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An approximate method for calculating the mean sojourn time of K-station production lines with no intermediate buffers
Int. J. Production Economics 54 (1998) 297—305
An approximate method for calculating the mean sojourn time of K-station production lines with no intermediate buffers H.T. Papadopoulos* Department of Business Administration, University of The Aegean, GR-821 00 Chios Island, Greece Received 19 September 1996; accepted 18 November 1997
Abstract Based on the analysis of the holding time model (HTM), an approximate method is proposed for calculating the mean sojourn time of a K-station production line with exponential service times, manufacturing blocking and no intermediate buffers between adjacent stations. Application of this method gives numerical results for the mean sojourn time for K"2, 3, 4, 5 and 6 station lines, that are within acceptable error levels (2—5%), compared against simulation. Also, for longer production lines (with 7—20 stations), it is shown that the approximate results underestimate the simulation ones by 6%. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Queueing networks; GERT networks; Production lines; Stochastic modelling; Blocking phenomenon; Sojourn time
1. Introduction and literature review A production line is modelled mathematically as an open queueing network with a serial topology (see [1] (Ch. 4), and [2] (Ch. 5)). The total time a part (or job) spends on such a network — which is known as sojourn time or time delay — is an important performance measure. Many researchers have dealt with this issue [3—9] among others. Some others worked on the same issue for more general topologies (non-serial
* Tel.: #30 273 39914; fax: #30 273 33896; e-mail: [email protected].
queueing networks) where overtaking may also take place, that is, a part may be passed by later arriving parts [10—12]. King [13] dealt with sojourn time distributions for particular customers in networks of queues. A detailed literature on this subject is given in his Ph.D. thesis. King developed an analytical model based on a Markovian state space to determine the sojourn time distribution. The exact distribution was shown to be a generalized hyper Erlang (for lines with exponential servers). However, the solution of the model was computationally infeasible even for relatively small scenarios as the number of paths through the state space explodes. Given the failure of the exact solution, King developed heuristic
0925-5273/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 0 0 6 - 1
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H.T. Papadopoulos/Int. J. Production Economics 54 (1998) 297—305
algorithms, based on the central limit theorem, with accurate results, concerning the mean and the variance of the sojourn time distribution. In this work, we exploit the findings of the holding time model (HTM) method [14—17], and we develop an algorithm for calculating the mean sojourn time of a K-station production line with exponential service times, manufacturing blocking (blocking after service (BAS)) and no intermediate buffers between any two successive stations. This algorithm is shown to give numerical results that are within acceptable error levels (2—5% for lines with up to six stations and 6% for longer lines with up to 20 stations) compared against simulation. This paper is organized as follows. In the next section, the HTM method is briefly recalled, then in the following section, the general algorithm is presented for calculating the mean holding times and the mean sojourn time. In the subsequent section, numerical results are tabulated for K"2, 3,2, 6,2, 20 station lines. Finally, the last section concludes the paper and recommendation for further research is suggested.
2. The holding time model method Production lines consisting of K single-machine stations linked in series are considered with no intermediate buffers. There is an unlimited supply of items at the first station so that this station can never be empty (idle). Each item enters the line at station 1, passes through all stations in order and leaves the Kth station in finished form. The service times S , i"1, 2,2, K, are independent exponentii ally distributed random variables and not identically distributed. It is assumed that the service stations are reliable and that they can service only one item at a time. Three periods are defined: the idle period, the busy period and the blocking period. Idle period is the time interval during which a station is idle; busy period is the time interval during which a station is servicing an item and blocking period is the time interval during which a station cannot provide service on an item, since the last item serviced at this station cannot move on to the next station because that station is busy or blocked. It is assumed that
the last station (Kth) can never be blocked (i.e., there is a buffer of unlimited capacity out of the last station that can accommodate all the released parts). The following random variables are defined: S (n): the service period of item n at station j, j B (n): the blocking period of item n at station j, j H (n): the holding period of item n at station j, j H (n)"S (n)#B (n), (1) j j j and I (n): the idle period of station j following the deparj ture of item (n!1). The following recursive relationships among the above random variables have been derived by Muth [14]: H (n)"max[S (n), R (n!1)], (2) j~1 j~1 j where R (n) is given by j R (n)"H (n)!I (n#1) for j"1, 2,2, K. (3) j j j~1 R (n) represents the time difference between the j departure of item n from station j and the arrival of the next item, (n#1), at station ( j!1), and hence this random variable can be either positive or negative. It is defined as R`(n)"max[0, R (n)] j j and
(4)
(5) R~(n)"min[0, R (n)], j j where R`(n) is the residual holding period of item j n at station j, following the entrance of item n#1 into station j!1, and the absolute value of R~(n) is j the time period during which both stations j and j!1 are starved together. Since station 1 is never starved, R (n)"H (n), so R (n) is a positive ran2 2 2 dom variable and R~(n)"0. 2 Another useful relationship is I (n)"max[I (n)#S (n)!H (n!1), 0] j j~1 j~1 j for j"1, 2,2, K.
