Machine Repair Model with Standbys,
Discouragement with Vacationing
Unreliable Repairmen
POOJA NITISH SINGH
M.Phil. Scholar, Department of Mathematics, AISECT University, BHOPAL (M.P) – INDIA
poojanitishsingh@yahoo.com
SUPRIYA MAHESHWARI
Department of Mathematics, Hindustan Collage of Science & Technology, Mathura, (U.P.) – INDIA
supmaheshwari@gmail.com
Abstract :
This investigation deals with a machine repair problem with warm standbys, discouragement and unreliable multi-repairmen following a vacation policy. The moment a machine is failed, it is promptly attended by an available repairman. In the absence of any failed machines in the queue seeking repair, the repairmen leave for a vacation of random length. On the completion of vacation period, the repairmen return and check for any failed machine in the queue. If the queue is non-empty, the repairmen start repairing of failed machines until the queue becomes empty, followed by another vacation. The repairmen may also become out of order with a subsequent restitution. The caretaker of the failed machines may renege from the system when all repairmen busy. The steady state queue size distribution and other important performance measures have been derived using matrix recursive approach. Numerical example provides a deep understanding of the sensitivity of parameters with respect to various performance measures.
Keywords: Machine repair, Unreliable repairmen, Warm standbys, Reneging, Multiple vacations, Matrix recursive approach, Queue size.
INTRODUCTION
The introduction of machining systems in human life in almost every field viz. communication systems, computer systems, manufacturing/production has made our life much easier; consequently our total reliance on the machines can not be denied. But this dependence is significantly affected when the machines become out of order. To avoid the inconvenience caused due to machine failures, the industrial managers attempt to make the provision of standby support and repair facility under techno-economic constraints.
A proficient machining system is attached with the facilities of sufficient standby support and expert repairmen to tackle any situation produced by machine breakdown. In this view many researchers have indicated towards the inclusion of these important concepts while modeling a sophisticated machining system. In modern perspective, the failure and repair are coupled actions in a typical machining system. Many researchers have recommended the provision of repair crew and spare part support to guarantee the desired efficiency of the machining system. The important contributions in the areas of machine repair problem are due to Gross et al. (1977), Sivazlian and Wang (1989), Wang (1995), Wang and Kuo (2000) and many others. Cost benefit analysis of series system with warm standby components was performed by Wang and Pearn (2003) and Wang et al. (2005). Ke and Lin (2007) studied a manufacturing system consisting of M operating machines and S spares under the supervision of a group of repairmen. Bieth et al. (2010) considered a standby system with two repair persons under arbitrary life and repair times using semi-Markov processes. Ke et al. (2013) analyzed a machine repair problem as a finite-state Markov chain with warm standbys and unreliable repairmen and obtained its steady state distribution by recursive matrix approach.
Wang et al. (2011) developed a machine repair model with identical machines and variable servers by taking into account balking concept. They deduced steady state solution using recursive technique and also constructed cost model to determine the optimal values of the number of busy servers and maintained the balking rate at a certain level. Sherif et al. (2013) illustrated the application of generating function technique to obtain the busy period density function of an M/M/1 queue with balking and reneging.
Sometimes a situation may be faced where the system becomes empty. This situation may arise if the unreliable repairmen are idle and leave the system for a vacation. The idle repairmen can utilize this time by leaving the system for a short time period (vacation) for some auxiliary work related to the system itself which may include proper placement of the required tools, preventive maintenance etc. The last two decades have observed a considerable amount of work to study queueing systems with vacations. Gupta (1997), Jain et al. (2004) and Ke and Wang (2007) examined machine repair problems with spares and multiple server vacations. Wang et al. (2009) studied the single server machine repair problem with working vacation in which the server works with different repair rates rather than completely terminating the repair during a vacation period. Hu et al. (2010) performed the probabilistic analysis of a three unit repairable system with one repairman operating single vacation. Recently, synchronous vacation machine repair models with standbys were investigated by Ke et al. (2011) and Ke and Wu (2012).
