State Dependent Models of Material Handling Systems in Closed ...

Closed Networks TND/MHS

typeset October 28, 2010

Smith & Kerbache

State Dependent Models of Material Handling Systems in Closed Queueing Networks ∗

Laoucine Kerbache † email:[email protected]

J. MacGregor Smith email: [email protected]

October 28, 2010 Abstract — A comprehensive algorithmic analysis of finite state dependent queueing models and exponentially distributed workstations is formulated and presented. The material handling system is modeled with finite state dependent queueing network M/G/c/c models and the individual workstations are modeled with exponentially distributed single and multi-server M/M/c queueing models. The coupling of these queueing models is unique via the material handling structure. The performance modeling of the systems for series, merge, and split and other complex network topologies are included so as to demonstrate the type of topological network design (TND) that is possible with these incorporated material handling systems (MHS). Of some importance, it is shown that these integrated M/M/c and M/G/c/c networks have a product form when the population arriving at the M/G/c/c queues is controlled. Numerous experimental results demonstrate the efficacy of our approach for a variety of contexts and situations. Keywords — Closed Networks, Multi-chain, Performance, Optimization,

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λ(Wk ) := throughput rate; µi := workstation service rates (e.g. III) index on conveyance

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Figure 1: Closed Loop Queueing Network System

∗ Department † HEC

of Mechanical and Industrial Engineering, University of Massachusetts Amherst Massachusetts 01003 School of Management, Paris, France

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I NTRODUCTION

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losed queueing networks are often used in modeling manufacturing and service systems because one can control the work-in-process (WIP). This is very valuable from a managerial perspective because it demonstrates the influence of the WIP on the performance measures of the system. Unfortunately, it makes the resulting system much more complex to analyze as opposed to open queueing network models [48]. In this paper, we also couple this with modelling of the the travel time movement of the products (entities/customers) from one work station to the next so that the influence of the material handling system along with the WIP is included. Thus, integrating the material handling system affords a comprehensive environment to design and control the production or service system. 1.1 Motivation Figure 1 illustrates the type of closed network model we wish to analyze and its related complex topologies (see Figure 6) of series, merge, and split systems. This paper proposes a novel and unique algorithm and associated methodology to incorporate the material handling system through finite state dependent queues along with the multi-server workstation components. The end result is a flexible software tool for modeling complex production and service oriented environments where the material handling or transport of commodities is an important element of the system. What is unique about this paper? We have developed approaches using our methodologies for finite buffer queueing systems in the past [22, 23], but we have found as described in this paper, another simpler approach. This new approach models the finite buffer as part of the material handling system and in this way provides a unique and different methodology to model finite systems whereas before, we associated the buffers strictly with the workstations. A most recent paper concerning this new approach appears in [40]. Most research studies using analytical methods either do not account for travel time between work stations or else assume it is instantaneous. However, the material handling system is a buffer and one should account for the travel time in the buffer. This is what we do. While this new approach will not allow us to model general service time distributions at the work stations (at least at this point in time in this paper), it will afford a practical approach to model finite buffer systems in closed queueing networks through an infinite queueing system approach. 1.2 Outline of Paper The structure of the paper is comprised of six sections. Section §1 includes this introduction, motivation and outline of the paper while §2 provides a background and overview of the relevant literature about the problem. §3 presents the details about the mathematical models involved in the research. §4 provides the algorithmic details of the methodological implementations and §5 includes experimental results where first the performance modeling of single chain systems are considered, then multi-chain(class) systems are included to round out the methodology. §6 concludes the paper with a summary and a set of open questions for further research. 2

P ROBLEM B ACKGROUND

Fundamentally, the initial work on closed queueing models begins with Koenigsberg, [24] who modelled single closed-loop or cyclic systems. Gordan and Newell [16, 17] examined the general closed queueing network model and their stochastic equivalents, and Basket-Chandy-Muntz and Palacios, [2] extended the range of queues to be used in product form networks. The algorithmic concepts of mean value analysis (MVA) as proposed by Reiser and Lavenburg [36] provided significant breakthrough in the solution of closed systems. Buzacott and Shantikumar extensively discuss the issues and advantages of open and closed queueing network models in manufacturing and material handling systems [8]. There are many papers on infinite systems such as Madu [26] who looks at maintenance repair centers, Solot and Bastos [41] who examine flexible manufacturing systems (FMS) with several pallet types, Di Mascolo, Frein, and Dallery [13] who examine kanban systems as closed queueing network models, and for finite buffer systems such as Suri and Diehl, [43, 44]. Additionally, certain works have been published regarding general service time distributions for finite queues including Akyilidiz [1], and Bouhchouch, Frein, and Dallery, [5], and also the work of Tempelmeier and Kuhn, [46]. Two fairly recent Ph.D dissertations have dealt with approaches to finite buffer closed queueing network models namely one by Edgar Gonzales [15] and the other by Mustafa Yuzukirmizi [49]. The literature on material handling systems will be roughly divided into two classes: i) discrete systems (transporters, carts, vehicles, etc.) and ii) continuous systems (conveyors, escalators, elevators, etc.) Discrete material handling systems where travel time has been modelled have seen a few analytical papers beginning with the work of Benson and Gregory [3] who extended the work of Koenigsberg [24] and considered DocNum

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closed cyclic systems with exponentially distributed transit times between successive stations. Posner and Bernholtz [33] followed Benson and Gregory where they modelled a closed queueing network of two stations and model the transfer time between stations (time lags) with a general distribution through a supplementary variables technique. Posner and Bernholz [34] extended their work to more stages and probabilistic transfers and eventually several classes of customers [35]. Nishball and Koenigsberg modelled cyclic queues for maritime fleet operations where they considered general service times at the port operations. In the semiconductor area, where automated guided vehicles (AGVs) are employed, there is a more extensive body of literature on analytical models. Since the focus of our paper is on conveyor models, we will not go into any detail on this literature. For interested readers a sample of the breadth of related papers includes the papers by Ting and Tanchoco, 2000, [45] who examine an analytical model for a single loop overhead material handling system, Curry et. al. [10] who examine a general queueing network model for AGV systems, Muduli, P.K. and T.M. Yegulalp who examine truck-shovel systems in mining operations [27], and Nazzal and McGinnis [29] who examine a discrete time Markov chain model for multi-vehicle performance in a single closed loop. We can model discrete material handling systems with our state dependent models, and some simple examples are included in this paper. Explicitly modeling the continuous material handling systems especially conveyors along with the workstations has not seen that much progress in the analytical model area, while simulation models of such systems abound. Gregory and Litton, 1975, [14] examined closed loop conveyors with a stochastic model with only unloading stations and Muth [28] studied a closed-loop model with random material flow. Sonderman, 1982, [42] modeled a recirculating conveyor with a single loading and unloading station as a GI/M/1/1 system. Schmidt and Jackman, 2000, [37] studied a recirculating conveyor as a network of queues accounting for blocking.

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M ATHEMATICAL M ODELS

In this paper, M/M/c queues are employed to measure the material processing delays while M/G/c/c queues are used to capture the travel times of the material handling flows in the systems. The combination of processing times and travel times in modelling these systems is one of the central ideas of this paper and we will show that what results is a powerful analytical tool. All the experiments used in the paper are validated with ARENA simulation runs.

