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3 506 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 3, JULY 2008 In order to construct a discrete-time Markov chain, i.e., uniformize, we must discretize time. We let denote a positive small unit of time. We then have the following definition. 1) Definition 3.1: For, let denote the maximum length of the time interval during which the probability of two or more arrivals of jobs, at the th machine, is less than, i.e., (2) where is the number of arrivals at the th machine during a time interval of length. Next we introduce some notation. Fig. 1. Schematic view of n-machine system. The performance metrics that we explore for evaluation optimization of the system are: 1) average number of jobs waiting at each machine; 2) probability that the number of jobs waiting at a machine when the AGV departs from the same machine exceeds a given value; 3) average long-run cost of running the system. III. MARKOV CHAIN MODEL A. System State We introduce a Markov chain model for analyzing the system. Let the set denote the entire system that contains machines (or pick-up points). Then Dropoff point, Machine 1, Machine 2 Machine A subset of, called, can be defined as follows: Dropoff point, Machine (1) for. The behavior of the above-described system can be modeled with a sequence of Markov chains associated with the following: The Markov chain associated with will be referred to as the th Markov chain. Because each machine will be analyzed independently, we consider Markov chains, one for each machine. Also, a nice property (Theorem 4.2) of the system is that one can use the invariant distribution (limiting probabilities) of the th Markov chain to compute the same for the th Markov chain. This provides for a simple recursive scheme that can be used to determine some important system performance measures associated with the entire system from the invariant distributions. Maximum number of jobs that can wait at the th machine. Time spent by the AGV in traveling from machine to machine where machine 0 is the dropoff point. This also includes the loading time on machine when the unloading time when. An integer multiple of such that Probability that jobs arrive at the th machine in a time interval of length. Number of states in the th Markov chain. (States are defined below in Definition 3.2.) Maximum capacity of the AGV. Actual available capacity of the AGV. (It is a rom variable for all machines but the first for which it equals.) Let denote the time between the th the ( )th arrival of jobs at machine. Then, since jobs continually arrive at each machine, for any We will observe the system after unit time, i.e.,, following a stard convention in the literature (see [18, p. 435]), after unit time, a new epoch will be assumed to have begun. Thus, if one specifies, the length (time duration) of any epoch in the th Markov chain will equal. The following phenomena will be treated as events for the th Markov chain: 1) A job arrives at a machine; 2) the AGV departs from the th machine; 3) the AGV arrives at the th machine. An event will signal the beginning of a new epoch. The probability of two or more arrivals in one epoch will be assumed to be negligible for small, the definition of ensures that. Furthermore, we will assume that if a job arrives during an epoch, we will move that event forward in time to coincide with the end of that epoch. Since is a small quantity, when is small, this should not pose serious problems. This approximation is necessary to ensure the Markov property (3) (4)

4 KAHRAMAN et al.: STOCHASTIC MODELING OF AN AUTOMATED GUIDED VEHICLE SYSTEM WITH ONE VEHICLE AND A CLOSED-LOOP PATH 507 for the system we consider. However, numerical results (presented later) will demonstrate that with the approximation we still have reasonably accurate results. Moreover, the approximation allows us to use the powerful framework of Markov chains. 2) Definition 3.2: Specify. A state of the th Markov chain in the th epoch is defined by the following 4-tuple: where denotes the number of jobs waiting at machine when the th epoch begins, denotes the available capacity in the AGV when the th epoch begins, if the AGV is traveling from the dropoff point to any machine while if the AGV is traveling from the th machine to the dropoff point when the th epoch begins. Since the th epoch could begin while the AGV is traveling either between the dropoff point the machine or between the machine the dropoff point, we let denote the number of multiples of that have elapsed until the beginning of the th epoch since the AGV s departure. Clearly, the AGV s departure could either be from the dropoff point or the th machine. B. Transition Probabilities With the above discretization of time, we have the following property. For any small value of for any, itis approximately true that (5) the probability of two or more arrivals converges to zero faster than either of the other two probabilities. In other words, for a small value of, one may practically ignore the event associated with more than two arrivals. With a small, we have for every a small enough, i.e., in the two properties above. Thus, with a sufficiently small duration of the epoch, one can safely ignore the probability of two or more arrivals for certain distributions. We will prove this when the interarrival time is exponentially distributed uniformly distributed. Note, however, that even for these distributions, e.g., exponential, our approach remains approximate. 1) Theorem 3.1: If the interarrival time is exponentially distributed with mean, Properties 1 2 are satisfied. Proof: thereby satisfying Property 1. Similarly by L'Hospital's rule This implies that for nonzero, but small,, our system can be approximately modeled by a Markov chain. It is to be noted that the Markov chains constructed are essentially parametrized by the non-negative scalar. However, to keep the notation simple, because is fixed, we will suppress this parameter. Thus, will be denoted by. By the definition, the th Markov chain is only concerned with job arrivals at the th machine. The definition of requires us to analyze whether the probability of two or more arrivals in an epoch can be ignored in comparison to that of zero or one arrival. We justify the use of a small with the following properties. Consider the following two properties assuming to be the duration of an epoch. Property 1: 2) Theorem 3.2: If the interarrival time is uniformly distributed with, Properties 1 2 are satisfied. Proof: For non-poisson arrivals, to determine the probability distribution of the number of arrivals, one has to compute, the -fold convolution of the distribution of the interarrival time. If denotes the time of the th arrival (of job), then for the uniform distribution Also (6) Then Property 2: (7) Then, it follows that: When the arrival distribution satisfies these two properties, one can ignore the probability of two or more arrivals in comparison to those of zero arrivals one arrival as tends to zero because