The deficiency of the works coupled with unreliable multi-repairmen problem with spares and server’s vacations motivated us to develop multi-repairmen machine repair model with multiple vacations. By incorporating two types of warm spares along with the conception of common cause failure and reneging, the model becomes more productive and realistic. Earlier, a similar M/M/R model was developed by Ke and Wang (2007), who used two types of warm spares to support the system, along with single and multiple vacation policies.
The paper has been structured in different sections as follows. In section II, the model has been described by stating the required assumptions and notations for mathematical formulation purpose. The state dependent failure and repair rates along with the system states notations are given in section III and the balance equations governing the model have been constructed using these rates. The matrix method to evaluate the probabilities has been presented in section IV. For various performance measures explicit expressions have been established in section V. Some special cases have been discussed in section VI. Numerical results and sensitivity analysis have been provided in section VII. Finally, the concluding remarks have been made in section VIII. I. MODEL DESCRIPTION
Consider a multi component machining system having operating machines as well as two types of warm spares and a repair facility consisting of R repairmen. For the mathematical formulation of the model, the following assumptions are made:
There is provision of Si (i=1,2) spare machines of ith type along with M operating machines.
The operating machines and ith type spare machines fail in Poisson fashion with rateand i , i = 1,2 respectively. Also
The machining system can also fail due to some common cause in Poisson fashion with rate c.
The failed unit is immediately replaced by an available spare of higher failure rate and sent to repair facility.
The switch over time from standby state to operating state is considered negligible.
After being repaired, the unit joins the operating group if the system works with less than M machines otherwise put with the standby group.
The repair time of failed machines is assumed to be exponentially distributed.
The repairmen repair the failed machines with rate till there are n<R failed machines in the system and with faster rate 1 when the number of failed machines becomes equal or exceeds the number of repairmen i.e. n≥R.
When the repairmen becomes idle, he leaves for a vacation; the vacation time is exponentially distributed with rate
The repairmen after returning from vacation check for any failed unit in the queue; if any, it is repaired, otherwise repairmen leave for another vacation. This fashion continues, until the repairmen find any failed unit in the queue, on returning from vacation.
The failure and repair rates are given by K S S M n S S , ) n K ( S S n S , ) n S S ( ) n S ( M S n 0 , S ) n S ( M 2 1 2 1 c 2 1 1 2 2 1 1 1 c 1 2 2 1 1 c
n (1)
And K n R , ) R n ( R R n 0 , n 1
n (2)
The system states are represented in the form (i,j) where, i denotes the number of busy repairmen and j is used to denote the number of failed machines.
We denote the probabilities of state (i,j) by Xi,j
II. THE GOVERNING EQUATIONS
The steady state equations governing the model for different states of the servers are constructed as follows:
(i) When all repairmen are on vacation i.e. i=0.
0
X
X
0,0 1,10
… (3)
1
K
n
1
,
0
X
X
X
)
C
(
n
0,n
n 1 0,n 1
1,n
… (4)0
X
X
X
)
C
(
0,K
K 1 0,K 1
1,K
… (5) (ii) When some repairmen are on vacation i.e. 1 i R-1, 0 X ) 1 i ( X ] ) 1 i R [( X i X ) i
(i i,i i,i 1 i 1,i i 1,i 1
inR1 … (6)
, 0 X X ] ) 1 i R [( X X i X ] ) i R ( i
[n i,n i,n 1n 1 i,n 1 i1,n i 1,n
1
R n
1 (7)
R , i 1 1 R , 1 i 1 R , i 1
R i (R i) ]X [(R i 1) ]X [i (R i) ]X
[
, 0 X
Xi,R 2 i 1,R 1 2
R
1nR1 … (8)
n , 1 i n , i 1
n (i (n i) ) (R i) ]X [(R i 1) ]X
[
… (9)
K i K
i
R
i
X
X
i
R
i
K
i
1(
)
)
(
)
]
,[(
1
)
]
1,[(
,
0
X
X
i,K 1 i 1,K1
K
…. (10)(iii) When all repairmen are busy in providing repair, i.e. i =R.
