Closed Networks TND/MHS
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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,
I
µ2
II
λ(Wk )
µ3
µ1
VI
III
µ4
µ6
V
λ(Wk ) := throughput rate;
µi := workstation service rates
(e.g. III) index on conveyance
µ5
IV
Wk := finite population
⇒ := material handling
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
C
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
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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.
3
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:
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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)
Figure 2: Representation of M/G/c/c queue
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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)/Ḡ(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)
(1)
(a)
1-5
(b)
Mean walking speed (m/s)
(c)
1-0
(f)
(d)
(a)
0-5
(e)
0
0
1
2
3
4
Crowd Density (P/m2)
Figure 3: Empirical Pedestrian Speed-Density Curves
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70
60
Mean Speed (mph)
50
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.
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M/M/c
M/G/c/c
µj
µ(Wk )
M/G/c/c
M/G/c/c
µ(Wk )
µ(Wk )
Smith & Kerbache
M/G/c/c
M/M/c
M/G/c/c
µ(Wk )
µj
µ(Wk )
M/M/c
M/G/c/c
M/M/c
µj
µ(Wk )
µj
M/G/c/c
µ(Wk )
M/M/c
M/G/c/c
µj
µ(Wk )
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)
µ(c − 2)
(c − 2, 1)
(c − 1, 0)
µ2
µ(c − 3)
(0, c − 1)
(c − 3, 2)
µ2
µ(1)
µ2
µ2
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
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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
1
30
2
40
3
50
density
4
flow rate
60
5
70
Figure 6: Conveyor Speed-Density Curve
Wk = c − 1
M/G/c/c
M/M/c
µj
µ(Wk )
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)
3
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.
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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.
Density
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Inter−Departure Time Histogram
(Poisson(lam=3) )
0
1
2
3
4
Inter−departure time from the system, s
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,
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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 =
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Πni=1 µi
p0 , for n = 1, · · · c
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and
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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.
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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 )]
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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
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M/M/1
M/G/c/c
Smith & Kerbache
M/M/1
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
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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
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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
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n
P3 (n)
P6 (n)
P1 (n)
0
0.237
0.237
0.812
Smith & Kerbache
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.
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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
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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
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µ8
VIII
Smith & Kerbache
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
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n
P3 (n)
P6 (n)
P1 (n)
0
0.059
0.059
0.764
Smith & Kerbache
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
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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
typeset October 28, 2010
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
Smith & Kerbache
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
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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
Smith & Kerbache
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.
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