(6)
H (n), I (n) and S (n) are defined to be identically j j j zero for j"0 and K#1. It also holds I (n)"0, 1
(7)
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H.T. Papadopoulos/Int. J. Production Economics 54 (1998) 297—305
From Eqs. (3) and (5),
since the first station can never be idle, and H (n)"S (n), (8) K K since the last station can never be blocked. It may be shown that the following relationships hold: H (n)"max[S (n), R` (n!1)] j j j`1 and
2.1. The distribution functions of the holding times In this paragraph, the integral equations that relate the random variables (r.v.) R (n), I (n) and j j H (n) are given as well as the cumulative distribuj tion function of the r.v. H (n) (holding time) at each j individual station of the line. This is denoted by F (t). (In general, by F ( ) ) we denote the distribuHj X tion function of the r.v. X.) If all service times have finite mean values, then R (n), I (n) and H (n) converge in distribution to the j j j random variables R , I and H as n tends to infinj j j ity. If H (n!1) is statistically independent of j I (n) [14], then from Eqs. (3) and (4) (see [15]): j~1 F ` (t)" Rj = 1! F (x!t) dF (x) if t*0, Hj Ij~1 x/.!9*0,t+/t 0 otherwise,
G
P
(11)
or by changing the variables inside the arguments: F ` (t)" Rj
G
1! 0
=
P
x/.!9*0,~t+/0
F (x) dF (x#t) Ij~1 Hj
G
F ~ (t)" Rj
if t*0, otherwise. (12)
1!
P
F (x!t) dF (x) Ij~1 Hj
x/0
1
(9)
I (n)"max[S (n)!R (n!1), 0]. (10) j j~1 j The relationships among the above variables can be conveniently displayed and derived through the use of activity (GERT) networks, which have proved useful for describing the sample path behaviour of any production line.
= if t)0, otherwise. (13)
If R` (n!1) is statistically independent of S (n), j`1 j then from Eq. (9), F (t)"F (t)F ` (t) Hj Sj Rj`1 and from Eq. (10),
(14)
F (t)" Ij = 1! F ~(x) dF (x#t) Sj~1 Rj x/~t = ! F ` (x) dF (x#t) if t*0, Rj Sj~1 x/0 0 otherwise.
G
P
P
(15)
Substitution of Eq. (14) into Eqs. (12) and (13) gives, for all j"1, 2,2, K, F ` (t)" Rj
G
=
P
1!
x/.!9*0,~t+/0
(x) d(F (x#t)F ` (x#t)) Sj R j`1 Ij~1 if t*0, (16)
0
otherwise.
F ~ (t)" Rj = 1! F
G
F
P
1
(x!t) d(F (x)F ` (x)) if t)0, Sj R j`1 Ij~1
x/0
otherwise.
Distribution functions of the holding times: F (t): Eq. (2) for j"2, gives H1 H "max[S (n), R`(n!1)] 2 1 1 and from Eqs. (9) and (4) R`(n!1)"max[0, R (n!1)], 2 2
(17)
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where
3. The algorithm for the mean sojourn time
R (n!1)"H (n!1)!I (n)"H (n!1). 2 2 1 2 From Eq. (14) it follows that (18) F (t)"F (t)F` (t). R2 S1 H1 Since the first station is never starved it holds R`"H , and using Eq. (14) again 2 2 (19) F (t)"F (t)F (t)F `(t). S1 S2 R3 H1 To compute F (t), F ` (t) has to be determined first R3 H1 from the simultaneous Eq. (12) and Eqs. (15), (16) and (17). F (t), j"2,2, K!1: For j"2, 3,2, K!1 Hj and t*0, after some algebraic manipulation, we have derived the general expression F (t)"F (t)F (t) Hj j Hj`1 = = F (x ) dF (x #x ) #F (t) Hj 2 j~1 2 1 j x1/0 x2/0 ]dF (x #t) Hj`1 1 = = = = F (x ) dF #F (t) Hj~1 4 j~2 j x1/0 x2/0 x3/0 x4/0 ](x #x )dF (x #x ) dF (x #x ) 2 j~1 2 1 4 3 Hj 3 ]dF (x #t) Hj`1 1 #
P P
P P P P
F # =
=
P P
=
For simplicity, throughout this analysis, we apply the approximate method of the HTM to derive the distribution functions of the holding times at each individual station of the line. The approximation relies on the assumption that R (t)"R`(t), 3 3 i.e., the event of two adjacent stations starving simultaneously cannot occur (the probability of its occurrence is supposed to be zero). This is not a restrictive assumption as the first station is always busy and therefore the line is saturated with the probability of having two adjacent stations idle at the same time being almost zero. In practice, this approximation means that we ignore all the terms where R~(n) is involved (e.g., in Eq. (15)). As a result j of this