3.1 Assumptions We assume Exponential service time distributions in the closed network so that given the queueing types, the intra-arrival (throughput) and departure processes will be assumed to be Poisson. We further assume that there is a single class of customer for all product chains in the network. Since, however, the M/G/c/c queue can cause blocking in the system, we have to further limit the finite population and the service rates so that the capacity of the M/G/c/c queues is not violated. A property of the network we will analyze will be presented to constrain the type of networks we examine. This restriction is not felt to be a major impediment to the analysis. If it is an impediment, then closed models with blocking should be pursued.

3.2 Notation This section presents most of the notation we need for the paper: DocNum

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Variable A ai (mi ) E(σ) G G k λℓk mi µj µC c N ρ τℓ,k θ(Wk )

wℓk Wk y(ℓ, k)

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Description Number of product chains in the network. Service time function in queue i at steady-state. Expected service time of unit in an M/G/c/c queue. General service time distribution in the M/G/c/c queue Normalization constant. Index on a chain in the network. Poisson arrival rate to node ℓ in a class (k) closed network. Number of customers in queue i Exponential mean service rate at node j. Conveyor speed rate for an M/G/c/c queue with fixed conveyor length. Number of servers. Number of stations in the network. λ (µc) ,Proportion of time each server is busy. Average service time at queue ℓ in a class (k) closed network. Throughput of the closed queueing network as a function of the population Wk . Average delay at node ℓ in a class (k) closed network. Number of products(customers) i.e. the population in a single chain(class). The arrival rate (throughput) of class k products at queue ℓ relative to the arrival rate of class 1 products at queue 1.

3.3 M/G/c/c State Dependent Models There are many papers where we have described M/G/c/c performance modelling and optimization models for vehicular and pedestrian traffic flows. The difference in this paper is that we will examine closed queueing network models and apply the M/G/c/c model for modeling the flow of parts on conveyors and also incorporate the interactions of M/G/c/c queues with M/M/c queues. Graphically, the abstract representation (top diagram) and the iconic representation (bottom diagram) with a flow of parts of an M/G/c/c queue is given below in Figure 2. M/G/c/c λ

θ µ(Wk )

Width

Velocity (meters/sec)

Input rate (units/sec)

Output rate (units/sec)

Length (meters)

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Figure 3 illustrates the type of empirical state dependent rate for pedestrian studies which we have incorporated into many of our models. The letters in the graph (a),(b),(c), etc. refer to empirically derived studies which underscore the state dependent behavior of the speed-density curve, see Tregenza for more details, [47]. In the graph relating speed and density, the y-axis reflects that the speed of a unit in an M/G/c/c queue decreases with increasing density of pedestrians along the x-axis. Figure 4 illustrates the state dependent model that is appropriate for modeling vehicles or transporters such as occur in material handling networks. The letters in the graph (a),(b),(c), etc. also refer to empirically derived studies for vehicular speed-density curves. That both these systems have this exponential decay rate attests to the generality of M/G/c/c models for certain types of material handling and transport systems. This type of vehicular state-dependent curve will be used to model automated-guided vehicle systems (AGVS) later on in the paper in §5.4. An important concept of an M/G/c/c queue is the probability distribution of the number of parts in the queue which is given by the following theorem. Technically speaking, there is no queue, but since the capacity of the queue is C, when we refer to the number of parts in the queue, we are concerned with the work-in-process (WIP) which includes those in service. Suppose customers arrive according to a Poisson process having rate λ. Any arrival that finds all C servers busy does not enter the system but is lost to the system. Further, let us assume that service times of the customers are distributed according to a general distribution G. The service rate is dependent on the number of customers in the system, given that there are n people in the system, each server processes work at rate f (n). In other words, if there is an arrival, the service rate will change to f (n + 1), or if there is a departure, the service rate will change to f (n − 1). ¯ We shall suppose that G is a continuous distribution having density g and failure rate µ(t) = g(t)/G(t). Loosely speaking, µ(t) is the instantaneous probability intensity that a service of t units old will end. By letting the state at any time be the amount of work already performed on customers still in the system, that state will be x = (x1 , x2 , . . . , xn ), x1 ≤ x2 ≤ . . . ≤ xn . If there are n customers (n ≤ C) in the system and x1 , x2 , . . . , xn is the amount of work already performed on these customers, the process of successive states will be a Markov process in the sense that the conditional distribution of any future state will depend only on the present state. In the formula which follows E(σ) is the mean service time of a lone occupant flowing through a transport mechanism of length L. Theorem [Cheah and Smith , 94][9] For an M/G/c/c state dependent queue, the number of customers in the system has the distribution: P (n) =

1+

[λE(σ)]n n!f (n)...f (2)f (1) PC λE(σ)]i i=1 [ i!f (i)...f (2)f (1)

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40 (b) 30 (c) 20 (d) 10

(a) (e)

(f) 0 0 20 40 60 80 100 120 140 160 180 200 Density (veh/mi./lane)

Figure 4: Empirical Vehicular Speed-Density Curves The previous property will be utilized to calculate the number of parts on conveyors or the transport mechanism. Thus, our M/G/c/c model can be used to model individual transporters as well as parts traveling along accumulating conveyors in a MHS. Figure 5 represents the type of closed network series, merge, and split topology M/M/c workstation and M/G/c/c conveyor modules and their combinations we wish to examine so that we can predict the work-in-process (WIP) and throughput θ of these interacting systems. One might be skeptical of the speed-density effect of parts on accumulating conveyors. In practice, there is a complex relationship between the size of the parts, friction between the parts, friction between the parts and the sides and the materials of the conveyor which causes the decay in the velocity of parts along the conveyor. We have searched for the equivalent empirical curves for conveyors, but there do not seem to be any available within the literature. However, there is a group of physicists from China who have done research on granular material on conveyors, investigating the relationship between the opening of a chute diameter in a conveyor and the blocking probability affecting the movement of the granules on the conveyor [12]. One equation they developed investigates 2 the relationship between density ρ (parts/m ), velocity v (m/sec), and flow rate Q (parts/sec) and the opening size of the chute R. f := ρvR = Q (2) Solving for v, we have v=

Q ρR

(3)

Fixing the opening of the chute outlet d = R/16 due to the diameter of the part size, then one can generate a plot, see Figure 6, where one sees that the relationship between the velocity-density function of the granular parts is exponential as one expects. 3.4 Closed System Properties As mentioned earlier, given the finiteness of the M/G/c/c model, we need to limit the number of customers (population) within network models we want to examine. This is referred to as a stability condition for the closed queueing networks examined. Assuming that there are A chains of products labeled by a = 1, . . . , A. Chain k contains Wk products k = 1, 2, . . . , A. While there could be several classes of products, for this paper, we will assume we only have one class of products. If the average population level summed across the chains in the closed network model and eventually passing through each M/G/c/c queue is so large that at a single finite queue of the M/G/c/c type overflows, then, the parts (customers) will be lost. We want to avoid this situation. DocNum

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Figure 5: M/M/c & M/G/c/c Queue arrangements Let’s assume we have a simple closed network model of two queues, one an M/G/c/c queue and the other a single-server station. Let’s further assume a finite population of c − 1 customers. Figure 7 is a good example of how the integration of the M/G/c/c and M/M/c models work so that the output process of the M/G/c/c queue provides Poisson arrivals to the M/M/c queue and since there is no blocking in the M/G/c/c queue there is no interruption of the flow processes. If we develop the probability distribution for this system using the local balance equations, we can see that the probability distribution of the number of customers in the M/G/c/c queueing network can be bounded below its capacity. Thus, pc = 0. µ(c − 1)