5 508 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 3, JULY 2008 Property 1 is thus satisfied. Similarly, for Property 2, we have by L'Hospital's rule We show that the conditions (6) (7) hold for the triangular distribution in Appendix I. For interarrival distributions (of jobs) that satisfy (6) (7), we can develop expressions for the transition probabilities with a finite but small value of. The transitions are characterized by 12 conditions. For each Markov chain that we consider, we will assume, for the time being, that the available capacity is known. Subsequently, we will present a result (Theorem 4.2) to compute the distribution of. The state in the th epoch will be denoted by. The one-step transition probability will be denoted by. Fig. 2 presents a pictorial representation of the underlying Markov chain in our model. The available capacity of the AGV,, equals when it arrives at the first machine is a rom variable for all other machines. Then,, if, if (see 3.2),. The transition probabilities for this Markov chain will now be defined for 12 conditions (cases). Cases 1 6(7 12) are associated with the AGV traveling from the dropoff point to a machine (from a machine to the dropoff point). We now describe all the cases in detail. Consider the following scenario: the AGV travels from the dropoff point to machine during the th epoch, the arrival of the AGV at machine does not occur by the end of the epoch, the buffer at machine is not full. Then, if no job arrives during the th epoch, we have the following. Case 1: If,,,,,,, then Fig. 2. Markov chain underlying our model. B in figure sts for. Since no job arrival occurs in unit time, the above transition probability is equal to the probability of no job arrival. Similarly, if a job arrival does occur, we have the following. Case 2: If,,,,,,, then Case 3: If,,,,,,, then Since job arrival occurs in unit time, the above transition probability is equal to the probability of job arrival. Consider the following scenario: the AGV travels from the dropoff point to the machine during the th epoch, the arrival of the AGV at machine does not occur by the end of the epoch the buffer at machine is full. Then, we have the following. Since the buffer at machine is full, we do not take job arrivals into consideration. Consider the following scenario: the AGV travels from the dropoff point to the machine during the th epoch, it arrives at machine by the end of the epoch, the buffer at machine is not full. Then, if no job arrives during the current epoch, we have Case 4, if a job arrives we have Case 5.

