1. Introduction

With the advancement of modern technology, machining system has pervaded every nook and corner of our lives thus ensuring our utmost dependence on them. As time proceeds, a machine may fail due to wear and tear or due to some unpredicted fault and thus requires a corrective measure by providing the repair facility, after which it can again start working properly. If at any time, the failed machines need the repairman’s attention; a queue of failed machines may develop due to unavailability of idle repairmen. This phenomenon needs attention of mathematicians as such machine repair models have captivated the interest of many renowned researchers working in the area of queueing theory.

System efficiency can also be improved by providing sufficient spare part support in case of machine failure. In queueing theory a lot of work has been done on machine repair models with spares provisioning. Reference[1] presented a Markovian model for machine repair system having cold standbys. Reference[2] analyzed multi-standby system considering repair and replacement policy and developed the

Balking is a realistic phenomenon in many real life congestion situations and can be incorporated in case of multi- component machining system wherein failed machines may not join the queue in case of long queue or insufficient waiting space. It is also common observation that in many situations, the caretaker of waiting failed machines leaves the queue after waiting sometimes without getting service. This is termed as reneging and may be incorporated while analyzing the performance of real time machining system. In order to decrease the backlog and to check the balking & reneging behavior of the failed machines, the server may increase their service rate after a threshold level of the queue of failed machines. The facility of spares can also be helpful to reduce the discouraging behavior. These considerations give the model a realistic touch that’s why in the present study, we consider the balking & reneging effect, provision of spares and faster service rate after a threshold level while developing Markovian model of machine repair problem. Reference[12] derived some theorems on single server queue with balking. Reference[13] investigated M/M/R machine repair problem with reneging and spares. Reference[14] suggested cost analysis of the M/M/R machine repair problem with balking, reneging and server breakdowns. Reference[15] investigated the machine interference model with balking, reneging and spares. Reference[16] studied the repairable system with warm standbys, reneging and balking. Reference[17] analyzed M/M/R machine interference model with balking, reneging, spares and two modes of failure. Reference[18] worked out performance modeling of machining system with mixed standby components, balking and reneging. Reference[19] analyzed the M/M/R machining system with mixed standbys, switching failures, balking, reneging and additional removable repairmen for transient state. Reference[20] had used Newton's method for optimal management of the machine repair problem with working vacation. Reference[21] investigated machine repair problem in production systems with spares and server vacations. Reference[22] studied multi-product transfer lines subject to random failures. Reference[23] gave some balking strategies for the equilibrium in the single server Markovian queue with catastrophes.

In many real life congestion scenario of machine repair problem due to techno-economic constraint, the server activates at threshold level N of failed machines and continues till there is no waiting failed machine for service. When a unit fails or server breakdowns, the startup/setup time is required i.e. the server as well as the repairman spends some time for preparation before they can start the service of machines and servers. A vast number of papers in literature deal with a detailed analysis of N policy queueing models in different frame-works where the service is initiated by the accumulation of N jobs and ends when queue becomes empty. Reference[24],[25] studied N policy for M/G/1 queueing system. Reference[26] considered a queueing system with removable server and presented proof for the optimality of the best N policy. Reference[27] discussed a Poisson input queue under N policy and with a general startup time. An optimal N policy for production system with early set up was studied in[28]. N policy with setup time was considered in[29],[30]. Reference[31] dealt with the operating characteristics of an M/M/1 queue with an exponential setup time under N policy and obtained various state probabilities for queue size distribution at various points of time. Reference[32] analyzed N policy for a steady state bulk queue with multiple vacations, setup times and close down times. Reference[33] studied N-policy machine repair system with mixed standbys and unreliable servers. Reference[34] had performed the performance analysis of the finite source retrial queue with server breakdowns and repairs. Reference[35] used the maximum entropy method to analyze a queue having a randomized N-policy.