0 X ) R ( X ) ( X ) R
(R 1 R,R R 1,R 1 R,R 1
… (11)
1 n , R 1 n , 1 R n , R 1
n (R (n R) )]X ( )X [R (n 1 R) ]X
[
, 0 XR,n 1 1
n
R 1nK1 … (12)
0 X X ) ( X ] ) ) R K ( R
[ 1 R,K R 1,K K 1 R,K 1
III. MATRIX RECURSIVE METHOD
Consider an irreducible Markov chain with transition probability matrix Q which can be represented in the following block tri-diagonal structure.
C C 1 C 1 C 1 C 2 C 2 C 2 C 3 3 3 2 2 2 1 1 1 0 0D
L
0
0
U
D
L
0
U
D
L
U
D
L
U
D
L
0
0
U
D
L
0
0
0
U
D
Q
The matrix Q is a square matrix and of order (K+1) (R+1)-R(R+1)/2 and is composed of following sub-matrices
) R ( 0 0 0 0 0 0 0 ) R ( 0 0 0 0 0 0 0 0 0 0 0 0 0 ) R ( 0 0 0 0 ) R ( 0 0 0 0 0 D 1 K 1 K 2 i 2 i 1 i 1 i i i 0 R i 1 , v u 0 0 0 0 0 0 0 w v u 0 0 0 0 0 0 0 w v u 0 0 0 0 0 0 0 w v 0 0 0 0 0 0 0 0 0 w v u 0 0 0 0 0 0 0 w v u 0 0 0 0 0 0 0 w v u 0 0 0 0 0 0 0 w v D i , K i , K i , 1 K i , 1 K i , 1 K i , 2 K i , 2 K i , 2 K i , 3 K i , 3 K i , 3 i i , 3 i i , 3 i i , 2 i i , 2 i i , 2 i i , 1 i i , 1 i i , 1 i i , i i , i
i
Where
R j i if , i 1 R j i 0 if , i K j if , i ) i R ( 1 K j R if , i ) i R ( 1 R j 1 i if , ) i j ( i ) i R ( v 1 j j 1 1 j j i , j
w
,
if
i
j
K
1
j i ,
And
K j R ,
) i j ( i
1 R j 1 i ,
i u
1 i , j
For 0 ≤ i ≤ R-1
) i R ( 0
0 0 0 0
0
0 )
i R ( 0 0 0 0
0
0 0
0 0
) i R ( 0
0 0
0 )
i R (
0 0
0 0
Ui
And
, 0 i R
0 0
0
0 0 0
0
0 0 0
0
0 0 z
L i
i
Where,
R i i
1 R i 1 i z
1 i
The dimensions of D0, Di, Ui and Li are (K+1)(K+1), (K+1-i)(K+1-i), (K+1-i)(K-i) and (K+1-i)(K+2-i), respectively.
The stationary probability vector X = [X0, X1, X2,…, XC], where, Xi ={Xi,i , Xi,i+1, Xi,i+2 ,…, Xi,K} for 0 i R is a 1(K+1-i) vector and is evaluated using XQ=0 along with the normalizing condition
R
0 i
ie 1
X … (14)
Here e is column vector with all elements equal to one. Computation of stationary probabilities
The balance equations (3)-(13) can be written in matrix form as follows:
X0D0+X1L1=0 … (15) Xi-1Ui-1+Xi Di+Xi+1Li+1=0, 1iR1 … (16) XR-1UR-1+XRDR=0 … (17) Xi can be obtained by solving the above equations in terms of X0 as
1 R 1 R 1 R
R X U D
X … (19) Putting i=1 in equation (18) and using equation (15), we have
X0 (D0+L1η1) =0 … (20) where
1 1 i 1 i i 1 i
i
U
(
D
L
)
; for i= 1,2,..,R-1 … (21)Also we have R UR1DR1 … (22) X0 can be obtained by using equation (14). With the help of equations (18) and (19), we evaluate the probability vector Xi (i=1,2,…,R).