approximation will be the under estimation of the various performance measures of the lines under investigation. This is shown in the next section where the numerical results obtained from the proposed approximate method are compared against simulation. The approximation is adopted to save time from the complex mathematical expressions involved. In this section, the general algorithm is presented for calculating the E[H ]’s for any K-station proi duction line, the summation of which provides the required mean sojourn time. The proposed approximate method can be summarised as follows. To obtain the mean holding times of the K-station reliable production lines with exponentially distributed processing times and no intermediate buffers, the proposed method involves the following steps:
=
P P
#F (t) 2 F (x ) j H2 l x1/0 x2/0 xl~1/0 xl/0 hggggiggggj l/2(j~1) ]dF (x #x ) 1 l l~1 ]dF (x #x ) dF (x #x )2 H3 l~1 l~2 2 l~2 l~3 ]dF (x #x )dF (x #t). (20) j~1 2 1 Hj`1 1 F (t): HK F (t)"F (t),F (t)"1!e~kKt. (21) HK SK K
Steps of the algorithm Step 1: Derivation of the integral equations defining the cumulative distribution functions (CDFs) of the holding periods of the i stations, i"1, 2,2, K. General expressions giving these CDFs are Eqs. (19)—(21). Step 2: Solution of the system of integral equations. Step 3: Calculation of the mean holding times by direct integration. General expressions giving these expected values are Eqs. (23)—(28), given below.
H.T. Papadopoulos/Int. J. Production Economics 54 (1998) 297—305
Step 4: Determination of the mean sojourn time by summing the mean holding times at all stations of the line (see Eq. (22)). Comments: For the exponential case as well as any phase-type distribution for the service times, the integral equations of the first step can be easily transformed to algebraic non-linear equations (this is done in Step 2). These are then solved by applying conventional numerical techniques, such as the well-known Newton—Raphson method. The drawback of the proposed approach is that tedious mathematical calculations are involved. To facilitate the procedure, we have succeeded to provide general expressions which give the mean holding times at each station of the production line, for any number, K, of stations. These are Eqs. (23)—(28) mentioned in Step 3 and which are given below. Notation: The expected value of the holding time r.v. of the ith station of the line is denoted by E[HKwj], given that its length (the maximum numi ber of stations, K) is greater than or equal to j. Let ¹ represent the sojourn time, i.e., the time a part spends in the system (except for the time at the first queue). This is a random variable with expected value (see Step 4) K~1 E[¹]"E[S ]# + E[H ], K i i/1 where
(22)
(23) E[S ]"k~1, K K #k )~1], (24) E[HKw3]"k~1 #A[k~1!(k K K~1 K K~1 K~1 2,1 C E[HKw4]"k~1 # + B K~2 K~2 i1 i1/1 K #k )~1] ] + [k~1!(k i K~2 i i/K~1 #B [(k #k )~1 C2,1`1 K~1 K !(k #k #k )~1], (25) K~2 K~1 K C3,1 E[HKw5]"k~1 # + D K~3 i1 K~3 i1/1 K #k )~1] ] + [k~1!(k i K~3 i i/K~2
301
C3,1`C3,2 K # + D + [(k #k )~1 i2 i j i2/C3,1`1 i,j/K~2 jEi !(k #k #k )~1] K~3 i j #D C3,1`C3,2`1 ][(k #k #k )~1 K~2 K~1 K !(k #k #k #k )~1], K~3 K~2 K~1 K (26) C4,1 E[HKw6]"k~1 # + D K~4 K~4 i1 i1/1 K ] + [k~1!(k #k )~1] i K~4 i i/K~3 K C4,1`C4,2 D + [(k #k )~1 # + i2 i j i, j/K~3 i2/C4,1`1 jEi !(k #k #k )~1] K~4 i j C4,1`C4,2`C4,3 # D + i3 i3/C4,1`C4,2`1 K [(k #k #kl)~1 ] + i j l i, j,l /K~3 EjEi !(k #k #k #kl)~1] K~4 i j #D C4,1`C4,2`C4,3`1 [(k #k #k #k )~1 K~3 K~2 K~1 K !(k #k #k K~4 K~3 K~2 #k #k )~1], (27) K~1 K F E[HKw3]"k~1#k~1!(k #k )~1 1 1 2 1 2 K CK~2,1 # + D + [k~1!(k #k )~1 i 1 i i1 i/3 i1/1 !(k #k )~1#(k #k #k )~1] 2 i 1 2 i CK~2,1`CK~2,2 K # D + [(k #k )~1 + i2 i j i, j/3 i2/CK~2,1`1 jEi !(k #k #k )~1!(k #k #k )~1 1 i j 2 i j #(k #k #k #k )~1] 1 2 i j
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H.T. Papadopoulos/Int. J. Production Economics 54 (1998) 297—305