In some cases, a standby unit might not be able to switch over to a primary failed unit successfully. Reference[36] first introduced the concept of the standbys switching failures in the reliability with standby system. Reference[37] performed comparisons of reliability and the availability between four systems with warm standby units, reboot delay and standby switching failures for specific values of distribution parameters. Reference[38] developed profit model to determine optimal operating conditions of the machine repair problem with balking, reneging and standby switching failures. Reference[39] derived expressions for system reliability and mean time to system failure of the system with multiple unreliable service stations and standby switching failures. Using the supplementary variable technique, Reference[40] obtained the explicit expressions for the steady state availability of three systems with general repair time, reboot delay and switching failures. Reference[41] performed throughput assessment of mixed-model flexible transfer lines with unreliable machines. By using Markov chain approach and system cost analysis, Reference[42] obtained the stationary distribution of queue size of the queueing systems and provided algorithms in order to identify the equilibrium strategies for the fully and partially observable models. They derived the equilibrium threshold balking strategies and the equilibrium social benefit for all customers for the fully and partially observable system respectively, both with server breakdowns and delayed repairs. Reference[43] studied a finite-source parallel queue system by maximizing the throughput.

No authors in past had investigated switching failure of different types of spares under N policy and different service rate for servers with impatience behavior of machine during waiting for its turn for service. In this paper we investigate the machine interference problem with mixed spares, switching failure, balking and reneging under N policy and heterogeneous service rate using cost function. The server breakdown and its different types of service rates are taken into considerations. The rest of the paper is outlined as follows: In section 2, we formulate the problem in detail by stating requisite assumptions and notations. State probabilities are computed using governing Chapman-Kolmogrov difference equation of the model in section 3. The solution technique SOR has been discussed in section 4 which is followed by the performance measures in section 5. The cost has been analyzed in section 6. For realistic and deep insight about the investigation, we present numerical results in section 7. In the section 8, the conclusion and future scope are remarked.

2. Problem Statement

Consider a multi-component machining system consisting of machines, single server and two types of heterogeneous repairmen in the repair facility. The present model is appropriate for production system, manufacturing system and computer & communication system. The assumptions and notations for developing the model and constructing the governing equations are as follows.

● The system consists of identical operating machines having life time as exponentially distributed with parameter

● The rest machines are worked as standbys where, are cold spares having failure rate zero and are warm spares whose life time is exponentially distributed with parameter. The failure of warm spares is independent of state of the others.

● Whenever one of these machines fails, it is immediately replaced firstly by an available cold spares and then by warm spare. When a spare moves into an operating state, its failure characteristics will be that of an operating machine.

● The switching of spare to operating state may not be perfect. It is not unusual to consider the switching of machine has a failure probability. If a spare fails to switch to an operating machine, the next available spare attempts to switch. This process continues until switching is successful or all the spares have failed.

● Whenever an operating machine or spare fails, it is immediately sent to a repair facility where failed machines are repaired in the order of their failures.

● When both type of spares are used, the failed machines may balk; the joining probability is when i machines have already failed.

● The repair facility has one server who renders service to failed machines according to the N policy i.e. the server starts the service on the accumulation offailed machines and continues till the system is empty.

● The server takes setup time of random duration which is exponentially distributed with parameter.

● The time-to-repair of a single server is exponentially distributed with parameter in case when at least one cold spare is available, when all cold spares are exhausted but at least one warm spare is available; when all spares are exhausted.

● When server is busy, the failed machine may renege in exponential fashion with rate and if at least one cold spare is available, all cold spares are utilized but some warm spares are available and all spares are exhausted, respectively.

● As a machine is repaired, it is as good as new one and goes into standby or operating state immediately.

● The server is prone to breakdown; the lifetime of server follows exponential distribution with parameter.

● The primary repairman takes some time before starting the repair of broken down server as setup period which is exponentially distributed with the rate.

● After taking setup time, primary repairman may provide the essential repair or primary service to broken down server in Bernoulli fashion with parameter.

● The time-to-essential repair is exponentially distributed with the parameter whereas the time-to-primary service is also exponentially distributed with the parameter. Each repair is independent of the state of the other.

● After primary repair, secondary repairman provides the secondary service. The time-to-secondary service follows exponential distribution with parameter and service is independent of the state of the other.

● The failure of machine is possible only when the system is in operation i.e. the server is in working state.