6 KAHRAMAN et al.: STOCHASTIC MODELING OF AN AUTOMATED GUIDED VEHICLE SYSTEM WITH ONE VEHICLE AND A CLOSED-LOOP PATH 509 Case 4: If,,,,,,, then Case 10: If,,,,,,, then Case 5: If,,,,,,, then Consider the scenario in which the AGV travels from the dropoff point to machine during the th epoch, arrives at machine by the end of the th epoch, the buffer at machine is full. Then, we have the following. Case 6: If,,,,,,, then Consider the scenario in which the AGV travels from machine to the dropoff point during the th epoch, the arrival to the dropoff point does not occur by the end of the epoch the buffer at machine is not full. Then, if no job arrives during the th epoch, we have Case 7, if a job does arrive we have Case 8. Case 7: If,,,,,,, then Case 8: If,,,,,,, then Consider the scenario in which the AGV travels from machine to the dropoff point during the th epoch, the arrival to the dropoff point does not occur by the end of the th epoch the buffer at machine is full. Then, we have the following. Case 9: If,,,,,,, then Case 11: If,,,,,,, then Finally, consider the scenario in which the AGV travels from machine to the dropoff point during the th epoch, it arrives at the dropoff point by the end of the epoch, the buffer at machine is full. Then, we have the following. Case 12: If,,,,,,, then IV. PERFORMANCE MEASURES AND OPTIMIZATION From Definition 3.2, the number of states in the th Markov chain, i.e.,, can be computed as The following well-known result allows us to determine the invariant distribution (limiting probabilities) of the underlying Markov chains. 1) Theorem 4.1: Let denote the one-step transition probability matrix of a Markov chain. If the matrix is aperiodic irreducible, if, whose th element is denoted by, denotes a column vector of size, then solving the following system of linear equations yields the limiting probabilities of the Markov chain: From the definition of our transition probabilities, it is not hard to show that each state is positive recurrent aperiodic that there is a single communicating class of states. Then, the above result holds we may use it to compute the limiting probabilities. We next define a function,, that assigns an integer value in the set to each state in the th Markov chain Consider the scenario in which the AGV travels from machine to the dropoff point during the th epoch, it arrives at the dropoff point by the end of the epoch, the buffer at machine is not full. Then, if no job arrives during the th epoch, we have Case 10. If a job arrival occurs during the th epoch, we have Case 11. where denotes the state in the th Markov chain at a given epoch,. Note that, the epoch index, is suppressed here from the notation of (5) to increase clarity. Our transition probabilities for the th Markov chain were computed under the

7 510 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 3, JULY 2008 Fig. 3. Computational scheme based on Theorem 4.2. assumption that capacity of the AGV when it arrives at the th machine is known ( ). We now need to compute the transition probabilities of the entire system the transition probabilities that we need for our performance evaluation model. We will provide a result, Theorem 4.2, to this end. The main idea underlying the result is as follows. The transition probabilities of the th Markov chain yield the distribution of the AGV s available capacity of the th Markov chain. Because the AGV starts empty, for the first Markov chain, the available capacity is known to be. From that point onwards, one can compute the transition probabilities in a recursive style. The computational scheme based on the next result is depicted in a flowchart in Fig. 3. 2) Theorem 4.2: Let the limiting probability of state in the th Markov chain be denoted by. Let denote the available capacity of the AGV when it arrives at the th machine, further let denote the limiting probability for the state in the th Markov chain when. Then, we have that Proof: For the proof, we need to justify (8) (10). Equation (8) follows from the fact that the AGV empties itself at the dropoff point (see Fig. 1). Hence, when it comes to Machine 1, the available capacity is not a rom variable but a constant equal to, which is the maximum capacity of the AGV. Equation (9) follows from the fact that when the AGV arrives at stations numbered 2 through, its capacity is a rom variable whose distribution has to be calculated. The distribution is given in (10). For the last statement, we argue as follows: When the AGV arrives at the th station, for, its capacity is the available capacity of the AGV when it leaves the th station. The distribution for the available capacity on the AGV when it leaves the th station can be computed from the limiting probabilities associated with the th Markov chain of those states in which the AGV has departed the th machine is on the way to the th machine. (Note that denotes the set of states in the th Markov chain in which the AGV has departed the th machine its available capacity is. Similarly, denotes the set of states in the th Markov chain in which the AGV has departed from the th machine its available capacity assumes all possible values.) Exploiting the transition probabilities, one can derive expressions for some useful performance measures of the system. They are considered next. 3) Average Inventory at Each Machine: If denotes the average number of jobs waiting at the th machine, then where if if (11) for,, where (8) (9) (10) 4) QoS Performance Measure Associated With Downside Risk: The probability that the number of jobs waiting at the th machine, denoted by, when the AGV departs from the th machine exceeds can be calculated as follows: (12) where in which are defined as follows: 5) Average Long-Run Cost: The average long-run cost of running the system is composed of two elements, which are: the holding cost of the jobs near the machine the cost of operating the AGV. To calculate the holding cost, we will need the average number of jobs waiting at each machine. Then, the average cost for operating a system described in this paper, with