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For example, let’s say that we have a small network as in Figure 7 with Wk = 2 customers. The M/G/c/c queue DocNum

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Graph of Velocity vs. Density(rho) and Flow Rate(Q)

1.2 1 0.8 vel

0.6 0.4 0.2 20

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flow rate

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Figure 6: Conveyor Speed-Density Curve Wk = c − 1 M/G/c/c

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Figure 7: Proposition Illustration has capacity c = 5 and say that the state dependent service rate follow the curves in Figure 3 with a maximum E(σ) = 1.5m/sec. Finally, the service rate of the M/M/c queue is µ = 1. There are only three states for this system, viz. {(2, 0), (1, 1), (0, 2)}, we have the following intensity rate matrix: (2,0) (2,0)

Q=

(1,1) (0,2)



−3  1 0

(1,1)

(0,2)

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0

−5 2

3 2

1

−1

 

Solving for the exact steady state probabilities, we have, P(2,0) =

2 6 9 ; P(1,1) = ; P(0,2) = 17 17 17

Calculating the average number in each queue, we have: Lmgcc = g

10 mmc 24 ;L = 17 q 17

So, if we can keep the average number of customers in the M/G/c/c below c, then there will be no queueing possible and we shall have no blocking. Thus, we have: Proposition 1: In order for the M/G/c/c queues not to overflow and cause blocking in the network, a restriction of the number of customers visiting the finite queues is required so that the blocking probability in each M/G/c/c queue is pc = 0, A X λℓk wℓk ≤ cℓ − 1 ∀ ℓ M/G/c/c queues k=1

PA Proof. k=1 λℓk wℓk represents the average number of customers arriving to the M/G/c/c queue. Given the finite population in the network and summing over all chains incident to the M/G/c/c queue, if this sum is below c, there can be no blocking in the queue node, and no customers will be lost, so pc = 0. DocNum

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One might think that this limitation is too restrictive, however, a conveyor acts also as a buffer besides being a transport mechanism and usually maintains a fairly large capacity. Even in many other applications, the finite capacity of an M/G/c/c queue can be quite large so this is not a problem. We also assume that the network contains N queues with associated servers. As will be discussed in the next sections, these queues are used to represent physical entities such as workstations, facilities, warehouses, manufacturing processes, material handling, warehouse staging, packaging, and shipping and delivery activities. This stability condition is to insure that the customers do not overflow the state dependent queues, because then they would block the M/M/c queues which would be unacceptable in the methodology for this paper. Since we have a finite population circulating in the system, we insure that the number of customers at the M/G/c/c queues is below the threshold value for the blocking probability. We could allow for blocking, but this will be treated in other papers. Probably the most important property of these state dependent queues is that they are quasi-reversible which implies that they act independent of one another as shown in the following property. Corollary 1: [Cheah and Smith, 1994] [9] In the M/G/c/c state dependent model, the departure process (including both customers completing service and those that are lost) is a Poisson process at rate λ. In order to illustrate the implications of this property, let’s examine an M/G/c/c queue in isolation and see what the property means. We will simulate an example M/G/c/c queue with L=8 meters;W=2.5 meters, so c = 5LW=100 and λ = 3 according to the Figure 3 state dependent curve. This arrival rate will generate some blocking, but we are interested in the inter-departure time distribution, since we assume that the inter-arrival time distribution is Poisson. Utilizing a simulation model for M/G/c/c queues, we can capture the statistics so that the inter-departure time distribution can be represented. Tabular results for the simulation are given in the following table. Measures pc θ Eq Ets

Mean 0.204928 2.367667 69.581052 29.716352

95% lower 0.139494 2.174105 56.262485 21.973235

95% upper 0.270361 2.561229 82.899618 37.459469

Table 1: Performance of M/G/c/c Queue Figure 8 illustrates the inter-departure time probability density for this M/G/c/c queue which is in fact a Poisson process.

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Figure 8: Inter-departure Time Distribution Together with Proposition 1 which insures that no customers are actually lost in the network, this latter property leads to the fact that these state dependent queues can be incorporated into a product form network along with M/M/c queues, since the output processes of these queues are all Poisson (along with the stability condition). We also assume that the network contains N queues with associated servers. As will be discussed in the next sections, DocNum

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these queues are used to represent physical entities such as workstations, facilities, warehouses, manufacturing processes, material handling, warehouse staging, packaging, and shipping and delivery activities. As a result of the previous property, the M/G/c/c queues are quasi-reversible. This is a necessary assumption to have a product form solution in the closed queueing network model. In fact, there is a nice property from Kelly’s book about closed queueing networks that is appropriate here. Theorem 3.12[[19], page 86] A closed network of quasi-reversible queues has the following properties: i) The equilibrium distribution is of the form: π(x1 , x2 , . . . , xN ) = G(M (1), M (2), . . .)π1 (x1 )π2 (x2 ) · · · πJ (xN ) where G(M (1), M (2), . . .) are normalizing constants so as π sums to unity. ii) Under time reversal, the system becomes another closed network of quasi-reversible queues. iii) When a customer of a given class arrives at a queue, the disposition of the other customers in the system is distributed in accordance with the equilibrium distribution which would obtain if they were the only customers in the system. Kelly’s Loss Networks have been shown to have the product-form property[20]. This leads to the final critical property needed for our algorithms. The proposition that follows is based upon an assemblage of previous results and the properties of M/G/c/c nodes just presented, namely the quasi-reversibility and the stability condition. In general, the product-form result follows from the definition of quasi-reversibility and the results of Baskett, Chandy, Muntz, Palacios (BCMP) networks [2]. BCMP requires that the service rates have a rational Laplace transform, but since most of the state dependent curves have decaying service rates that are Exponential, this is not a problem. The incorporation of the speed-density curves in the probability distribution of the M/G/c/c queue is detailed below in the proof. Posner and Bernholtz showed in their closed network models where the time lags represented the material handling moves from one station to another, that the state probabilities of the system are independent of the form of the time lag distribution [33]. Thus, even though the service distribution of the M/G/c/c queue can be general, the independence of the state probabilities of the M/M/c queues is maintained. Finally, with Proposition 1 controlling the population and the arrival rate to the M/G/c/c queues, the blocking probability for the M/G/c/c queue is pc = 0. Proposition 2: The integrated M/M/c and M/G/c/c queueing models will result in a network which has a product form solution: 1 π(S) = ΠA yk (Wk )ΠN (4) i=1 ai (mi ) G k=1 where π(S) is the probability distribution of the number of customers and the network state S, G is a normalizing constant, Wk is the total network population in chain k, the function yk , 1 ≤ k ≤ A, is defined in terms of the network parameters and functions ai (mi ) depend on the state of the system and the type of service center i, 1 ≤ i ≤ N. Proof. For an M/G/c/c queue, the steady state probabilities are normally generated by the following equations: pn =

λ0 λ1 . . . λn−1 p0 µ1 µ2 . . . µn

(5)

such that c X 1 λ0 λ1 . . . λn−1 =1+ p0 µ1 µ2 . . . µn n=1

(6)

In the context of our investigation, the arrival rates are not influenced by n, and thus, we define λ, such that λ = λ0 = λ1 = · · · = λc which yields: pn = DocNum

