Performance analysis and bu'er allocations
in some open assembly systems
N. Hemachandraa;∗ , Sarat Kumar Eedupugantib
a
IE and OR Interdisciplinary Programme, Indian Institute of Technology Bombay, Mumbai 400 076, India
b
Tata Consultancy Services, Sector 19, Udyog Vihar, Gurgaon 122016, India
Abstract
This paper considers a 3nite capacity fork–join queueing model for open assembly systems with arrival
and departure synchronizations and presents an approach for enumerating the state space and obtaining the
steady state probabilities of the same for such a model under exponential assumptions. Several performance
measures like the throughput of the system, fraction of arrivals that actually enter into the system, utilizations
of the work stations, etc., are obtained. Further, design of such fork–join systems in terms of bu'er size
con3gurations for maximizing the throughput of the system, or for minimizing the mean waiting time of
a typical job, or for minimizing the Work-In-Process of the system is done using the above performance
prediction approach. Such optimal con3gurations exhibit some interesting features.
Scope and purpose
Analytical models are often used to understand the performance of manufacturing systems and also the
associated decision making trade-o's that arise in their operations. We consider a model of a system with
two assembly lines and a single join operation. It takes into account the 3nite capacity of the system and
considers certain types of randomness in the inter-arrival times of the orders and processing and assembly of
the tasks. Most models either consider that there are no external arrivals that correspond to orders or assume
that assembly is instantaneous. We propose an analytical model to numerically obtain useful performance
measures like fraction of orders that are actually accepted, mean number of tasks in the system, etc. We next
use this technique to design such systems by 3nding bu'er con3gurations that maximizes the throughput of
the system, etc. We note some interesting features of such optimal designs.
Keywords: Performance evaluation; Optimal bu'er allocations; Fork–join queueing models; Open assembly systems
696
1. Introduction
Quantitative models are useful to better understand the performance of manufacturing systems and
also the associated trade-o's involved in decision making arising in operations of such systems. In
view of the uncertain environment in which such systems operate, stochastic models, in particular
Markovian and queueing models, are used for this purpose. This has been the focus of Viswanadham
and Narahari [1], Buzacott and Shanthikumar [2], Gershwin [3] and Altiok [4]. The chapter by Suri
et al. [5] also gives a good overview of many such models.
Many manufacturing industries, like refrigeration plants, etc., involve assembly of semi-3nished
parts. Fork–join queueing models are appropriate models for such systems; see Chapter 6 of Altiok
[4] for many models of such systems. Also, the 3nite capacity of the manufacturing systems can be
captured in a fork–join queueing system with 3nite bu'ers. Altiok [4] considers fork–join models
for manufacturing systems, but the models assume in3nite supply of raw material with no external
arrivals. We interpret the external arrivals as the orders placed by customers to the plant. Kim and
Agrawala [6] consider a fork–join queue with external arrivals but the bu'ers are assumed to be of
in3nite capacity and the 3nal assembly is assumed to be instantaneous.
In the rest of this section, we describe our model and notation. Section 2 has analysis of this
model. In Section 3, we show how one can use this numerical performance tool for 3nding the
optimal allocation of bu'er capacity for di'erent criteria. We make some observations about such
optimal con3gurations.
The fork–join queueing with 3nite capacity that is considered, shown in Fig. 1, is a queueing
system which consists of two work stations operating with the synchronization constraint on the
arrivals and one main assembly work station operating after the join primitive of the tasks. Each
work station has 3nite bu'er capacity and operates independently according to FIFO discipline. For
this fork–join queueing system we have four matching bu'ers. Matching bu'er 1 is part of the
subassembly work station 1, while matching bu'er 3 is the part of subassembly work station 2.
Both matching bu'ers 2 and 4 are part of the main assembly work station 3.
Every job arriving at the system consists of two tasks. A job is immediately split into two tasks
which are simultaneously placed on corresponding subassembly lines i.e., in bu'ers 1 and 3, when
the bu'ers at both work stations 1 and 2 are not full. Otherwise, assume that the arrivals are lost.