CK~2,1`CK~2,2`CK~2,3 D # + i3 i3/CK~2,1`CK~2,2`1 K ] + [(k #k #kl)~1 i j i,l j,l/3 EjEi !(k #k #k #kl)~1 1 i j !(k #k #k #kl)~1 2 i j #(k #k #k #k #kl)~1]# 1 2 i j F #D [(k #2#k )~1 K iK~2 3 !(k #k #2#k )~1 1 3 K !(k #k #2#k )~1 2 3 K #(k #k #k #2#k )~1], 1 2 3 K
A
B
E[HK/3]"3#B(1!2#1), 2 1 1 2 3 9#2B " , 6
(28)
2#B . E[HK/3]" 2 2
(33) (34) (35)
Applying the value 0.82 to B, we get E[HK/3]"1.77, E[HK/3]"1.41 and E[HK/3] 3 2 1 "E[SK/3]"1. Hence, E[¹K/3]"4.18. 3
where i "C #C #2#C K~2 K~2,1 K~2,2 K~2,K~2 "2K~2!1
B
1 # , (31) k #k #k 1 2 3 1 1 1 E[HK/3]" #B ! , (32) 2 k k k #k 2 3 2 3 where B is a real number, between 0 and 1 (0.82 is its value which is obtained from the solution of two non-linear equations). 2. Case K"3 with identical servers (balanced line): If k "k "k "k"1, then Eqs. (31) 1 2 3 and (32) give
(29)
4. Experimentation and numerical results
and, in general, for n"1, 2,2, m,
AB m
C "C(m, n)" . m,n n
(30)
All the coefficients (A, B , C , D ) contained in the i i i above expressions obey to the rule that they make the cumulative distribution functions of the various random variables involved, valid distribution functions (e.g., R`(t) for j"3, 4,2, K). j Example: We show the application of the proposed method to find the mean holding times of a line with 3 stations. More results are available from the author. 1. Case K"3 with non-identical servers (unbalanced line): Applying formulas (28) and (24) we calculate E[HK/3] and E[HK/3], respectively, 1 2 1 1 1 E[HK/3]" # ! 1 k k k #k 1 2 1 2 1 1 1 #B ! ! k k #k k #k 3 1 3 2 3
A
A
B
To calculate the mean sojourn time of a K-station line with exponential service times and no intermediate buffers, we have to apply Eq. (22) in conjunction with Eqs. (23)—(28). However, to determine the coefficients involved in E[H ]’s, numerical i computations are involved. More specifically, we applied the Newton—Raphson method to solve the non-linear equations to which the integral equations of Step 2 are transformed. A S/W programme was written in Fortran, for this particular problem. By applying the HTM approximate method, the following numerical results were obtained for the mean holding times, E[H ]’s and the mean sojourn i time spent in the line by a part (job), E[¹ ]HTM (see K Table 1), as well as for the variances of the respective holding times (Var(H )’s, see Table 2), for lines i with K"2, 3, 4, 5, and 6 stations. For comparison reasons, the balanced line cases were treated (with k "k"1, for all i). A simulation programme was i written in Simscript to derive numerical results for the mean sojourn time (this is denoted by E[¹ ]SIM) K as opposed to the same result derived by the proposed method (E[¹ ]HTM). The last column of K
303
H.T. Papadopoulos/Int. J. Production Economics 54 (1998) 297—305 Table 1 Mean holding times and mean sojourn time of balanced lines K
E[H ] 1
E[H ] 2
E[H ] 3
2 3 4 5 6
1.50 1.77 1.95 2.08 2.18
1.00 1.41 1.65 1.82 1.94
1.00 1.39 1.62 1.78
E[H ] 4
1.00 1.38 1.61
Table 2 Variance of the holding times of balanced lines Var(H ) Var(H ) Var(H ) Var(H ) Var(H ) Var(H ) 1 2 3 4 5 6
2 3 4 5 6
1.25 1.37 1.42 1.47 1.52
1.00 1.23 1.37 1.45
1.00 1.24 1.37
1.00 1.24
1.00 1.38
E[H ] 6
E[¹ ]HTM K
E[¹ ]SIM K
PD
1.00
2.50 4.18 5.99 7.90 9.89
2.56 4.38 6.33 8.28 10.31
2% 5% 5% 5% 4%
Table 3 Mean holding times and mean sojourn time of longer balanced lines
K
1.00 1.24 1.33 1.44 1.52
E[H ] 5
1.00