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 c  X 1 λn   =1+ p0 n=1 Πni=1 µi

(8)

In developing the exponential congestion model, µn , we assume that the service rate of each of the n occupied servers, is related to the number of parts or products on the transport device by an exponential function. The explicit form of the exponential function is based on the speed density curves relevant to the transport device. The exponential state dependent delay curve we utilize to fit the material handling speed or transport velocity is derived in the following way. We assume a relationship of the form: γ    n−1 Vn = A exp − β where Vn is the velocity of the nth customer, A := free-flow speed of an occupant, and β and γ are parameters. β and γ are determined algebraically by solving for the following equations   ln(V −a/A) ln ln(Vb /a)   γ= ln a−1 b−1

(9)

b−1 a−1 β=   γ1 =   γ1 ln(A/Va ) ln(A/Vb )

(10)

With β, γ then the service rate which is used in the MVA algorithm is:   γ  A (n − 1) µn = n exp − L β

(11)

Then substituting µn into Equation 7 and 8 we obtain pn =

and where

λn   γ  p0 , for n = 1, · · · c (n−1) exp − Πni=1 i A L β

c  X 1  =1+ p0 n=1

λn   γ  (n−1) A n Πi=1 i L exp − β



(12)

(13)

Since we have no blocking of the M/M/c queues by the M/G/c/c queues and we thus have the required independence of the queues in order to have a product form solution. Output from the M/M/c models are Poisson and input to the M/G/c/c models. Therefor, the population in the network is controlled so that average number PA k=1 λℓkt wℓk passing through an M/G/c/c queue ≤ cℓ − 1, there is no blocking in the M/G/c/c nodes, so the output from the M/G/c/c nodes is Poisson and is input to the M/M/c nodes. The notation will be expanded below and this notation that follows is based on the model and algorithm which originally appeared in [39]. We allow queues to adopt one of the following three queueing disciplines: 1. First Come First Served (FCFS)- If a queue ℓ has a FCFS discipline, products are served in order of arrival; the queue may contain Mℓ identical servers, Mℓ = 1, 2, . . .. Service times are restricted to an exponential distribution with an average given by τℓk = τℓ for a class k product. We introduce the notation µ(i) = min(i, Mℓ). In subsequent sections, this type of model is referred to as an M/M/c queue (c = Mℓ ) 2. Infinite Server (IS)- If queue ℓ has an IS discipline, products are delayed independently of each other; the queue behaves as if each product has his own server. Service times for a class k product may come from an arbitrary distribution. The average service time for a class k product is tℓk but could differ from class to class. For notational purposes, we introduce µ(i) = i. This type of node is referred to as a M/G/∞ queue. DocNum

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3. State Dependent Queue (M/G/c/c )- If queue ℓ has an M/G/c/c discipline, products are delayed depending upon the number of products within the queue. Each product has its own server. Service times for a class k product may come from an arbitrary distribution. In general, upon receiving service from queue ℓ, a class (k, t) product proceeds to queue m and transforms into a class (k, s) product with probability p(ℓ, m, t, s, k). Whenever we have a single class Tk = 1 is true (for this paper), this probability is abbreviated as p(ℓ, m, k). One can visualize the routing of chain k products as described by a discrete time Markov process. Hence, all results stemming from the theory of such processes apply. In particular, the mean number of visits a product of class k makes to queue n between successive visits to queue 1 as a class 1 product satisfies the equations: A X y(ℓ, k) = y(m, k)p(ℓ, m, k) m=1

with ℓ = 1, . . . , N,

k = 1, . . . , A

y(1, k) = 1;

k = 1, . . . , A

One can also think of y(ℓ, k) as the arrival rate (throughput) of class k products at queue ℓ relative to the arrival rate of class 1 products at queue 1. Let (X1 , . . . , XN ) be the state vector of the network, where the Xi are given by Xi = (mi1 , . . . , mi1 , mi2 , . . . , miA ) PN and mik is the number of class k products in queue i. Note the restriction i=1 mik = Wk . Under our assumptions, the equilibrium probability of being in state (X1 , . . . , XN ) is given by π(S) =

1 Π1 (X1 ) · · · ΠN (Xn ) G

where Πi (Xi ) = ai (mi )mi !ΠA k=1 ([y(i, k)τik ]

ai (mi ) =

i Πm j=1

1 , µi (j)

mi =

A X

mik ) mik ! mik

k=1

and G is the normalizing constant. 4

A LGORITHM

The performance measures that are of interest in our networks are the cycle times, throughputs, utilization of resources, and queue lengths. In the next section, these measures will be related to the following performance measures which can be determined directly from the queueing network model: expected queue lengths, throughputs, and expected delays of class k products at each of the queues. Reiser and Lavenberg [36] developed an efficient algorithm for obtaining these performance measures from product form networks. Their algorithm assumes that there is a single class for all chains k. Algorithm This algorithm determines the performance measures for product form network based upon the assumption that all FCFS queues have a single server (M/M/1) Variables: i := (i1 , i2 , . . . , iA ) a vector of the chain population. For the network with a given population vector i: nℓ (i) := is the expected length of queue ℓ, λℓ,k := is the throughput of class k products at queue ℓ, wℓ,k := is the expected delay of class k products at queue ℓ, The algorithm is based on the three fundamental equations: 1. Little’s equation for queues: A X nℓ (i) = λℓ,k (i)wℓ,k (i) (14) k=1

2. Little’s equation for product chains: ik λ1,k,1 (i) = PN [ ℓ=1 wℓ,k (i)yℓ,k (i)]

(15)

3. Reiser and Lavenberg’s property of product form networks:

wℓ,k (i) = τℓ,k [1 + nℓ (i − ek )] DocNum

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where ek is a vector of all zeroes except in the k th component which is set to 1. The algorithm is initialized with a zero population vector, then incrementally updates the waiting times, throughputs, and, finally, the average number in the queues. The details of the pidgin algorithm are given in the following:

MVA Algorithm [Initialization] for ℓ ∈ {1, 2, . . . , N} do nℓ (0) ← 0 [Obtaining the Performance Measures] for i1 ∈ {0, 1, . . . , W1 }, i2 ∈ {0, 1, . . . , W2 }, . . . , iA ∈ {0, 1, . . . , WA } do for ℓ ∈ {1, 2, . . . , N} do for k ∈ {1, 2, . . . , A} do if ik = 0 then wℓ,k ← 0 else wℓ,k ← τℓ,k [1 + nℓ (i − ek )] for k ∈ {1, 2, . . . , A} do P  λ1k ← ik / N ℓ=1 wℓ,k y(ℓ, k) for ℓ ∈ {1, 2, . . . , N} do   PA nℓ (i) ← k=1 λℓ,k wℓ,k 5