● The switchover time from spare state to operating state, from failure to setup, from setup to repair, or from repair to standby state (or operating state if the system is short) is instantaneous.

3. State Probabilities

The Chapman-Kolmogrove difference equations for different states are constructed by using conservation of rate principle from figure 1.

Let be the number of failed machines in the repair facility at the time t and let the state of the server be denoted by at time t, defined as follows:

Then, is a Markov process with state space

We define the state probabilities of the system as follows:

and limit as

for steady-state analysis.

The governing equations of the model are obtained as follows:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

4. Solution Technique

The steady-state governing difference equations of the present model can be expressed in the form of

(22)

where A is the coefficient matrix of an order 5K+1, X is the column vector having elements

and 0 is the null column vector of order 5K+1.

Using the normalizing condition

(23)

the system of linear equations in (22) can be expressed as

(24)

where A* is the matrix A replacing the last row with a row vector having all unit elements and B is the column vector of the form of order 5K+1.

The system of linear equations (24) have been solved using the numerical technique ‘Successive Over Relaxation (SOR) method with parameter value 1.25’ in MATLAB 7.1 since it gives more efficient and accurate solution than any other matrix method.

5. Performance Measures

In this section, the probabilities obtained in the section 4 are used to establish some performance indices as follows:

■ The expected number of failed units in the system is given by

(25)

■ The throughput of the system is

(26)

■ The expected number of operating units in the system is given by

(27)

■ The expected number of spare machines in the system is given by

(28)

■ The Probability that the server is in the idle state is

(29)

■ The Probability that the server is in working state is

(30)

■ The Probability that the server is in the broken down is

(31)

■ The Probability that the server is under primary repair state is

(32)

■ The Probability that the server is under secondary repair state is

(33)

■ The steady-state availability of the system is

(34)

■ The steady-state failure frequency of the system is

(35)

■ The Effective balking rate is

(36)

■ The Effective reneging rate is

(37)

■ The Average rate of failed machine loss is

(38)

■ The Average switching failure rate is

(39)

6. Cost Analysis

In order to determine the optimal threshold parameter N, we construct a steady-state expected cost function per unit time and impose constraint on the availability of the system. The optimal value of N, say N*, is obtained by minimizing the cost function subject to the system availability constraint. The cost per unit time of each machine in different states and other cost elements are defined as follows:

C_O = Cost per unit time of one machine in the operating state

C_S= Cost per unit time of one unit that function as a spare

C_W= Cost per unit time when server is in working state

C_B= Cost per unit time when server is in broken down state

C_PS= Cost per unit time when the server is under primary repair state

C_SS= Cost per unit time when the server is under secondary repair state

C_L= Cost per unit time due to failed machines.

C_SR= Cost per unit time due to switching failure.

C_I = Cost per unit time that server is idle.

Now, we construct the expected cost function per unit time as

(40)

where E (O), E(S), SR, LR, P_S(SS), P_S(PS), P_S(I), P_S(W) and P_S(B)are derived in the previous section. The cost parameters are assumed to be linear in the expected number of indicated quantity since it is most general and realistic. Under only some special circumstances, costs show non-linear behaviour. In order to maintain the availability of the system at a certain level, we present the cost minimization problem in order to determine the optimal threshold parameter N* as follows:

(41)

(42)

where A is the steady-state availability of the system and can be determined as described in the previous section and A₀ is the pre-specified level of the availability of the system. To minimize cost function, until the availability constraint is satisfied, we search optimal N using a direct search method since it is a discrete quantity.

7. Numerical Results

For the performance indices obtained in the previous section 6, we perform the sensitivity analysis to explore the effects of changes in the specific parameter values of the system characteristics. The computational results are obtained by coding the computer program in the software MATLAB 7.1. We summarize the numerical results for the and in tables 1-12 by varying, , , , , , , , , , and, respectively. The trends of the performance indices namely, are displayed in figures 2-4, respectively. For computation purpose, the default parameters are fixed as: and.