Table 1, gives the percentage difference (denoted by PD"(E[¹ ]SIM!E[¹ ]HTM)/E[¹ ]SIM) between K K K these two results. This difference lies between 2% and 5% and it is caused by the omission of the realisations of certain events (that two adjacent stations cannot be starved simultaneously). Taking into account these probabilities would increase the complexity of the mathematics involved 100%, something that cannot be justified from the slight improvement in the performance evaluation. We also ran a heuristic algorithm developed by King [13], the so-called DPL (enhanced deterministic simulation heuristic). Comparing King’s results (these are not tabulated) against ours and simulation, we observed that our method gives better results for the mean sojourn time. Finally, to investigate the applicability of the proposed approximation to longer production lines, we performed simulation runs for 5000—10000 time units each, for systems with 7—20 stations with exponential servers and no intermediate buffers. For the termination of the simulation runs we used as a rule of thumb the exact numerical result for the mean throughput of the lines as we know it from
K
E[H ]SIM K
E[H ]HTM K
PD
7 8 9 10 11 12 13 14 15 16 17 18 19 20
12.34 14.52 16.72 18.75 20.86 22.83 25.24 27.52 29.47 31.75 33.61 35.79 38.13 40.10
11.85 13.79 15.88 17.81 19.85 21.70 23.99 26.13 28.01 30.15 31.58 34.01 35.83 38.11
4% 5% 5% 5% 5% 5% 5% 5% 5% 5% 6% 5% 6% 5%
the Markovian state model for lines with up to 14 stations. The simulated results matched perfectly well this exact result. As it may be seen from Table 3, these results deviate from those derived from the HTM method from 5% to 6%. We also observed that the average delay due to blocking, at the workstations of these balanced lines (with k "k"1, for all is) started exceeding the average i processing time for lines with more than 12 stations. This delay for K"2, 6, 12, 20 accounts for 55%, 86%, 99% and 106% of the mean processing time, respectively, of each station of the balanced line. Concerning the variability of the output process of serial production lines, the interested reader is referred to Hendricks [18] and Hendricks and McClain [19] who dealt with this important issue.
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They proved that this process has a negative correlation, which means that an especially long sojourn time with respect to the average at one station leads to shorter times at the succeeding station, while a short sojourn time leads to a longer stay at the succeeding station. This negative correlation causes the variance of the overall sojourn time to obviously be less than the sum of the individual station sojourn variances.
5. Conclusion and further research Exploiting the findings of the holding time model method, a general algorithm was developed for calculating the mean sojourn time of a K-station production line with exponential service times, no intermediate buffers between any two successive stations and manufacturing blocking (blocking after service). Comparing the numerical results for K"2, 3, 4, 5 and 6 stations against simulation we observed that the proposed approximation method gives numerical results for the mean sojourn time with a good accuracy. Their deviation from the simulation estimates lies between 2% and 5%. Finally, to investigate the applicability of the approximation method to longer production lines, we conducted simulation runs for 5000—10000 min for systems with 7—20 stations and we verified that the approximate results underestimate the mean sojourn time by 6%, the maximum. We also observed that the average delay due to blocking, at each workstation of these balanced lines (with k "k"1, for all is) started exceeding unity (the i average processing time), for lines with more than 12 stations. This delay for K"2, 6, 12, 20 accounts for 55%, 86%, 99% and 106%, of the mean processing time, respectively. As areas for further research we recommend the use of the exact HTM method in conjunction with curve fitting, in order to determine easily the open coefficients involved in the expected holding times at the various stations of the production line. That would facilitate a lot, making the proposed algorithm quite attractive, especially if this could be extended to handle production lines with intermediate buffers of non-zero finite capacity.