The job enters in assembly work station 3 (join primitive) after both its tasks are serviced at stations
Matching Buffer 1
Work Station 1
Matching Buffer 2
1
Work Station 3
Arrivals
3
Departures
2
Fork
Matching Buffer 3
Work Station 2
Join
Matching Buffer 4
Fig. 1. Fork–join queuing system with 3nite bu'er.
697
1 and 2, respectively and if the main assembly work station 3 is free. Otherwise, jobs wait for
service at station 3 in their respective matching bu'ers if both these bu'ers are not full. Due to the
3nite bu'er capacity of the bu'ers 2 and 4, either stations 1 and 2 can be blocked individually or
simultaneously. For this model, we assume that the blocking situation is of blocking after processing
(BAP) type [1]. The assembly operation makes the blocking phenomenon that arises in production
lines and assembly systems di'erent. The job departure at a blocking station in a production line
guarantees termination of blocking. However, in our system a departure from main assembly work
station 3, while it is blocking subassembly work station 1 will not terminate the blocking, if the
matching bu'er 4 is empty. In this case, it is the process completion at subassembly work station 2
that terminates the blocking at subassembly work station 1.
Finally, for this model, we assume that the work stations are failure-free and arrivals to the system
are Poisson and the work station service times are exponential. In this system, the main assembly
work station is never blocked, because we assume that the 3nished product warehouse has unlimited
capacity.
Notation
Ni
j
ni
Uj
frac
e'
ANi
L
W
capacity of the matching bu'er i, i = 1; : : : ; 4,
arrival rate to the system,
service rate of the work station j, j = 1; 2 and 3,
number of tasks in the matching bu'er i, i = 1; : : : ; 4,
utilization of work station j, j = 1; 2 and 3,
fraction of arrivals that actually enter the system,
e'ective arrival rate to the system,
average number of the tasks in the matching bu'er i, i = 1; : : : ; 4,
average number of jobs in the system,
average waiting time of a typical job in the system.
Here, the possible values to number of tasks in the matching bu'er are:
ni = 0; 1; : : : ; Ni ;
ni = 0; 1; : : : ; Ni
i = 1; 3;
and
Ni∗ ; i = 2; 4:
N2∗ and N4∗ refer to the state when work stations 1 and 2 are, respectively, blocked. The state
space can be partitioned into four sets of states:
(a) no blocking in the system; then ni ∈ {0; 1; : : : ; Ni } for i = 1; : : : ; 4.
(b) 3rst subassembly line blocked; then ni ∈ {0; 1; : : : ; Ni } for i = 3 and 4 and n1 ∈ {1; : : : ; N1 } and
n2 = N2∗ .
(c) second subassembly line blocked; then ni ∈ {0; 1; : : : ; Ni } for i = 1 and 2 and n3 ∈ {1; : : : ; N3 } and
n4 = N4∗ .
(d) 3rst and second subassembly lines both blocked; then ni ∈ {1; 2; : : : ; Ni } for i = 1 and 3, while
n2 = N2∗ and n4 = N4∗ .
It is clear, under our assumptions, that the dynamics of the assembly system can be captured as
a continuous time Markov chain. Let (n1 n2 n3 n4 ) be a typical state of this Markov chain with
698
P(n1 n2 n3 n4 ) being its steady state probability. The chain is irreducible and being a 3nite state
chain, it is positive recurrent and hence P(n1 n2 n3 n4 ) exits.
We observe that at any time instant (in particular, in steady state), the total number of tasks in the
entire 3rst subassembly line is equal to the total number of tasks in the entire second subassembly
line. This is because of the synchronization of the arrival epochs of new jobs into the system as
well as at the service commencement and departures from the main assembly by work station 3.
Thus, the total number of tasks in both subassembly lines is same i.e., n1 + n2 = n3 + n4 .
We use this fact as well as the above partition of the state space for automatic machine enumeration of the states and for writing down the transition rate matrix Q. We use LU decomposition method [7] to obtain these steady state probabilities by numerically solving PQ = 0 with
n1 ;n2 ;n3 ;n4 P(n1 n2 n3 n4 ) = 1, where P is the vector of {P(n1 n2 n3 n4 )}. In terms of these probabilities, we compute many relevant performance measures. Later we use this for optimal bu'er
allocation in the system.