Table 1. Performance measures with respect to M and λ0 M λ0 E(O) E(S) RR SR F(N) 5 0.05 4.93 4.52 0.0076 0.0037 187.26 0.45 4.91 2.32 0.0132 0.0080 158.95 0.85 4.78 1.07 0.0096 0.0082 141.62 10 0.05 9.86 4.00 0.0096 0.0052 310.79 0.45 9.75 0.97 0.0091 0.0081 813.49 0.85 9.50 0.29 0.0046 0.0060 2318.35 15 0.05 14.81 3.65 0.0109 0.0060 490.01 0.45 14.59 0.46 0.0059 0.0069 2511.47 0.85 14.36 0.13 0.0030 0.0045 6770.89

Table 2. Performance measures with respect to Y and λ0 Y λ0 E(O) E(S) RR SR F(N) 4 0.05 9.84 3.19 0.0075 0.0052 270.80 0.45 9.73 0.86 0.0069 0.0078 297.91 0.85 9.49 0.27 0.0040 0.0057 433.34 5 0.05 9.86 4.00 0.0096 0.0052 310.79 0.45 9.75 0.97 0.0091 0.0081 813.49 0.85 9.50 0.29 0.0046 0.0060 2318.35 6 0.05 9.87 4.85 0.0114 0.0052 503.09 0.45 9.77 1.07 0.0116 0.0084 5462.03 0.85 9.51 0.31 0.0053 0.0063 20677.35

Now we examine the effects of, , , and on various performance indices which are as follows:

(i) Effects of number of operating units and threshold number of failed machinesfor the server to come in action: In tables 1 and 4, we see that, and are increasing whereas is decreasing with respect to M and N respectively. In tables 5 and 9, and show the increasing trend but is decreasing as M and N increase. It is also found that is first increasing and then decreasing with respect to. In table 8, and are decreasing but and are increasing; and are almost constant with respect to. In table 12, and increases but and decrease. We see that remains almost constant with respect to. Figures 2(a), 3(a) and 4(a) reveal that the, and respectively show an increasing trend with respect to whereas there is no change in these performance indices with respect to as clear from figures 2(d), 3(d) and 4(d), respectively.

Table 3. Performance measures with respect to S and λ0 S λ0 E(O) E(S) RR SR F(N) 1 0.05 9.85 3.17 0.0093 0.0053 270.29 0.45 9.71 0.72 0.0098 0.0070 297.67 0.85 9.48 0.23 0.0045 0.0052 436.28 2 0.05 9.86 4.00 0.0096 0.0052 310.79 0.45 9.75 0.97 0.0091 0.0081 813.49 0.85 9.50 0.29 0.0046 0.0060 2318.35 3 0.05 9.87 4.85 0.0101 0.0050 518.15 0.45 9.78 1.22 0.0109 0.0089 5258.85 0.85 9.52 0.33 0.0075 0.0065 20507.34

Table 4. Performance measures with respect to N and λ0 N λ0 E(O) E(S) RR SR F(N) 2 0.05 9.84 4.25 0.0088 0.0052 307.86 0.45 9.75 0.97 0.0091 0.0081 810.84 0.85 9.50 0.29 0.0046 0.0060 2318.23 3 0.05 9.86 4.00 0.0096 0.0052 310.79 0.45 9.75 0.97 0.0091 0.0081 813.49 0.85 9.50 0.29 0.0046 0.0060 2318.35 4 0.05 9.87 3.74 0.0101 0.0052 316.51 0.45 9.75 0.97 0.0091 0.0081 816.56 0.85 9.50 0.29 0.0046 0.0060 2318.49

(ii) Effects of number of standby units, and: In tables 2-3, 6-7 and 10-11, all performance indices show increasing trend with respect to and, respectively. and are increasing gradually whereas all other indices are increasing remarkably. Figures 2(b)-2(c), 3(b)-3(c) and 4(b)-4(c) reveal that the, and respectively show an increasing trend by increasing the number of standby units i.e. and.