8 KAHRAMAN et al.: STOCHASTIC MODELING OF AN AUTOMATED GUIDED VEHICLE SYSTEM WITH ONE VEHICLE AND A CLOSED-LOOP PATH 511 machines to be served by an AGV whose capacity is given by,is (13) where are constants representing the operating cost per unit time of an AGV having unit capacity the holding cost per a unit time for one job, respectively; denotes the fixed cost of the vehicle In the uniformization procedure, the probability of two or more arrivals in one epoch is neglected. Thus, a few arrivals are unaccounted for. This causes a reduction in the values of both performance measures, i.e.,, for the first machine. This error affects the capacity distribution results in a reduced available capacity for the AGV when it visits the subsequent machines. This error in the capacity distribution partly negates the error introduced by uniformization. Hence, at all machines but the first, where there is no error in estimating the capacity, the overall error in the performance-measure values is low. At the first machine, the capacity is equal to the maximum capacity, so this reduction does not occur, thereby producing a higher error. This is reflected in the numerical results presented later. Also note that one has to find the limiting probabilities of different Markov chains, one chain associated with a specific machine in the system. However, to find the limiting probability of the specific state in a Markov chain associated with machine, where, one has to find the limiting probabilities of the different Markov chains associated with machine (see Theorem 4.2), since the available capacity of the AGV can assume any value between zero when it arrives at any machine but the first machine. (Note that since the available capacity of the AGV is when it arrives at the first machine, the limiting probabilities of the Markov chain associated with the first machine can be found without the use of Theorem 4.2.) Hence, the computational work associated with any given machine but the first constitutes of the computational work of calculating the limiting probabilities of Markov chains plus the limiting probabilities of the Markov chain associated with the given machine via Theorem 4.2. Therefore, the complexity of the Markov chain model is a first-order polynomial in the number of machines. The advantage of our computational scheme is that the state space collapses for each Markov chain, thereby making the computation of the limiting probabilities feasible. 6) AGV Capacity Optimization: We are interested in optimizing the AGV s capacity with respect to the cost of operating the system. The optimization problem considered in this paper is to determine to minimize the expression in (13). Other ways for optimization could be considered depending on what the manager desires. One example is to minimize such that where are set by managerial policy. Optimization is performed by an exhaustive enumeration of the AGV s maximum capacity variable,. Since the state space of this discrete optimization problem, which has a single decision variable, is very small, an exhaustive enumeration is feasible. V. COMPUTATIONAL RESULTS Simulation is by far the most extensively used tool for performance evaluation of AGV systems, because it provides us with very accurate estimates of performance measures. As a result, it is essential that we compare our results to those obtained from a simulation model. A. Simulation Model Consider a probability space, where denotes the (universal) set of all possible round trips of the AGV, denotes the sigma field of subsets of, denotes a probability measure on. Using a discrete-event simulator, it is possible to generate rom samples from the measurable space. The samples can then be used to estimate values of all the performance measures derived in the previous section. Let denote the number of jobs waiting near the th machine at time in the simulation sample. Then, from the strong law of large numbers, with probability 1 Also, with probability 1 (14) (15) where denotes the number of occasions in which the number of jobs left behind at machine (i.e., jobs not picked up by the AGV because it is full) equals or exceeds in visits to the machine in the simulation sample. B. Performance Tests for Markov Model We conducted numerical experiments with our model to determine the practicality of the approach to benchmark its performance with a simulation model that is guaranteed to perform well but is considerably slower. The error of our model with respect to the simulation estimate is defined by Error (%) where denotes the estimate from the Markov chain model denotes the same from the simulation model. We present results on two-machine five-machine systems, along with a simple example to illustrate our methodology. Tables I II define the parameters for the systems with two machines, Tables XI XII define the parameters for the systems with five machines studied. In Table I, denotes the rate of arrival of jobs at machine. We used the exponential gamma to model the distribution of the interarrival time of jobs. The performance metrics of the systems with two machines five machines are shown in Tables III VIII Tables XIII XX, respectively. These tables also show the simulation estimates the corresponding error values which are calculated as defined above. Convergence was achieved with