2. Performance evaluation
We are mostly interested in performance measures of the system in time or customer average sense.
Due to the regenerative nature of the Markov chain with 3nite mean cycle lengths, these averages
can be computed in terms of the stationary distribution of the Markov chain; see for example, Wol'
[8]. For example, the long run fraction of time that the system is able to capture arrivals is
N N N
3 2 4
P(N1 n2 n3 n4 )
frac = 1 −
n2 =0 n3 =0 n4 =0
+
N3 N4
∗
P(N1 N2 n3 n4 ) +
n3 =0 n4 =0
+
N3
P(N1 N2∗ n3 N4∗ ) +
N
N2 N4
1 −1 P(n1 N2∗ N3 n4 ) +
n1 =1 n4 =0
+
P(n1 n2 N3 n4 )
n1 =0 n2 =0 n4 =0
N
N4
1 −1 N
1 −1
P(N1 n2 n3 N4∗ )
n2 =0 n3 =1
n3 =1
+
N3
N2 N
N2
1 −1 P(n1 n2 N3 N4∗ )
n1 =0 n2 =0
P(n1 N2∗ N3 N4∗ ) :
n1 =1
By Poisson arrivals, see time averages property, PASTA [8], the fraction of arrivals that enter the
system is also frac and hence, the rate of the actual arrivals to the fork–join queueing system is
e' = frac :
Little’s law can be used to obtain the mean waiting time at various queues and utilization of
various servers, if certain conditions are satis3ed. Let Ak be the arrival epoch of kth task into the
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system under consideration, (say a server), Dk be its departure from the system and Wk := Dk − Ak
the time spent in the system by the kth task. Let Q(s)
t be the number in the system at time s. The
time average of number in the system, limt →∞ (1=t)
0 Q(s) ds is the mean number in the system L.
In a regenerative system, this is also given by
npn , where pn is the steady state probability of
having n number in the system; see Wol' [8]. In our fork–join model, these pn ’s can be calculated
in terms of P(n1 ;
n2 ; n3 ; n4 ). The average waiting time in the system W is de3ned as the customer
average limn→∞ nk=1 Wk =n. While W can also be found in terms of a limiting distribution [8], we
can use Little’s law to 3nd it. Whitt [9] shows that L = W a.s. if: (a) job is in the system for the
entire period [Ak ; Dk ], (b) limk Ak =k = −1 ∈ (0; ∞) a.s., (c) L ¡ ∞ a.s. and (d) Wk =k → 0 a.s.; see
also Stidham [10].
Consider U3 , the utilization of server 3. Using Little’s law, we show now that U3 = e' (1=3 ). In
our model, (a) is satis3ed. For this single server system, L 6 1 and hence (c) is satis3ed. Since, Wk
is the service time of kth task, which is an exponential, (d) is also satis3ed. We now verify (b).
−1
Let tn , n ¿ 1 be the arrival epochs of jobs to the system and we have tn =n → e'
a.s., where e' is
the rate of the e'ective arrivals to the system. For n ¿ 1, let 1 wn be the time elapsed for nth piece
since arrival in line 1 before it reaches the head of the matching bu'er 2. Similarly, de3ne 2 wn for
n ¿ 1. Let Bn , n ¿ 1 be the time between (n − 1)th and nth visits of the Markov chain to state
(0; 0; 0; 0) with B0 = 0. Then, i wn 6 Bk(n) and n ¿ k(n), n ¿ 1, i = 1; 2, where k(n) is that cycle of
the Markov chain in which nth task is in system. Thus, i wn =n 6 Bk(n) =k(n),n ¿ 1, i = 1; 2. By strong
law of large numbers and positive recurrence of Markov chain, we have ni=1 Bi =n → f0 ¡ ∞ a.s.,
where f0 is the mean return time to state (0; 0; 0; 0). Hence, Bn =n → 0 a.s. Now, k(n) → ∞ a.s.
as n → ∞, as state (0; 0; 0; 0) is visited in3nitely often in the positive recurrent Markov, and hence
i
wn =n → 0 a.s. Then, t˜n := max(tn + 1 wn ; tn + 2 wn ), n ¿ 1 is the epoch of commencement of service
−1
of the nth task at the assembly. Since max(·; ·) is a continuous operator, we have t˜n =n → e'
a.s.
so that (b) is veri3ed.