Figure 1. Transition diagram with following abbreviations

Figure 2. Expected number of failed units in the system by varying λ0 for different values of (a) (b) (c) (d)

(iii) Effects of failure rate of operating units: Tables 1-4 show that and decrease but increases; alsoand first increase and then decrease with respect to. In table 1, is decreasing for lower value of whereas increasing for higher values of with respect to. In tables 2-4, increases with respect to. Figures 2(a)-2(d) and 3(a)-3(d) depict that and respectively increases whereas figures 4(a)-4(d) show that decreases with respect to, which is quite obvious and desirable.

Figure 3. Availability of the system with respect to λ0 for different values of (a) (b) (c) (d)

Figure 4. Throughput of the system by varying λ0 for different values of (a) (b) (c) (d)

(iv) Effects of failure rates of standby units: In tables 5 and 6, increases whereas decreases with respect to. It is noticed that, and decreases for lower values of wher eas increase for higher value of, and increases for lower values of whereas decreases for higher values of with respect to. In table 7, , ,and decrease. We can also see that increases for lower values of whereas decreases for higher values of. It is quite clear that first increases and then decreases with respect to. In table 8, all performance indices decrease with respect to.

Table 5. Performance measures with respect to M and α0 M α0 E(O) E(S) RR SR F(N) 5 0.15 4.93 3.91 0.0154 0.0058 180.58 0.55 4.95 3.19 0.0178 0.0042 170.32 0.95 4.96 2.51 0.0181 0.0034 160.41 10 0.15 9.92 2.85 0.0175 0.0074 1756.89 0.55 9.92 2.20 0.0171 0.0062 1370.04 0.95 9.91 1.70 0.0156 0.0054 1261.37 15 0.15 14.88 1.90 0.0158 0.0082 3833.37 0.55 14.86 1.45 0.0141 0.0071 3909.53 0.95 14.84 1.16 0.0125 0.0063 4213.19

Table 6. Performance measures with respect to Y and α0 Y α0 E(O) E(S) RR SR F(N) 4 0.15 9.86 1.96 0.0095 0.0072 277.94 0.55 9.88 1.62 0.0092 0.0060 266.70 0.95 9.88 1.35 0.0088 0.0051 258.87 5 0.15 9.90 2.40 0.0133 0.0073 459.96 0.55 9.90 1.91 0.0130 0.0061 402.53 0.95 9.90 1.53 0.0120 0.0053 374.82 6 0.15 9.92 2.85 0.0175 0.0074 1756.89 0.55 9.92 2.20 0.0171 0.0062 1370.04 0.95 9.91 1.70 0.0156 0.0054 1261.37

Table 7. Performance measures with respect to S and α0 S α0 E(O) E(S) RR SR F(N) 1 0.15 9.89 2.27 0.0178 0.0072 462.05 0.55 9.89 1.97 0.0181 0.0063 431.39 0.95 9.89 1.69 0.0181 0.0056 409.19 2 0.15 9.92 2.85 0.0175 0.0074 1756.89 0.55 9.92 2.20 0.0171 0.0062 1370.04 0.95 9.91 1.70 0.0156 0.0054 1261.37 3 0.15 9.94 3.47 0.0178 0.0075 11171.38 0.55 9.94 2.44 0.0165 0.0062 7479.46 0.95 9.93 1.78 0.0145 0.0054 7854.48

Table 8. Performance measures with respect to N and α0 N α0 E(O) E(S) RR SR F(N) 4 0.15 9.92 2.81 0.0175 0.0075 1861.95 0.55 9.92 2.19 0.0171 0.0062 1415.44 0.95 9.91 1.70 0.0156 0.0054 1279.53 5 0.15 9.92 2.77 0.0174 0.0075 1992.38 0.55 9.92 2.17 0.0170 0.0062 1468.84 0.95 9.91 1.70 0.0156 0.0054 1300.03 6 0.15 9.91 2.74 0.0173 0.0074 2147.65 0.55 9.92 2.17 0.0170 0.0062 1529.28 0.95 9.91 1.69 0.0156 0.0054 1322.45