9 512 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 3, JULY 2008 TABLE I SYSTEM (SYS.) PARAMETERS. VALUE OF IS 0.05 FOR EACH SYSTEM AND EACH SYSTEM HAS TWO MACHINES TABLE IV AVERAGE NUMBER OF JOBS WAITING AT MACHINE 2 FOR POISSON ARRIVALS TABLE II SYSTEM (SYS.) PARAMETERS (CONTINUED) TABLE V AVERAGE NUMBER OF JOBS WAITING AT MACHINE 1 FOR GAMMA-DISTRIBUTED INTERARRIVAL TIME TABLE III AVERAGE NUMBER OF JOBS WAITING AT MACHINE 1 FOR POISSON ARRIVALS 1000 trips per replication we used ten replications. The mean estimate was assumed to have converged when it remained within 0.05% in the next iteration. The mean reported is an average over all replications. The stard deviation was no more than 0.1% of the mean in each case. 1) 2-Machine Systems: Tables III IV provide for Poisson arrivals for, respectively. Similarly, Tables V VI show the corresponding values for a gamma-distributed interarrival time. In these systems, i.e., in the systems with two machines, for the gamma distribution we used the same arrival rate, but the parameter was set to eight. Tables VII VIII show the values of with Poisson arrivals for, respectively; Tables IX X show the corresponding values for a gamma-distributed interarrival time.

10 KAHRAMAN et al.: STOCHASTIC MODELING OF AN AUTOMATED GUIDED VEHICLE SYSTEM WITH ONE VEHICLE AND A CLOSED-LOOP PATH 513 TABLE VI AVERAGE NUMBER OF JOBS WAITING AT MACHINE 2 FOR GAMMA-DISTRIBUTED INTERARRIVAL TIME TABLE VIII PROBABILITY THAT AGV LEAVES TWO OR MORE JOBS BEHIND FOR POISSON ARRIVALS TABLE VII PROBABILITY THAT AGV LEAVES TWO OR MORE JOBS BEHIND FOR POISSON ARRIVALS TABLE IX PROBABILITY THAT AGV LEAVES TWO OR MORE JOBS BEHIND FOR GAMMA-DISTRIBUTED INTERARRIVAL TIME 2) Example: Consider System 5, whose parameters are given in Tables I II. In this system, jobs arrive at the rate of 2 per unit time at each machine. We choose. Here, for. The AGV travels from the dropoff point to Machine 1, from Machine 1 to Machine 2, from Machine 2 to the dropoff point in time equaling,,, respectively. The maximum capacity of the AGV is 2, the buffer capacities of each machine are 4. We provide some sample transition probabilities for the Markov chain associated with Machine 1 Case 1: Case 3: Case 5: Case 10: The capacity of the AGV when it arrives at machine 1 is, the capacity distribution of the AGV when it arrives at Machine 2 is calculated using Theorem 4.2 Then, the transition probabilities for the second Markov chain can be calculated from the distribution of. The limiting probabilities can then be used to calculate the values of the performance measures, such as the average inventory the downside risk.

11 514 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 3, JULY 2008 TABLE X PROBABILITY THAT AGV LEAVES TWO OR MORE JOBS BEHIND FOR GAMMA-DISTRIBUTED INTERARRIVAL TIME TABLE XIII AVERAGE NUMBER OF JOBS WAITING AT EACH MACHINE FOR POISSON ARRIVALS TABLE XIV TABLE XIII (CONTINUED) AVERAGE NUMBER OF JOBS WAITING AT EACH MACHINE FOR POISSON ARRIVALS TABLE XI SYSTEM (SYS.) PARAMETERS. VALUE OF IS 0.05 FOR EACH SYSTEM, INTERARRIVAL TIMES ARE i. i. d., EXPONENTIALLY DISTRIBUTED AND EACH SYSTEM HAS FIVE MACHINES TABLE XV AVERAGE NUMBER OF JOBS WAITING AT EACH MACHINE FOR GAMMA DISTRIBUTED INTERARRIVAL TIMES TABLE XII SYSTEM (SYS.) PARAMETERS. VALUE OF IS 0.05 FOR EACH SYSTEM, INTERARRIVAL TIMES ARE i. i. d., GAMMA DISTRIBUTED AND EACH SYSTEM HAS FIVE MACHINES. G DENOTES GAMMA 3) Five-Machine Systems: Tables XIII XIV list the values of for, Tables XVII XVIII list the values of for for Poisson arrivals. The corresponding values for a gamma-distributed interarrival time are shown in Tables XV XVI (for the average number waiting) Tables XIX XX (for the probability). Tables XXI XXII present the results from optimization performed with the Markov chain model. Fig. 4