Acknowledgements The author would like to express his sincere thanks to the anonymous referees for their valuable comments which improved significantly the appearance of the paper. References [1] H.T. Papadopoulos, C. Heavey, J. Browne, Queueing Theory in Manufacturing Systems Analysis and Design, Chapman Hall, London, 1993. [2] J.A. Buzacott, J.G. Shanthikumar, Stochastic Models of Manufacturing Systems, Prentice Hall, Englewood Cliffs, NJ, 1993. [3] E. Reich, Waiting times when queues are in tandem, Annales of Mathematical Statistics 28 (1957) 768—773. [4] E. Reich, Note on queueing in tandem, Annales of Mathematical Statistics 34 (1963) 338—341. [5] J. Walrand, P. Varaija, Sojourn times and the overtaking condition in Jacksonian networks, Advances in Applied Probability 12 (1980) 1000—1018. [6] B. Melamed, Sojourn times in queueing networks, Mathematics of Operations Research 7 (2) (1982) 223—244. [7] B. Melamed, M. Yadin, Randomization procedures in the computation of cumulative time distributions over discrete state Markov processes, Operations Research 32 (1984) 926—944. [8] I. Mitrani, Response time problems in communication networks, Journal of the Royal Statistical Society, Series B 47 (3) (1985) 396—406. [9] C. Knessl, On the sojourn time distribution in a finite capacity processor shared queue, Journal of the Association for Computing Machinery 40 (5) (1993) 1238—1301. [10] B. Simon, R.D. Foley, Some results on sojourn times in acyclic Jackson networks, Management Science 25 (1979) 1027—1034. [11] I. Mitrani, A critical note on a result by Lemoine, Management Science 25 (1979) 1026—1027. [12] G. Fayolle, R. Iasnogorodski, I. Mitrani, The distribution of sojourn times in a queueing network with overtaking: reduction to a boundary value problem, in: A.K. Agrawal, S.K. Tripathi (Eds.), Performance ’83, North-Holland, Amsterdam, 1983, pp. 477—486. [13] R.E. King, Sojourn distributions for particular customers in networks of queues, Ph.D. Dissertation, Department of Industrial and Systems Engineering, University of Florida, USA, 1986. [14] E.J. Muth, Stochastic processes and their network representations associated with a production line queueing model, European Journal of Operational Research 15 (1984) 63—83. [15] A. Alkaff, E.J. Muth, The throughput rate of multistation production lines with stochastic servers, Probability in the Engineering and Informational Sciences 1 (1987) 309—326.
H.T. Papadopoulos/Int. J. Production Economics 54 (1998) 297—305 [16] H.T. Papadopoulos, The throughput of multistation production lines with no intermediate buffers, Operations Research 43 (4) (1995) 712—715. [17] H.T. Papadopoulos, An analytic formula for the mean throughput of K-station production lines with no intermediate buffers, European Journal of Operational Research 91 (1996) 481—494.
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[18] K.B. Hendricks, The output processes of serial production lines of exponential machines with finite buffers, Operations Research 40 (6) (1992) 1139—1147. [19] K.B. Hendricks, J.O. McClain, The output processes of serial production lines of general machines with finite buffers, Management Science 39 (10) (1993) 1194—1201.
An approximate method for calculating the mean sojourn time of K-station production lines with no intermediate buffers H.T. Papadopoulos* Department of Business Administration, University of The Aegean, GR-821 00 Chios Island, Greece Received 19 September 1996; accepted 18 November 1997
Abstract Based on the analysis of the holding time model (HTM), an approximate method is proposed for calculating the mean sojourn time of a K-station production line with exponential service times, manufacturing blocking and no intermediate buffers between adjacent stations. Application of this method gives numerical results for the mean sojourn time for K"2, 3, 4, 5 and 6 station lines, that are within acceptable error levels (2—5%), compared against simulation. Also, for longer production lines (with 7—20 stations), it is shown that the approximate results underestimate the simulation ones by 6%. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Queueing networks; GERT networks; Production lines; Stochastic modelling; Blocking phenomenon; Sojourn time
1. Introduction and literature review A production line is modelled mathematically as an open queueing network with a serial topology (see [1] (Ch. 4), and [2] (Ch. 5)). The total time a part (or job) spends on such a network — which is known as sojourn time or time delay — is an important performance measure. Many researchers have dealt with this issue [3—9] among others. Some others worked on the same issue for more general topologies (non-serial
* Tel.: #30 273 39914; fax: #30 273 33896; e-mail: [email protected].
queueing networks) where overtaking may also take place, that is, a part may be passed by later arriving parts [10—12]. King [13] dealt with sojourn time distributions for particular customers in networks of queues. A detailed literature on this subject is given in his Ph.D. thesis. King developed an analytical model based on a Markovian state space to determine the sojourn time distribution. The exact distribution was shown to be a generalized hyper Erlang (for lines with exponential servers). However, the solution of the model was computationally infeasible even for relatively small scenarios as the number of paths through the state space explodes. Given the failure of the exact solution, King developed heuristic
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algorithms, based on the central limit theorem, with accurate results, concerning the mean and the variance of the sojourn time distribution. In this work, we exploit the findings of the holding time model (HTM) method [14—17], and we develop an algorithm for calculating the mean sojourn time of a K-station production line with exponential service times, manufacturing blocking (blocking after service (BAS)) and no intermediate buffers between any two successive stations. This algorithm is shown to give numerical results that are within acceptable error levels (2—5% for lines with up to six stations and 6% for longer lines with up to 20 stations) compared against simulation. This paper is organized as follows. In the next section, the HTM method is briefly recalled, then in the following section, the general algorithm is presented for calculating the mean holding times and the mean sojourn time. In the subsequent section, numerical results are tabulated for K"2, 3,2, 6,2, 20 station lines. Finally, the last section concludes the paper and recommendation for further research is suggested.