Table 9. Performance measures with respect to M and γ1 M γ1 E(O) E(S) RR SR F(N) 5 0.10 4.93 4.11 0.0145 0.0066 183.33 0.14 4.93 4.14 0.0147 0.0066 183.73 0.18 4.93 4.16 0.0149 0.0066 184.08 10 0.10 9.92 3.03 0.0173 0.0079 1897.15 0.14 9.92 3.09 0.0178 0.0080 1895.87 0.18 9.92 3.14 0.0183 0.0081 1895.45 15 0.10 14.88 2.03 0.0161 0.0086 3889.60 0.14 14.89 2.11 0.0172 0.0088 3768.84 0.18 14.89 2.18 0.0181 0.0091 3663.52

Table 10. Performance measures with respect to Y and γ1 Y γ1 E(O) E(S) RR SR F(N) 4 0.10 9.86 2.05 0.0094 0.0077 281.19 0.14 9.86 2.09 0.0100 0.0078 282.11 0.18 9.86 2.12 0.0105 0.0079 282.95 5 0.10 9.89 2.53 0.0132 0.0078 478.64 0.14 9.89 2.58 0.0138 0.0079 480.29 0.18 9.90 2.62 0.0143 0.0080 481.85 6 0.10 9.92 3.03 0.0173 0.0079 1897.15 0.14 9.92 3.09 0.0178 0.0080 1895.87 0.18 9.92 3.14 0.0183 0.0081 1895.45

Table 11. Performance measures with respect to S and γ1 S γ1 E(O) E(S) RR SR F(N) 1 0.10 9.89 2.35 0.0176 0.0075 470.69 0.14 9.89 2.38 0.0178 0.0076 470.49 0.18 9.89 2.40 0.0180 0.0077 470.32 2 0.10 9.92 3.03 0.0173 0.0079 1897.15 0.14 9.92 3.09 0.0178 0.0080 1895.87 0.18 9.92 3.14 0.0183 0.0081 1895.45 3 0.10 9.93 3.76 0.0176 0.0081 13003.47 0.14 9.93 3.85 0.0185 0.0083 13044.11 0.18 9.94 3.93 0.0192 0.0084 13094.51

Table 12. Performance measures with respect to N and γ1 N γ1 E(O) E(S) RR SR F(N) 4 0.10 9.91 2.98 0.0173 0.0079 2023.95 0.14 9.92 3.03 0.0178 0.0080 2025.30 0.18 9.92 3.08 0.0183 0.0081 2027.21 5 0.10 9.91 2.93 0.0172 0.0079 2183.71 0.14 9.91 2.98 0.0178 0.0080 2188.54 0.18 9.92 3.03 0.0183 0.0081 2193.53 6 0.10 9.91 2.89 0.0171 0.0079 2376.72 0.14 9.91 2.95 0.0176 0.0080 2385.91 0.18 9.91 3.00 0.0181 0.0081 2394.77

(v) Effects of reneging rate of failed machines: In table 9, all performance indices increase except as it increases for lower values of whereas decreases for higher value of with respect to. In tables 10 and 12, all performance indices show the increasing trend with respect to. In table 11, all performance indices increases except for and as they reveal the decreasing trend with respect to.

Based on the numerical results obtained above, we conclude that:

● As important in real time machining system, the number of operating units should be determined under cost constraints to make the system more available.

● It is quite obvious that the system designer must concentrate on the number of warm and cold spares carefully to maintain the grade of service (GoS) and just-in time model as is seen in real life situations.

● The system designer must use more number of cold spares in comparison to the respective number of warm spares.

● To minimize the cost, the repair rate of the operating units should be kept reasonably high whereas the reneging does not affect the cost much.

8. Discussion

In this investigation, we have discussed the machine repair problem with unreliable server. To increase the reliability, the facility of spares and repairmen is also considered. The realistic phenomenon of switching failure of spares and impatient behavior of failed machines has also been incorporated in this investigation. In order to decrease the backlog and to control the balking & reneging behavior of the failed machines, the server may increase their service rate after a threshold level of the queue of failed machines. Various performance indices have been developed to analyze the characteristics of the system and to perform the sensitivity analysis. The model that we studied is realistic. The future scope of the paper is to study mixed type spares facility in present model to make it more realistic and to find optimal service rate of heterogeneous servers and repairmen. We can extend this paper to search optimal number of operating and spares units at minimum cost.