12 KAHRAMAN et al.: STOCHASTIC MODELING OF AN AUTOMATED GUIDED VEHICLE SYSTEM WITH ONE VEHICLE AND A CLOSED-LOOP PATH 515 TABLE XVI TABLE XV (CONTINUED) AVERAGE NUMBER OF JOBS WAITING AT EACH MACHINE FOR GAMMA DISTRIBUTED INTERARRIVAL TABLE XIX PROBABILITY THAT AGV LEAVES TWO OR MORE JOBS BEHIND FOR GAMMA DISTRIBUTED INTERARRIVAL TIMES TABLE XVII PROBABILITY THAT AGV LEAVES TWO OR MORE JOBS BEHIND FOR POISSON ARRIVALS TABLE XX TABLE XX (CONTINUED). PROBABILITY THAT AGV LEAVES TWO OR MORE JOBS BEHIND FOR GAMMA DISTRIBUTED INTERARRIVAL TIMES TABLE XVIII TABLE XVII (CONTINUED) PROBABILITY THAT AGV LEAVES TWO OR MORE JOBS BEHIND FOR POISSON ARRIVALS TABLE XXI OPTIMIZED CAPACITY (Z ) AND OPTIMAL COST (C ) FOR SYSTEMS WITH TWO MACHINES. HERE, c =\$300, c = \$550, AND c = \$10000 TABLE XXII OPTIMIZED CAPACITY (Z ) AND OPTIMAL COST (C ) FOR SYSTEMS WITH FIVE MACHINES. HERE, c = \$300, c = \$550, AND c = \$10000 shows the cost values for the different capacities the need for optimization. Figs. 5 6 show the effect of a bigger value of on the value of for Systems 30 31, respectively. Figs. 7 8 show the effect of the same on for Systems 30 31, respectively. As seen from these graphs, the accuracy of the Markov model decreases as increases. Clearly, as decreases, the discrete-time approximation gets closer to its continuous limit [see (2) (4)]. The major conclusion from our experiments is that the Markov chain model produces a solution with a good quality does so in a very reasonable amount of computer time, which is less than 2 minutes on a Pentium processor PC with 2-GHz CPU frequency 250-MG RAM size. The simulation model in comparison takes about four times as much time.

13 516 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 3, JULY 2008 Fig. 4. Total cost versus capacity of AGV for System 2. Fig. 7. Effect of a bigger 1 on P [D(n) > 1] in System 30. Errors in other parameters, i.e., P [D(1) > 1], P [D(2) > 1], P [D(5) > 1] are even smaller for plotted values of 1. Fig. 5. Effect of a bigger 1 on [W (n)] in system 30. Errors in other parameters, i.e., [W (4)] [W (5)] are even smaller for plotted values of 1. Fig. 8. Effect of a bigger 1 on P [D(n) > 1] in System 31. Errors in other parameters, i.e., P [D(1) > 1], P [D(2) > 1], P [D(4) > 1], P [D(5) > 1] are even smaller for plotted values of 1. VI. CONCLUSION Fig. 6. Effect of a bigger 1 on [W (n)] in System 31. Errors in other parameters, i.e., [W (4)] [W (5)] are even smaller for plotted values of 1. The simulation model the theoretical model make identical assumptions about the system. The computer codes can be requested from the first author. In this paper, we studied a version of a problem commonly found in small-scale manufacturing industries, e.g., pharmaceutical firms, which require no more than one AGV that operates in a closed-loop path. We used a Markov chain approach for developing the performance-analysis model. Some special properties of the system were exploited to define a simple structure to the complete problem; the structure allows us to decompose the -machine system (Markov chain) into individual systems (Markov chains) provides us with a mechanism to simplify the computations. This leads to a state-space collapse that is computationally enjoyable. Also, the discretization of the state space in our approach yielded simple expressions for the transition probabilities; a common criticism of many transition-probability models is that they have complicated expressions with multiple integrals (that require numerical integration, which is slow) as transition probabilities.

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