N ETWORK E XPERIMENTS

In the following section, we present some preliminary results for the basic arrangements of M/M/c and M/G/c/c queues. We need to compare our analytical results with an Arena simulation of the same system since the simulation model does not incorporate the speed-density curves in the travel time along the conveyor. So our theoretical results only approximate the simulated values and vice-versa. In the first experiment, see Figure 9, we have two single-server work stations each with a service rate of µi = µj = 1 connected by a single conveyor a 75 × 1 foot conveyor, the capacity of the conveyor is C = 75 and has a conveyor speed of 40 ft/min. Each package is a rectangular part of 1 square foot occupying one virtual cell on the conveyor. The conveyor width was assumed to be one foot wide (.3048 m). We choose these part dimensions since Arena [21] requires the part to occupy discrete cells on the conveyor. The M/G/c/c model is not limited to this restricted width and is more general with regards to part diameter(maximum width), however, since we are using Arena’s conveyor model, this fixed part width was requisite. In the M/G/c/c model we chose to make the state dependent function a constant one, since there is little interaction between the parts, the parts and the walls of the conveyor and little interference in general in the part flows. Someone might argue that if you use a constant speed, that an M/G/∞ queue could be used to represent the material handling system. This is true for this restrictive dimensional part type, but if the part diameter is not equal to the width of the conveyor, then the M/G/∞ model would not be appropriate, however, the M/G/c/c model would be appropriate. 5.1 Series Systems Table 2 illustrates the results of the first experiment compared to an Arena simulation of the same closed network configuration. The top half of the table has the analytical results for the four performance measures while the bottom half of the table has the simulation results. While the confidence interval information is not shown for all the statistics, the half-width was deemed to be acceptable for the thirty replications. If one looks at the Cycle Time of the two systems, they are extremely close as well as the throughputs and mean number at each workstation and along the conveyor. Thirty replications of the experiments were done with a warmup period of 1000 time units and 100,000 time units run for each replication. Now let’s perturb the population and the service rates of the two M/M/1 systems and see how the number of parts on the conveyor changes as well as the overall performance measures of the system. These results are shown in Tables 3, 4, and 5. The results comparing the analytical model and the simulation model are quite close. The next two tables, Tables 6 and 7 represent an attempt to place more commodities on the conveyor and to see if the M/G/c/c model still gives an accurate representation of the system. This is carried out by increasing DocNum

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µj µ(Wk ) Figure 9: Two single-server workstation system µi

Perf. Wα θα ρ Lα Ws θs ρ Ls

Qi 4.6779 0.8904 0.8904 4.1652 4.6916 0.8901 0.8897 4.1760

Conveyor 1.8750 0.8904 0.223 1.6695 1.8850 0.8901 0.0220 1.6470

Qj 4.6779 0.8904 0.8904 4.1652 4.7933 0.8901 0.8899 4.2665

Total 11.2309 0.8904 – 10.000 11.2340 0.8901 – 10.000

Table 2: First Experiment Wk = 10, µ = 1 Perf. Wα θα ρα Lα Ws θs ρs Ls

Qi 14.5958 0.9657 0.9657 14.0947 14.637 0.9661 0.9666 14.107

Conveyor 1.8750 0.9657 0.0241 1.8106 1.8853 0.9661 0.0238 1.7875

Qj 14.5958 0.9657 0.9657 14.0947 14.531 0.9661 0.9658 14.072

Total 31.066 0.9657 – 30.000 31.053 0.9661 – 30.000

Table 3: Second Experiment Wk = 30, µ = 1 the service rates and the finite circulating population. This is where Proposition 1 is important and we cannot predict whether the M/G/c/c queues will overflow just by increasing the population or changing the service rates. Fortunately, there was no overflow of the M/G/c/c queues as the conveyor capacity was adequate for the perturbed system parameters. Even with the increased loading on the system, the overall Cycle Time and Throughput are very accurate in both cases and the other measures especially the WIP are very close. 5.2 Split Systems For the split system, we wish to model a configuration as depicted in Figure 10. As a proof of concept of our approach, we can compare our algorithmic approach with the results of Posner and Bernholtz [34] pg. 973, where they model an ore mining example. This is the only closed network example with travel times which we have found to compare our results. Suppose the two M/M/1 queues represent ore faces and the central server represents a

Wα θα ρ Lα Ws θs ρ Ls

Qi 2.208 4.7688 0.9538 10.5293 2.2115 4.7667 0.9540 10.423

Conv. 1.8750 4.7688 0.1192 8.9415 1.8887 4.7667 0.1176 8.8221

Qj 2.208 4.7688 0.9538 10.5293 2.1920 4.7667 0.9540 10.498

P

6.2910 4.7688 – 30.000 6.2923 4.7667 – 30.000

Table 4: Third Experiment Wk = 30, µ = 5 DocNum

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Wα θα ρ Lα Ws θs ρ Ls

Qi 0.6949 9.1889 0.9189 6.3854 0.6907 9.1807 0.9182 6.3230

Conv. 1.8750 9.1889 0.2297 17.2292 1.8907 9.1807 0.2265 16.987

Qj 0.6949 9.1889 0.9189 6.3854 0.6863 9.1807 0.9178 6.3162

Smith & Kerbache

P

3.2648 9.1889 – 30.000 3.2677 9.1807 – 30.000

Table 5: Fourth Experiment Wk = 30, µ = 10 Wα θα ρ Lα Ws θs ρ Ls

Qi 2.1352 9.7633 0.9763 20.847 2.1380 9.7638 0.9765 20.832

Conv. 1.8750 9.7633 0.2441 18.306 1.8910 9.7638 0.2409 18.066

Qj 2.1352 9.7633 0.9763 20.847 2.1160 9.7638 0.9759 20.684

P

6.1454 9.7633 – 60.000 6.1451 9.7638 – 60.000

Table 6: Fifth Experiment Wk = 60, µ = 10 central ore dump. Suppose that there are four shuttle carts in the system and the carts are filled in FCFS order at stations 1 and 2 then dump their loads at the central server also in FCFS order. The service rates at the ore faces are µ3 = µ6 = 0.125 and the service rate at the central depot is µ1 = 1. Carts are routed upon completion at the central server towards station r(r = 1, 2) with an equal probability 12 . We will instead of carts, use chain conveyors (with carts/buckets) to model the travel time. The reason for this is that we would have to know the free-flow speed of the carts, their size, etc. to estimate the speed-density curve and no information on cart speed is provided by Posner-Bernholtz. In Posner and Bernholz, they assume that the travel time lags are constant for all the arcs connecting the stations and we will impose also a constant travel time of one minute for all arcs assuming that the arc lengths are 40 feet since the conveyor speed we are using is 40 fpm. Thus, our travel times are roughly equivalent. We must use seven queues in our model, three for the work stations and four for the arcs for the travel times. We have changed the index on the Posner-Bernoltz to the queue numbering in our system so that they are comparable. In the PosnerBernholtz methodology they only derive the probability distribution of the number of carts in the system. The average number in the system WIP or L as shown in Table 8 for both the Posner-Bernholtz and the queues in our system #1, 3, 6 are very close and it is felt that the use of the conveyors for the travel time is roughly the difference or it may even be the numerical precision of our algorithm versus Posner-Bernholtz. Notice also that we get the number in each of the conveyors and the utilization of the queues, something not directly obtainable from the Posner-Bernholtz supplemental variable approach. Let’s model some similar systems as in Posner-Bernholtz, but actually simpler, as in Figure 11. Here the central server is the material handling system which we shall treat as an M/G/c/c conveyor system. In this series of experiments, we will actually vary the conveyor speed to see how it affects our four basic performance measures.