IV. PERFORMANCE MEASURES
Various performance measures of machining system in terms of probabilities can be obtained. Now we establish some results as follows:
The expected number of failed machines in the system is
R
0 i
K
i n
n , i nX )
N (
E … (23)
The expected number of failed machines in the queue is
R
0 i
K
i n
n , i
q) (n i)X
N (
E … (24)
The expected number of operating machines in the system is
R
0 i
K
1 s s
n , i 2
1 j
j
2 1
X ) S n ( M
) O (
E … (25)
The expected number of busy repairmen in the system is
R
1 i
K
i n
n , i X i )
B (
E … (26) The expected number of vacationing repairmen in the system is obtained as
R
0 i
K
i n
n , i X ) i C ( )
V (
E … (27) Machine availability is evaluated as
K
)
N
(
E
1
.
A
.
M
… (28) The effective failure rate is given by
R
0 i
1 K
i n
n , i n
eff X … (29)
V. SPECIAL CASES
Here we present some earlier existing results as special cases, which have been deduced by setting appropriate parameters in our model.
Case 1: By setting = 0, 1, 2 = and, our model matches with the M/M/R machine repair model with warm spares, which was developed by Sivazlian and Wang (1989).
spare and server multiple vacations considered by Gupta (1997).
Case 3: If c=0, =0, =0 and =0 then our model corresponds to machine repair model with two type spares and multiple vacation policy which was recently developed by Ke and Wang (2007).
VI. NUMERICAL EXPERIMENT
In this section, the effects of varying parameters on E(L) and M.A. have been displayed graphically. We set default parameters as M=10, R=4, S1=1, S2=2, c=0.05, 1= 0.3, 2= 0.2. Figs 1 and 2 depict, the effect of on E(L) and M.A. by varying the values of and respectively. For the increasing values of , E(L) shows a sharp increasing trend while M.A. displays decreasing trend. Fig. 1(a) shows a reasonable increment in E(L) with the increasing spare’s failure rate (). As the vacation rate () and repairmen’s failure rate () start increasing, a notable downfall in queue length can be observed in fig. 1(b) and fig. 1(c) respectively. In fig. 1(d), we note that the queue size is not much affected by the increasing values of reneging parameter and it comes down slightly. Figs 2(a)-(d) exhibit the trend of M.A. for increasing values of
and, respectively. From fig. 2(a), it is noted that M.A. gets lessen for increasing values of Whenthe
repairmen start returning from vacation frequently (recovers with rate ) i.e. () increases, M.A. also increases remarkably, as can be observed from fig. 2(b) and 2 (c). In fig. 2(d), the impact of varying reneging parameter is very slight which was also observed in case of its effect on E(L) in fig. 1(d).
As we expect, the queue of failed machines build up speedily with the increasing failure rate, whereas the availability of operating machines and spares comes down as such more repairmen become busy. As the rate of returning of vacationing repairmen rises, the repair process speeds up which helps in the
reduction of queue length; as a result, the operating machines and spares start accumulating.
VII. CONCLUDING REMARKS
In this investigation, a queueing model for a multi-component machining system with multiple repairmen and common cause failure has been developed. For the smooth running of the machining system, the provision of two types of warm spares has been made by keeping in mind the realistic economical and physical constraints. The inclusion of reneging and common cause failure concepts in our model fits it in real time framework. The consideration of multiple vacations for unreliable repairmen have an important role in reducing the cost involved as vacationing repairmen can do some auxiliary work during vacation period. Various performance measures, which play a significant role in deciding the performance of the system, have been established. To explore the effects of different parameters on performance measures, numerical experiment performed may be quite helpful. The model developed may be applied by the system engineers, management personal etc. in industrial organizations, which encounter complexity in the maintenance of the system while making decision regarding the setting up of the number of machines and the repairmen for continuance of the system operation.
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(a) (b)
(c) (d)
Fig 1: E(L) vs for varying (a) b) (c) d
0 1 2 3 4 5 6 7 8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
E(
L
)
0 1 2 3 4 5 6 7 8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
E(
L
)
0 1 2 3 4 5 6 7 8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
E(
L
)
0 1 2 3 4 5 6 7 8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
E(
L
)
(a) (b)
(c) (d)
Fig 2: M.A. vs for varying (a) b) (c) d
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
MA
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
MA
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
MA
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
MA