2. The holding time model method Production lines consisting of K single-machine stations linked in series are considered with no intermediate buffers. There is an unlimited supply of items at the first station so that this station can never be empty (idle). Each item enters the line at station 1, passes through all stations in order and leaves the Kth station in finished form. The service times S , i"1, 2,2, K, are independent exponentii ally distributed random variables and not identically distributed. It is assumed that the service stations are reliable and that they can service only one item at a time. Three periods are defined: the idle period, the busy period and the blocking period. Idle period is the time interval during which a station is idle; busy period is the time interval during which a station is servicing an item and blocking period is the time interval during which a station cannot provide service on an item, since the last item serviced at this station cannot move on to the next station because that station is busy or blocked. It is assumed that
the last station (Kth) can never be blocked (i.e., there is a buffer of unlimited capacity out of the last station that can accommodate all the released parts). The following random variables are defined: S (n): the service period of item n at station j, j B (n): the blocking period of item n at station j, j H (n): the holding period of item n at station j, j H (n)"S (n)#B (n), (1) j j j and I (n): the idle period of station j following the deparj ture of item (n!1). The following recursive relationships among the above random variables have been derived by Muth [14]: H (n)"max[S (n), R (n!1)], (2) j~1 j~1 j where R (n) is given by j R (n)"H (n)!I (n#1) for j"1, 2,2, K. (3) j j j~1 R (n) represents the time difference between the j departure of item n from station j and the arrival of the next item, (n#1), at station ( j!1), and hence this random variable can be either positive or negative. It is defined as R`(n)"max[0, R (n)] j j and
(4)
(5) R~(n)"min[0, R (n)], j j where R`(n) is the residual holding period of item j n at station j, following the entrance of item n#1 into station j!1, and the absolute value of R~(n) is j the time period during which both stations j and j!1 are starved together. Since station 1 is never starved, R (n)"H (n), so R (n) is a positive ran2 2 2 dom variable and R~(n)"0. 2 Another useful relationship is I (n)"max[I (n)#S (n)!H (n!1), 0] j j~1 j~1 j for j"1, 2,2, K.
(6)
H (n), I (n) and S (n) are defined to be identically j j j zero for j"0 and K#1. It also holds I (n)"0, 1
(7)
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From Eqs. (3) and (5),
since the first station can never be idle, and H (n)"S (n), (8) K K since the last station can never be blocked. It may be shown that the following relationships hold: H (n)"max[S (n), R` (n!1)] j j j`1 and
2.1. The distribution functions of the holding times In this paragraph, the integral equations that relate the random variables (r.v.) R (n), I (n) and j j H (n) are given as well as the cumulative distribuj tion function of the r.v. H (n) (holding time) at each j individual station of the line. This is denoted by F (t). (In general, by F ( ) ) we denote the distribuHj X tion function of the r.v. X.) If all service times have finite mean values, then R (n), I (n) and H (n) converge in distribution to the j j j random variables R , I and H as n tends to infinj j j ity. If H (n!1) is statistically independent of j I (n) [14], then from Eqs. (3) and (4) (see [15]): j~1 F ` (t)" Rj = 1! F (x!t) dF (x) if t*0, Hj Ij~1 x/.!9*0,t+/t 0 otherwise,
G
P
(11)
or by changing the variables inside the arguments: F ` (t)" Rj
G
1! 0
=
P
x/.!9*0,~t+/0
F (x) dF (x#t) Ij~1 Hj
G
F ~ (t)" Rj
if t*0, otherwise. (12)
1!
P
F (x!t) dF (x) Ij~1 Hj
x/0
1
(9)
I (n)"max[S (n)!R (n!1), 0]. (10) j j~1 j The relationships among the above variables can be conveniently displayed and derived through the use of activity (GERT) networks, which have proved useful for describing the sample path behaviour of any production line.
= if t)0, otherwise. (13)
If R` (n!1) is statistically independent of S (n), j`1 j then from Eq. (9), F (t)"F (t)F ` (t) Hj Sj Rj`1 and from Eq. (10),
(14)
F (t)" Ij = 1! F ~(x) dF (x#t) Sj~1 Rj x/~t = ! F ` (x) dF (x#t) if t*0, Rj Sj~1 x/0 0 otherwise.