Wα θα ρ Lα Ws θs ρ Ls

Qi 1.1001 24.538 0.9815 26.996 1.0988 24.528 0.9817 26.972

Conv. 1.8750 24.538 0.6135 46.009 1.9067 24.528 0.6051 45.386

Qj 1.1001 24.538 0.9815 26.996 1.0711 24.528 0.9812 26.293

P

4.0752 24.538 – 100.00 4.0768 24.528 – 100.00

Table 7: Sixth Experiment Wk = 100, µ = 10 DocNum

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µ4

Ore Face 1 µ3

µ2

Wk = 4 Central Ore Station µ1

µ5 µ6 Ore Face 2

µ7

Figure 10: Ore Mining Closed Network Model In the first experiment, we assume a finite population of ten customers and a conveyor speed of 40 fpm. Service rates at the M/M/1 queues are set to µi = µj = 1.

M/M/1

M/G/c/c

µ1 M/M/1

µ(Wk ) µ2 Figure 11: Basic Split System In Table 9 there is no difference in the throughput and a very marginal difference in the cycle times of the analytical and simulation models. In a second experiment, we reduce the conveyor speed to 20 fpm, keep the same finite population and the same service rates and analyze the same performance measures. In the next experiment we increase the conveyor speed to 80 fpm with the same population and service rates at the workstations. As shown in Table 10 we do pretty well even at this higher speed rate. The Cycle Times and throughputs are very close. Finally, for one last split experiment, (see Table 11) we keep the conveyor speed to 80 fpm and increase the population to 100 customers. Thus, we could violate the property here for the capacity of the conveyor. In fact, the utilization of the two M/M/1 queues is approaching their capacity. Fortunately, we do not violate the conveyor capacity, and the results are very good. 5.3 Cyclic Systems Now let’s examine a single closed loop system similar to the following diagram in Figure 12 where we have four workstations and four separate conveyors connecting them for a total of eight nodes. We will assume a population of 100 parts and service rates of µi = 1, ∀ i workstations and a conveyor speed of 40 fpm. We will provide an abbreviated output table comparing the analytical model and the simulation model. Table 12 includes the detailed performance measures of the four work stations and an average value on the conveyors since they are basically the same for the analytical and simulation outputs. In the first experiment, most of the product is located at the workstations because the service rate is smaller relative to the service rate of the conveyors. The Cycle Time and Throughput of the system is extremely accurate. There is a lot of variation in the WIP and delays at the workstations, but the overall results are very good. As a final experiment for this configuration, let’s DocNum

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n P3 (n) P6 (n) P1 (n)

0 0.237 0.237 0.812

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1 2 3 4 0.236 0.226 0.192 0.109 0.236 0.226 0.192 0.109 0.155 0.028 0.005 0.000 Posner-Bernholtz Results

Population Vector Chain Number Queue # Waiting Times 1- 1.1994 2- 1.0003 5- 1.0003 Queue #-Thruputs 1- 0.1903 2- 0.0951 5- 0.0951 Queue #-Utilizations 1- 0.1903 2- 0.0024 5- 0.0024 Queue #-Average WIP 1- 0.2282 2- 0.0952 5- 0.0952

WIP 1.700 1.700 0.226

4 1

3- 17.8218 4- 1.0003 6- 17.8218 7- 1.0003 3- 0.0951 6- 0.0951

4- 0.0951 7- 0.0951

3- 0.7611 6- 0.7611

4- 0.0024 7- 0.0024

3- 1.6956 6- 1.6956

4- 0.0952 7- 0.0952

Output from Algorithm Table 8: Table Comparison of Ore Mining Example Perf. Wα θα ρα Lα Ws θs ρs Ls

Qi 3.9233 0.8623 0.8623 3.3831 3.9303 0.8623 0.8636 3.3892

Qj 3.9233 0.8623 0.8623 3.3831 3.8916 0.8623 0.8604 3.3658

Conveyor 1.8750 0.8623 0.0431 3.2337 1.8871 0.8623 0.0425 3.1912

Total 5.7983 0.8623 – 10.00 5.7982 0.8623 – 10.00

Table 9: First Split Wk = 10, µ = 1, µC = 40f pm increase the service rates to µ = 10, ∀ i. This will shift the product onto the conveyors. Table 13 illustrates the results. Again, the Cycle Time and Throughput are very accurate while the other measures are not as accurate at the individual workstations, however, comparing for example the WIP at the first station, it is only off by 1.72%. To finalize the cyclic systems experiments, let’s double the single loop system so that there are eight work stations and eight conveyors, in the following single loop system. Figure 13 illustrates the double loop system. All the conveyors are 75 feet in length and run at 40 fpm. Workstations have a service rate of ten/minute and the population is Wk = 100 units. This is a very complex system. We will not provide a complete set of tabular results as it is really too large to present, but a portion of the output since many of the results are symmetric. Table 14 illustrates the results from the first loop of the system comparing the four workstations and four conveyors in the analytical and simulation models. Now, let’s examine some of the detailed outputs. For the Cycle Time in Table 14 we have from the analytical model a value of 20.300 minutes and the simulation provides a values of 20.392(±.00174) is the half width and the percentage error is roughly 0.453%. The throughput for the analytical model is 4.9262 while the simulation had a value of 4.9037(±.00042) is the half width and the percentage error is again roughly 0.45%. As a detailed example (see Table 14, the WIP on Conveyor #2 from the analytical is 9.2366 and from the simulation we have 9.0737(±.00079) and a percentage error of 1.79%. The run time for the simulation of thirty replications etc. was 81.15 minutes. Thus, the analytical model performed very well. DocNum

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Perf. Wα θα ρα Lα Ws θs ρs Ls

Qi 4.6779 0.8904 0.8904 4.1652 4.6833 0.8909 0.8909 4.1689

Qj 4.6779 0.8904 0.8904 4.1652 4.6650 0.8909 0.8892 4.1527

Smith & Kerbache

Conveyor 0.9375 1.7808 0.0223 1.6695 0.9425 1.7804 0.0220 1.6471

Total 5.6154 1.7808 – 10.00 5.6169 1.7804 – 10.00

Table 10: Third Split Wk = 10, µ = 1, µC = 80f pm Perf. Wα θα ρα Lα Ws θs ρs Ls

Qi 49.5721 0.9899 0.9899 49.072 51.034 0.9900 0.9911 50.517

Qj 49.5721 0.9899 0.9899 49.072 48.037 0.9900 0.9888 47.562

Conveyor 0.9375 1.9798 0.0247 1.8561 0.9427 1.9800 0.0244 1.8318

Total 50.50 1.9798 – 100.00 50.50 1.9800 – 100.00

Table 11: Fourth Split Wk = 100, µ = 1, µC = 80f pm 5.4 Manufacturing/Assembly AGVS System Let’s extend our approach to that of vehicles, in particular, automated guided vehicle systems (AGVS). These systems are increasingly being utilized especially in manufacturing, assembly, and warehouse material handling systems. Let’s say that we have the following layout of an AGVS in a factory layout where there is a central loop serving four machines. The AGVS vehicle and there can be as many as eight vehicles circulating between the machines given the geometry and dimensions of the circulation layout. Figure 14 (a) illustrates the layout for the AGVS system. We have developed a state dependent curve for the AGVS system as illustrated in Figure 14 (b) for a 1 × 1 meter wide vehicle assuming a normal flow speed of 150f t/min (45.72m/min) on the y-axis. The x-axis is the density of the number of vehicles in a 80’(24.384m) length of link. This is obviously an analytical approximation to the zone-based start-stop behaviour of AGVs, but as we will show in the simulation experiments, a reasonable approximation to the situation. We have built a simulation model and an analytical model for this system. First let us examine the layout with two AGVS vehicles (150f t/min), machine processing times that have a mean rate of 1/min. The lengths of the rectangle sides are a total of 80f t between machines. We will assume that there are 20 zones for the AGVS vehicles

