn+1−T˜n is exponentially distributed (1), then, for all ˜Tn6t ¡T˜n+1, we say that the status at time t is {1}. We can
then dene the status process at time t by
J(t) =
(
1 if the status is{1} at timet;
0 if the status is{0} at timet:
Note that the interarrival times in that queueing system are independent but they are not identically distributed. Also, the service requirements are i.i.d.; however, an interarrival time depends on the preceding service requirement.
We now apply level-crossing theory to compute the stationary density of V(t). We rst observe that the two-dimensional process{V(t); J(t): t¿0}is a Markov process in which the state space ofV(t) is continuous on [0;∞) and J(t) is either 0 or 1. Furthermore, we assume that conditions for stationarity prevail, so that the limiting distributions of V(t); J(t) and Z(t) are given, respectively, by those of V; J, and Z.
generator (or the innitesimal transition rate) of the Markov process {V(t); J(t)} asserts simply that Ex{0} can be reached (in a small interval of time) only fromEx{0} ∪E
x{1}. By level-crossing theory, the transition rate ofEx{0} →E
x{0} is the improper densityf0(x) which is the long-run average number of downcrossings
of level x per unit time byV(t), while the status of the system is {0}. Also, to analyze the transition rate of
Ex{1} →Ex{0}, we distinguish between two cases:
(i) x6a. As with the Pollaczek–Khintchine formula, the transition rate is 1R0xG(x−!) dF1(!), where, by
PASTA [22], F1(!) is the improper steady-state distribution of V when the status of the system is {1}.
(ii) x ¿ a. The transition rate is 1Rx −a
0 G(a) dF1(!) +1 Rx
x−aG(x−!) dF1(!). Similarly, to leave the set Ex{0} we distinguish between the same two cases:
(i) x6a. The transition rate is 0R
x
0 [1−G(x−!)] dF0(!), whereF0(!) is the improper steady-state
distri-bution ofV when the status of the system is{0}. (ii) x ¿ a. The transition rate is 0R
To summarize the above discussion, we now present the resulting Volterra integrals obtained by this level-crossing analysis: is F0+F1= 1, where Fi ≡Fi(∞). Observe that with this terminology a necessary and sucient condition for stationarity is [Ra
0 [1−G(x)] dx] −1¿
0F0+1F1.
Fig. 2. A typical density forZ for the case:−1= 2; a= 0:41; −1 0 = 1; −
1 1 = 0:5.
5. Cost function and optimization
From the numerical solution carried out in the appendix, various functionals can be determined which bear directly on the construction of a cost function to be investigated in this section. The main functionals required are Z ≡Pr(Z= 0); V ≡Pr(V = 0); EV andEZ. To obtain these, we must compute the various busy and idle periods for Z and V as BZ; IZ, and BV; IV, respectively.
We rst note that
Z=
EIZ
EBZ+EIZ
(5.1)
and
V =
EIV
EBV+EIV
; (5.2)
in which it is clear that
EIZ=EIV: (5.3)
By renewal theory, the expected length of an idle period in V is determined by
EIV =
f0(0)
f0(0) +f1(0)
1
0
+ f1(0)
f0(0) +f1(0)
1
1
= f0+f1
0f0+1f1
; (5.4)
using fi(0) =ifi; i= 0;1, where, invoking level-crossing theory, the ratio fi(0)=[f0(0) +f1(0)] is the
probability that an idle period of type i is initiated, and 1=i is the corresponding length of this idle period. Furthermore, since a mean cycle time is given by 1=[f0(0) +f1(0)] = 1=[0f0+1f1], it follows that
EBV =
1−f0−f1
0f0+1f1
Table 1
Optimal valuesa? for parameter set:= 20; = 100; h= 0:1; R
PR= 1; KSH= 1:5; KDT= 5; KF= 75
Case 1: −1 0 = 0:6;
−1
1 = 1 Case 2:
−1 0 = 1;
−1 1 = 0:5
−1 a? R(a?) −1
e z −
1 a? R(a?) −1
e z
0.25 0.22 −55:12 0.17 0.23 0.35 0.25 0.13 −51:29 0.11 0.18 0.41
0.50 0.39 −5:16 0.31 0.40 0.19 0.50 0.25 0:00 0.22 0.35 0.23
0.75 0.51 19:15 0.43 0.53 0.13 0.75 0.33 25:99 0.30 0.49 0.14
1.00 0.59 33:25 0.51 0.61 0.10 1.00 0.37 41:30 0.35 0.59 0.10
1.50 0.67 48:00 0.61 0.69 0.07 1.50 0.41 57:21 0.39 0.71 0.06
2.00 0.70 55:00 0.66 0.71 0.06 2.00 0.41 64:35 0.40 0.75 0.05
2.50 0.72 58:79 0.69 0.73 0.06 2.50 0.41 67:99 0.40 0.77 0.04
3.00 0.72 61:06 0.70 0.73 0.06 3.00 0.41 70:05 0.41 0.79 0.04
3.50 0.73 62:54 0.71 0.74 0.06 3.50 0.41 71:31 0.41 0.79 0.04
Case 3: −1 0 = 1;
−1
1 = 2 Case 4:
−1 0 = 2;
−1 1 = 1
−1 a? R(a?) −1
e z −
1 a? R(a?) −1
e z
0.50 0.78 −19:65 0.44 0.37 0.22 0.50 0.47 −32:90 0.35 0.22 0.36
1.00 1.06 19:47 0.75 0.55 0.12 1.00 0.72 15:88 0.59 0.42 0.19
1.50 1.18 34:62 0.94 0.61 0.10 1.50 0.79 37:11 0.70 0.55 0.12
2.00 1.22 42:09 1.05 0.63 0.09 2.00 0.80 47:88 0.74 0.62 0.09
2.50 1.27 46:22 1.13 0.66 0.08 2.50 0.79 53:87 0.75 0.66 0.08
3.00 1.29 48:82 1.18 0.66 0.08 3.00 0.78 57:49 0.75 0.69 0.07
3.50 1.30 50:56 1.22 0.67 0.08 3.50 0.77 59:83 0.75 0.70 0.07
using (5.4). Also,
Z=
f0+f1
1 + [1−f0−f1]
(5.6)
and V =f0+f1 are easily conrmed.