Consider now, say, server 1. Let Q(s) = 1 if this server is processing a task at time s and Q(s) = 0
if either there is no part at server 1 or server is blocked. Again, L 6 1. Since, Wk is the service
−1
time of the kth task, Wk =k → 0 a.s. As above, Ak =k → e'
a.s. so that U1 = e' (1=1 ). Similarly,
U2 = e' (1=2 ).
Using similar arguments, we have that Wn =n → 0 a.s. where Wn is the waiting time of nth job in
the entire system. The time average of the number in thesystem, L is 3nite a.s. as this is a 3nite
system. Thus, the customer average of the waiting times ni Wi =n → W ¡ ∞ a.s. and this is given
by L=e' .
Remark 1. It is clear that the above observations are valid for more general arrival and service
distributions if the long run arrival rate exists and the system is regenerative with 3nite mean cycle
lengths (L will be 3nite here as it is a 3nite capacity system).
Remark 2. The above observations are also valid for in3nite capacity systems if they are regenerative
with 3nite mean cycle lengths and 3nite mean number in the system.
Remark 3. Since 1 − Ui is the probability that the server i; i = 1; 2 is either blocked or starved and
Ui i = frac ; i = 1; 2; we have that these probabilities are same if 1 = 2 .
700
Remark 4. For arrivals from non-Poisson families; the e'ective arrival rate can be computed in
terms of Ui s.
Several other performance measures like the fraction of time the work stations are starved, ANi ,
i = 1; : : : ; 4, etc., can now be found by using these steady state probabilities.
3. Buer allocation
Many design and operation issues in manufacturing systems call for 3nding the con3guration of
a system, (i.e., the values of Ni , i = 1; : : : ; 4) that achieves the optimum value of a prede3ned performance measure (like maximizing the throughput rate of the system), given the system parameters
like the arrival and service rates and total bu'er space available.
The approach we use to solve such optimization problems is to use the above performance evaluation building block coupled with a search procedure. The performance evaluation building block
gives the system performance for a given set of system parameters by numerically solving the balance
equations of the Markov chain. The search procedure starts from an initial con3guration of bu'ers
and covers all possible con3gurations. At each step, the performance evaluation building block determines the performance of the system for a given con3guration of the system and compares it
with that of the previous best con3guration for optimum values of Ni . Many tools are available that
facilitate both the modeling and optimization of resources using higher level modeling, for example,
see [11].
We look at the assembly system design as captured by 3nding the capacity of matching bu'ers,
with three optimum objectives:
• bu'er allocation for maximization of frac ,
• bu'er allocation for minimization of W ,
• bu'er allocation for minimization of L.
While each of these objectives is a function of con3guration, they are related by Little’s law.
Since these performance measures are conOicting in some sense, we would also like to compare the
respective optimal con3gurations. The above optimization is attempted in the following two possible
scenarios:
1. bu'er capacity of each sub assembly line C1 = N1 + N2 and C2 = N3 + N4 , respectively, arrival
rate of the jobs and service rates of the work stations are given parameters,
2. total bu'er capacity of the system C = N1 + N2 + N3 + N4 , arrival rate of the jobs and service
rates of the work stations are given parameters.
3.1. Bu8er allocation for maximization of frac
Here, we look for the best con3guration that increases e' , the throughput, of a given system.
More numerical examples are in [12].