I

µ1

VI

µ2

λ(Wk )

µ4

λ(Wk ) := throughput rate; µi := workstation service rates

II

µ3

III

Wk := finite population ⇒:= material handling

Figure 12: Closed Loop Queueing System DocNum

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Q1 23.934 0.969 0.969 23.18 23.50 0.969 0.969 22.766

Perf. Wα θα ρ Lα Ws θs ρ Ls

Q2 23.934 0.969 0.969 23.18 24.17 0.969 0.969 23.417

Q3 23.934 0.969 0.969 23.18 23.70 0.969 0.969 22.959

Smith & Kerbache

Q4 23.934 0.969 0.969 23.18 24.30 0.969 0.969 23.542

Conv. 1.875 0.969 0.242 1.816 1.885 0.969 0.239 1.793

Total 103.24 0.969 – 100.00 103.22 0.969 – 100.0

Table 12: Cyclic Wk = 100, µ = 1, µC = 40f pm Perf. Wα θα ρ Lα Ws θs ρ Ls

Q1 0.875 9.092 0.909 7.954 0.861 9.080 0.908 7.819

Q2 0.875 9.092 0.909 7.954 0.862 9.080 0.908 7.824

Q3 0.875 9.092 0.909 7.954 0.862 9.080 0.908 7.828

Q4 0.875 9.092 0.909 7.954 0.865 9.080 0.908 7.858

Conv. 1.875 9.092 0.227 17.047 1.891 9.080 0.224 16.800

Total 10.992 9.092 – 100.0 11.012 9.080 – 100.00

Table 13: Cyclic Wk = 100, µ = 10, µC = 40f pm and each zone is 4f t because of the vehicle dimensions. The determination of the number of zones in each network link is critical to the congestion in the system and is regulated by the size of the vehicle. Stopping and starting of the vehicles and the resulting congestion is a function of the number of zones in the network links. Loading and unloading of the vehicles are incorporated in the machine processing times. Results of the comparison between the analytical and simulation model occur in Table 15 where the average and 95% half-width (δ) alone with the minimum and maximum values from the simulation of 30 replications and 100, 000 time units. The results in Table 15 are very accurate as one can see. Similar results with a larger number of vehicles also occurred for this single loop topology. 5.5 Multi-Chain Systems For the multi-class/chain systems we can compare our algorithm with the results of Posner-Bernholtz [35] and again examine the ore mining example, where now instead of four carts flowing through the entire system we have two carts in a dedicated routing scheme traveling to each of the separate ore faces. There are two carts associated with the first ore station #1(chain 1) and these are processed first-come-first-served (FCFS) and two carts associated with ore station #2(chain 2). Carts from both stations unload at the central depot in FCFS order then return to their respective stations. This is an example of a fixed routing schema since once the carts are finished at the central depot they are directed with probability 1 towards their respective workstations. We assume as in the previous example that travel times for the carts/conveyor are constant and that the conveyor lengths are 40 feet so that the travel time is a constant of one minute in our model, equivalent to the Posner-Bernholtz assumption. As can be seen in Table 16, our WIP values are very close to those of Posner-Bernholtz at stations #1,3, and 6. Furthermore, the utilization of the central depot is reduced but the ore face workstations are dramatically much Perf. Wα θα ρ Lα Ws θs ρ Ls

Q1 0.1937 4.9262 0.4926 0.9542 0.19135 4.9037 0.49034 0.93833

Q2 0.1937 4.9262 0.4926 0.9542 0.19132 4.9037 0.49029 0.93818

Q3 0.1937 4.9262 0.4926 0.9542 0.19119 4.9037 0.49027 0.93753

Q4 0.1937 4.9262 0.4926 0.9542 0.19140 4.9037 0.49042 0.93859

Conv. 1.875 4.9262 0.1232 9.2366 1.8887 4.9037 0.1210 9.0737

Total 20.300 4.9262 – 100.00 20.392 4.9037 – 100.00

Table 14: Double Cyclic Wk = 100, µ = 10, µC = 40f pm DocNum

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IX

µ2

I

II

X

µ7

µ5

λ(Wk )

µ1

µ3

λ(Wk )

V

VII

VI

µ6

VI

λ(Wk ) := throughput rate; µi := workstation service rates

III

µ4

Wk := finite population ⇒:= material handling

Figure 13: Closed Double-Loop Queueing System Speed Density/ State Dependent Curve for AGVS System

40

Speed meters/min

30

20

10

0

Figure 14: (a) AGVS Layout

5

10

15

20

25

30

35

40

45

50

(b) State Dependent Curve

busier (94.16%) respectively.

5.5.1 Multi-Chain Dual Loop System As another demonstration of the performance modelling of these closed systems, let’s examine a multi-chain (class) queueing network, one where we have multiple parts flowing through the system. We have two chains(classes) with populations respectively (10, 5) as shown in Figure 15.

Measure Cycle Time Throughput

Analy. 6.7852 0.2947

Simul. 6.7959 0.2943

δ 0.01056 4.571x10−4

Min 6.7524 0.29125

Max 6.8668 0.29620

Table 15: AGVS Comparison, Two vehicles DocNum

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n P3 (n) P6 (n) P1 (n)

0 0.059 0.059 0.764

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1 2 3 4 0.265 0.676 – – 0.265 0.676 – – 0.185 0.043 0.007 0.001 Posner-Bernholtz Results

Population Vector Chain Number Queue # Waiting Times 1- 1.2425 2- 1.0000 35- 1.0000 6Queue #-Thruputs 1- 0.1177 2- 0.1177 35- 0.0000 6Queue #-Utilizations 1- 0.1177 2- 0.0029 35- 0.0000 6Chain Number Queue # Waiting Times 1- 1.2425 2- 1.0000 35- 1.0000 6Queue #-Thruputs 1- 0.1177 2- 0.0000 35- 0.1177 6Queue #-Utilizations 1- 0.1177 2- 0.0000 35- 0.0029 6Queue #-Average WIP 1- 0.2925 2- 0.1177 5- 0.1177

WIP 1.617 1.617 0.296

Chain (2 2) 1 13.7507 4- 1.0000 20.9855 7- 1.0000 0.1177 0.0000

4- 0.1177 7- 0.0000

0.9416 0.0000

4- 0.0029 7- 0.0000

2 20.9855 4- 1.0000 13.7507 7- 1.0000 0.0000 0.1177

4- 0.0000 7- 0.1177

0.0000 0.9416

4- 0.0000 7- 0.0029

3- 1.6184 6- 1.6184

4- 0.1177 7- 0.1177

Table 16: Ore Mining Multi-Class Comparison

(W1 = 10) M/M/c

M/G/c/c

M/M/c

M/G/c/c

µ1

C2

µ2

M/M/c

M/G/c/c

M/M/c

µ3

C3

µ4

C1

(W2 = 5) Figure 15: Multi-Chain System This again is pretty complex because of the two chains(classes) of customers. We will provide another abbreviate table of results. The comparison of the utilization and the WIP at all the stations is very acceptable, see Table 17. Finally, for the Throughputs and the Cycle Times of the two chains(classes) we have the following comparison of the analytical and simulation models, see Table 17. Again, these results are very acceptable. For the throughputs we have a percentage error of .023 % on θ1 ; 0.156 % on t2 which is very close; and for the Cycle Times we have a percentage error of CT1 , 3.17% and 2.8% on CT2 which is not as close. In another experiment, let’s increase the population of the first chain to W1 = 100 and that of the second chain to W2 = 50 while increasing the service rate of the workstations to µi = 10. The rest of the parameters remain the DocNum