Because of the demonstrated equivalence between ˜FZ and ˜FV, we have fZ(x)=[1 −Z] =fV(x)=[1−
V(x)]; x ¿0, whence fZ(x)=fV(x) = [1 +=]={1 +=[1−f0−f1]}EZ=EV.
The optimization with respect to a will be carried out for a net expected revenue accumulation rate which includes the following functional components:
(i) Production revenue: Production yields revenue ofRPR per unit produced. The production rate is +,
and the portion of time that the system is ON is given by the ratio of ON time (EB0) to cycle time
(EBZ+EIZ). Since EB0=EBZ−EBV, then using (3.4) and (5.3), we have the net production revenue rate as RPR(+)(=)(1−f0−f1)=[1 + (=)(1−f0−f1)].
(ii) Linear holding cost: The holding cost rate is hEZ, where h is holding cost per unit time per item held in inventory.
(iii) Shortage cost: A cost of KSH is imposed for each demand arising during stockout. The corresponding
cost rate is KSHZ=KSH(f0+f1)=[1 + (=)(1−f0−f1)].
(iv) Repair and PM costs: A down machine incurs an ongoing cost of KDT per unit time for repair and
maintenance, while a breakdown incurs an additional cost ofKF. The number of failures per unit time is0F0
and the portion of time the system is OFF is given by the ratio of OFF time (EBV +EIV) to cycle time (EBZ+EIZ). As in (i) above, the net repair and PM cost rate is given byKF0F0+KDT=[1+(=)(1−f0−f1)].
problems were considered, with the results, and optimal values a? of a, summarized in Table 1. In the table, the mean eective operating time of the machine is given by−e1=Ra
0 [1−G(t)] dt, forGexponential (), and
= (0F0+1F1)=e is the analogue of the utilization factor for the associated queueing model. Reection
upon the results presented illustrates sensitivity with respect to changes in mean repair time (Cases 1 and 4) or mean PM time (Cases 2 and 3) as the mean time to failure, −1, varies. Of course, as −1 increases
beyond a?, the model stabilizes in the sense that well beyond a? we expect PM almost exclusively.
Acknowledgements
We wish to thank O.J. Boxma for his careful reading of the manuscript and for providing useful comments and suggestions. Also, the nancial support of the Natural Sciences and Engineering Research Council of Canada under grant OGP0004374 is gratefully appreciated.
Appendix Numerical solution
Dene densities fi(x) ≡fir (r= 0;1; : : : ;), where fir ≡ fi(r), in which is chosen so that a=N, with N arbitrary, but large. In addition, Gr ≡G(r) with Gr= 1−Gr and fi (i= 0;1) are the zero-level probability masses (system idleness). Rewriting the main equations (4.1) – (4.4) gives
f0n+1
From (A.1) and (A.2), for n= 1;2; : : : ; N, we can write the recursions
The same procedure can also be applied to Eqs. (A.3) and (A.4), leading to the vector recursion,
gn=A0 (A.10), we can nd these matrices from the resulting matrix recursions:
By numerical integration, we have the total probabilities of the system being in status 0 and 1 given by
g+
∞ X
r=0
gr=
"
F0
F1 #
;
where F0 andF1 are given in (A.5) and (A.6). Using gr=Hrg, this converts into the unique solution for g as
g=
"
I+
∞ X
r=0
Hr
#−1"
F0
F1 #
:
With g known, all gr are now determined, and hence the entire distribution is specied.
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