(a) We consider the e'ect of the arrival rate on the optimal con3gurations. From Table 1, we see
that if the arrival rate is low when compared to the service rates, the system design prefers to keep
701
Table 1
Optimal bu'er allocations for maximizing frac for varying with 1 = 4, 2 = 6, 3 = 2
C
N1
N2
N3
N4
frac
8
12
16
20
24
3
5
7
8
10
1
1
1
2
2
3
5
6
8
9
1
1
2
2
3
0.50
0.50
0.50
0.50
0.50
0.99315
0.99959
0.99997
1
1
8
12
16
20
24
2
3
4
4
5
2
3
4
6
7
2
3
4
4
5
2
3
4
6
7
3.00
3.00
3.00
3.00
3.00
0.55149
0.61216
0.64017
0.65451
0.66108
8
12
16
20
24
1
1
2
2
2
3
5
6
8
10
2
2
2
3
3
2
4
6
7
9
30.00
30.00
30.00
30.00
30.00
0.062642
0.065529
0.066344
0.066584
0.066646
more bu'er space in the 3rst and third matching bu'ers while for arrival rates that are comparable
with the service rates, the bu'er space is evenly shared among the two stages. On the other hand, for
arrival rates that are much higher than the service rates, the optimal con3guration is to keep more
bu'er space at the second and fourth matching bu'ers. This is so under both scenarios. Multiple
optimal con3gurations exist for some of these examples but we do not give the alternate solutions
here.
(b) The optimal con3gurations under scenario 2 will have e' that is at least as good as the one
given under scenario 1, if C = C1 + C2 . Optimal con3gurations under both scenarios may di'er; for
example, for system with = 8, 1 = 1, 2 = 15, 3 = 3 and C = 16, the optimal con3gurations are
(4 5 2 5), (4 5 3 4) and (4 5 4 3) with frac = 0:1249 while the optimal con3gurations under scenario 2
with C1 = C2 = 8 are (3 5 3 5) and (3 5 4 4) with frac = 0:1247.
(c) Consider an assembly system with = 1 = 2 = 3 = r for some r ¿ 0. For this system with
balanced service and arrival rates, we have examples which demonstrate that the optimal con3guration
is an unbalanced one; see Table 2 for some examples. The e' for balanced con3guration is also
given for comparison. Since the rate matrix for such systems are proportional to r, these optimal
con3gurations are independent of r. Also, these con3gurations are optimal under both scenarios.
In view of remarks in Section 2, this means that sum of starving and blocking probabilities of
server i; i = 1; 2 of the balanced con3guration is more than the sum of corresponding probabilities
of unbalanced ones. In view of such examples, we conjecture that for such systems with C = 4n,
the optimal con3gurations will be (n n n + 1 n − 1) and (n + 1 n − 1 n n).
To better understand this phenomena, we consider systems with 2 = 3 = = r and look at
their optimal con3gurations for di'erent values of 1 around r. From Table 3 we observe that for
such systems, the optimal con3guration is the same unbalanced one as for systems with 1 = r. A
related work is Tayur [13], where certain structural results about kanban allocations with respect
702
Table 2
Optimal bu'er allocations for maximizing frac for a given C with = 1 = 2 = 3
C
N1
N2
N3
N4
frac
8
8
8
2
2
3
2
2
1
2
3
2
2
1
2
0.54749
0.553
0.553
12
12
12
3
3
4
3
3
2
3
4
3
3
2
3
0.64269
0.64567
0.64567
16
16
16
4
4
5
4
4
3
4
5
4
4
3
4
0.70388
0.70561
0.70561
20
20
20
5
5
6
5
5
4
5
6
5
5
4
5
0.74684
0.74792
0.74792
24
24
24
6
6
7
6
6
5
6
7
6
6
5
6
0.77877
0.77948
0.77948
40
40
40
10
10
11
10
10
9
10
11
10
10
9
10
0.85268
0.85289
0.85289
to mean throughput-maximum inventory trade-o' in serial systems, having stochastic demand with
back ordering, are proved.
3.2. Bu8er allocation for minimization of W
For this fork–join queueing system, the factors which contribute to the waiting time of the typical
job in the system are: (1) delay time in the matching bu'er one or three, (2) service time of the
task at work station one or two, (3) blocking time of the work station one or two, (4) delay time
of task in matching bu'er 2 or 4 and (5) service time of the main assembly work station.
It is observed that the optimal con3guration to this fork–join queueing system for the 3rst scenario
is (1; C1 − 1; 1; C2 − 1), and for the second scenario is (1; C − 3; 1; 1) or (1; 1; 1; C − 3).