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Perf. ρα Lα ρs Ls

C1 0.0374 2.8066 0.0369 2.7684

Q1 0.8586 3.4029 0.8585 3.4016

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C2 0.0214 1.6092 0.0212 1.5881

Q2 0.8586 3.4029 0.8577 3.3871

Q3 0.6389 1.2905 0.6375 1.2865

C3 0.0160 1.1975 0.0157 1.1802

Q4 0.6389 1.2905 0.6377 1.2826

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Perf. θ1 θ2 CT1 CT2

Analytical 0.8585 0.6389 12.0516 8.0228

Simulation 0.8583 0.6379 11.681 7.7979

95% (c.i.) (±0.00116) (±0.00101) (±0.01984) (±0.01288)

Table 17: Multi-chain Wk = (10, 5), µ = 1, µC = 40f pm same. The results across the board in Table 18 are quite excellent as one can see. The throughputs and Cycle Times are also very accurate and the utilizations and WIP values are all very close. Therefore, even with a multi-class material handling system, the queueing network methodology seems very robust and accurate. Perf. ρ Lα ρ Ls

C1 0.4703 35.270 0.4636 34.770

Q1 0.9852 33.692 0.9850 33.417

C2 0.2462 18.466 0.2430 18.222

Q2 0.9852 33.692 0.9852 33.566

Q3 0.8965 6.038 0.8946 5.9078

C3 0.2241 16.804 0.2206 16.547

Q4 0.8965 6.038 0.8943 5.9109

Perf. θ1 θ2 CT1 CT2

Analytical 9.852 8.965 10.632 5.125

Simulation 9.8481 8.9430 10.590 5.1101

95% (c.i.) (±0.00543) (±0.00446) (±0.00804) (±0.00304)

Table 18: Multi-chain Wk = (100, 50), µ = 10, µC = 40f pm

5.5.2 Multi-Chain AGVS Multiple Loop System Finally, let’s examine an AGVS system with two separate loops interconnected by a dispatcher or common material handling system from a warehouse. Figure 16 illustrates the layout of the system. We will build the simulation model and the analytical model of these systems. Building the simulation model is much more laborious than the analytical model, but we hope to achieve a similar level of performance from the analytical model. We could do larger number of loops, but will not do so at the current moment. In this example with two loops, we allow five vehicles within each loop.

Figure 16: AGVS Two Loops Layout As one might expect, since the two-loop AGVS system has an identical number of vehicles in each loop, the performance analysis for the simulation and analytical models are expected to be similar. Table 19 illustrates the performance of the two-loop system for throughput and cycle time. These results are slightly different in the simulation model in comparison with the single-loop system but not statistically significant. The simulation run for the two loops took over ten minutes for 30 replications and 100,000 time units with 1000 time units startup while the analytical model was of course extremely fast. 6

S UMMARY AND C ONCLUSIONS

We have developed a comprehensive performance algorithm and methodology for modeling multi-chain/class M/M/c workstations and their material handling systems with M/G/c/c state dependent queues. We have shown DocNum

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Measure Loop #1 Cycle Time Loop #2 Cycle Time Loop #1 Throughput Loop #2 Throughput

Analy. 9.1024 9.1024 0.5492 0.5492

Simul. 9.1637 9.1456 0.5456 0.5467

δ 0.01367 0.01140 8.119x10−4 6.811x10−4

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Min 9.0786 9.0784 0.5407 0.5432

Max 9.2477 9.2051 0.5507 0.5507

Table 19: AGVS Comparison, Two-Loop, Two vehicles that including the M/G/c/c queues when the total population arriving to each M/G/c/c queues is controlled, results in a closed network model with product form. We feel that this is an important and practical result and consequently provides an important extension for closed queueing and perhaps even open queueing networks. The mean value analysis (MVA) algorithm which results is a very efficient method of computing the performance measures of complex topologies of series, merge and split queueing systems. While we have not shown how these systems can be optimized, we shall do so in separate paper(s). Also, we need to examine how one can incorporate general service time distributions at the work stations along with finite buffers at the workstations. Unfortunately, when we do this, we will probably lose the product form property of the networks. Nevertheless, the methodology surrounding our integration of M/M/c and M/G/c/c remains a viable tool for manufacturing and service system design where travel time between the work stations is critical to the performance of the entire system. Acknowledgements This paper is dedicated to Ernest Koenigsberg who spearheaded the development of queueing network models in material handling systems and inspired us to continue his quest. R EFERENCES [1] Akyildiz, I.F., 1988. “General Closed Queueing Networks with Blocking,” in P.J. Coutois and G. Latouche (eds.) Performance 87, North-Holland, Amsterdam, 282-303. [2] Baskett, F. K. Chandy, R. Muntz and F. Palacios, 1975. “Open, Closed, and Mixed Networks of Queues with Different Classes of Customers.” Journal of the ACM 22 (2), 248-260. [3] Benson, F. and G. Gregory, 1961. “Closed Queueing Systems: A Generalization of the Machine Interference Model.” Journal of the Royal Statistical Society, Series B 23(2), 385-393. [4] Bertsimas, D., I.C. Pascalidas, and J. N. Tsitsiklis, 1994. “Optimization of Multiclass Queueing Networks: polyhedral Nonlinear Characterization of Achieveable Performance.” The Annals of Applied Probabilty 4 (1), 43-75. [5] Bouhchouch, A., Y. Frein, and Y. Dallery, 1996. “Performance Evaluation of closed tandem queueing networks with finite buffers.” Performance Evaluation 26, 115-132. [6] Buitenhek, R., 1998. Performance Evaluation of Dual Resource Manufacturing Systems. Ph. D Dissertation. Febodruk, Enschede, The Netherlands. [7] Buzen, J., 1973. “Computational Algorithms for Closed Queueing Networks with Exponential Servers.” Communications of the ACM 16 (9), 527-531. [8] Buzzacott, J. and J.G. Shantikumar, 1993. Stochastic Models of Manufacturing Systems. Prentice-Hall, Inc. Englewood Cliffs, New Jersey. [9] Cheah, J. and Smith, J. MacGregor, 1994. “Generalized M/G/c/c State Dependent Queueing Models and Pedestrian Traffic Flows.” Questa 15, 365-386. [10] Curry, G. B.A. Peters, and M. Lee, 2003. “Queueing Network Model for a Class of Material-handling Systems.” International Journal of Production Research. 41, 3901-20. [11] Dallery, Y. and K.E. Stecke, 1990. “On the Optimal Allocation of Servers and Workloads in Closed Queueing Networks.” Operations Research 38 (4), 694-703. [12] De-Song, B., Zhang, X., Xu, G., Pan, Z. and Tang, X., 2003. “Critical Phenomenon of Granular Flow on a Conveyor Belt.” Phsical Review E. 67, 062301. DocNum

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