Since second and 3fth factors above are the unavoidable factors for the waiting time of a job
in the system, if the system con3guration has the least possible value of 1 at matching bu'er 1 or
3, then the contribution to the job’s waiting time in the system by the 3rst factor is eliminated in
scenario one. Also, bu'ers 2 and 4 will have more bu'er space and this will help in reducing the
contribution to waiting time due to blocking. This explains the nature of the optimal con3gurations
under scenario one.
For the scenario two, the optimum system design con3guration is as above because in addition
to the above factors, the fourth factor’s contribution to the waiting time is eliminated in scenario
703
Table 3
Optimal bu'er allocation for maximizing frac with = 2 = 3 = 200, C = 20
N1
N2
N3
N4
1
frac
5
5
6
5
5
4
5
6
5
5
4
5
180
180
180
0.72331
0.72452
0.72452
5
5
6
5
5
4
5
6
5
5
4
5
190
190
190
0.73632
0.73748
0.73748
5
5
6
5
5
4
5
6
5
5
4
5
200
200
200
0.74684
0.74792
0.74792
5
5
6
5
5
4
5
6
5
5
4
5
210
210
210
0.75529
0.75627
0.75627
5
5
6
5
5
4
5
6
5
5
4
5
220
220
220
0.76206
0.76293
0.76293
two. Also, in second scenario, optimal con3gurations for systems with C ¿ 4 will have same states
as the system with C = 4 except 2 states; states (1 1 1 1∗ ) and (1 1∗ 1 1∗ ) in system with C = 4 will
be replaced by states (1; 1; 0; 2) and (1; 1∗ ; 0; 2), respectively. Also, the rate matrices are same.
However, the waiting time of a job in these optimal systems will be same as the one in system with
C = 4 a.s. and hence, the mean waiting time is also same.
3.3. Bu8er allocation for minimization of L
The system design con3guration to this fork–join queueing system for minimizing the average
number of jobs in the system for the 3rst scenario is (1; C1 − 1; 1; C2 − 1), and for second scenario
(1; C − 3; 1; 1) or (1; 1; 1; C − 3).
This is because the con3guration having the least possible value of 1 at matching bu'ers one and
three reduces the number in the system.
Further, if the matching bu'ers two or four have the least possible value of 1, then the system
will be having least number of the jobs in the system, which is 2.
4. Conclusions
We consider an approach for performance analysis and bu'er allocation of two-stage and two-line
fork–join queueing model with 3nite bu'er capacity under exponential assumptions. Bu'er allocation
either for maximizing the throughput of the system, or for minimizing the mean waiting time of
a typical job, or for minimizing the WIP of the system is done using this performance prediction
704
approach. We note some features of such optimal con3gurations. For a given capacity, the optimal
con3gurations for minimization of mean waiting time as well as for minimizing mean number in
the system are the ones with minimal bu'er allocations to appropriate work stations. On the other
hand, our numerical experimentation suggests that the optimal con3gurations for maximizing the
throughput rate depend on both the service and arrival rates. Also, if = 1 = 2 = 3 and C = 20
then, the optimal con3guration for maximizing the throughput of the system is (5 5 6 4) or (6 4 5 5)
but not (5 5 5 5). Such features are also valid for a three-line, two-stage assembly system [12]. These
features could serve as reasonable guidelines for bu'er allocations in larger systems.
Acknowledgements
We thank Prof. Narayan Rangaraj and Dr. K Ravikumar for useful comments on a earlier draft.
We also thank a referee for useful comments. Sarat Kumar was supported by a scholarship by Govt.
of India while studying at IITB where this work was done.
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N. Hemachandra is Assistant Professor in IE and OR Interdisciplinary Programme at IIT Bombay. His academic
interests are primarily in stochastic models and applied probability and broadly in OR.
Sarat Kumar Eedupuganti is working as a Software Analyst in Tata Consultancy Services. He is currently working
in AIG as a Consultant (Architect=Designer to Web Application) from TCS. He did his Masters in IE and OR from IIT
Bombay. His primary interests are in the areas of Intranet, Internet, E-Commerce and Web-related Industrial Applications.