G
P
P
(15)
Substitution of Eq. (14) into Eqs. (12) and (13) gives, for all j"1, 2,2, K, F ` (t)" Rj
G
=
P
1!
x/.!9*0,~t+/0
(x) d(F (x#t)F ` (x#t)) Sj R j`1 Ij~1 if t*0, (16)
0
otherwise.
F ~ (t)" Rj = 1! F
G
F
P
1
(x!t) d(F (x)F ` (x)) if t)0, Sj R j`1 Ij~1
x/0
otherwise.
Distribution functions of the holding times: F (t): Eq. (2) for j"2, gives H1 H "max[S (n), R`(n!1)] 2 1 1 and from Eqs. (9) and (4) R`(n!1)"max[0, R (n!1)], 2 2
(17)
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where
3. The algorithm for the mean sojourn time
R (n!1)"H (n!1)!I (n)"H (n!1). 2 2 1 2 From Eq. (14) it follows that (18) F (t)"F (t)F` (t). R2 S1 H1 Since the first station is never starved it holds R`"H , and using Eq. (14) again 2 2 (19) F (t)"F (t)F (t)F `(t). S1 S2 R3 H1 To compute F (t), F ` (t) has to be determined first R3 H1 from the simultaneous Eq. (12) and Eqs. (15), (16) and (17). F (t), j"2,2, K!1: For j"2, 3,2, K!1 Hj and t*0, after some algebraic manipulation, we have derived the general expression F (t)"F (t)F (t) Hj j Hj`1 = = F (x ) dF (x #x ) #F (t) Hj 2 j~1 2 1 j x1/0 x2/0 ]dF (x #t) Hj`1 1 = = = = F (x ) dF #F (t) Hj~1 4 j~2 j x1/0 x2/0 x3/0 x4/0 ](x #x )dF (x #x ) dF (x #x ) 2 j~1 2 1 4 3 Hj 3 ]dF (x #t) Hj`1 1 #
P P
P P P P
F # =
=
P P
=
For simplicity, throughout this analysis, we apply the approximate method of the HTM to derive the distribution functions of the holding times at each individual station of the line. The approximation relies on the assumption that R (t)"R`(t), 3 3 i.e., the event of two adjacent stations starving simultaneously cannot occur (the probability of its occurrence is supposed to be zero). This is not a restrictive assumption as the first station is always busy and therefore the line is saturated with the probability of having two adjacent stations idle at the same time being almost zero. In practice, this approximation means that we ignore all the terms where R~(n) is involved (e.g., in Eq. (15)). As a result j of this approximation will be the under estimation of the various performance measures of the lines under investigation. This is shown in the next section where the numerical results obtained from the proposed approximate method are compared against simulation. The approximation is adopted to save time from the complex mathematical expressions involved. In this section, the general algorithm is presented for calculating the E[H ]’s for any K-station proi duction line, the summation of which provides the required mean sojourn time. The proposed approximate method can be summarised as follows. To obtain the mean holding times of the K-station reliable production lines with exponentially distributed processing times and no intermediate buffers, the proposed method involves the following steps:
=
P P
#F (t) 2 F (x ) j H2 l x1/0 x2/0 xl~1/0 xl/0 hggggiggggj l/2(j~1) ]dF (x #x ) 1 l l~1 ]dF (x #x ) dF (x #x )2 H3 l~1 l~2 2 l~2 l~3 ]dF (x #x )dF (x #t). (20) j~1 2 1 Hj`1 1 F (t): HK F (t)"F (t),F (t)"1!e~kKt. (21) HK SK K
Steps of the algorithm Step 1: Derivation of the integral equations defining the cumulative distribution functions (CDFs) of the holding periods of the i stations, i"1, 2,2, K. General expressions giving these CDFs are Eqs. (19)—(21). Step 2: Solution of the system of integral equations. Step 3: Calculation of the mean holding times by direct integration. General expressions giving these expected values are Eqs. (23)—(28), given below.
H.T. Papadopoulos/Int. J. Production Economics 54 (1998) 297—305
Step 4: Determination of the mean sojourn time by summing the mean holding times at all stations of the line (see Eq. (22)). Comments: For the exponential case as well as any phase-type distribution for the service times, the integral equations of the first step can be easily transformed to algebraic non-linear equations (this is done in Step 2). These are then solved by applying conventional numerical techniques, such as the well-known Newton—Raphson method. The drawback of the proposed approach is that tedious mathematical calculations are involved. To facilitate the procedure, we have succeeded to provide general expressions which give the mean holding times at each station of the production line, for any number, K, of stations. These are Eqs. (23)—(28) mentioned in Step 3 and which are given below. Notation: The expected value of the holding time r.v. of the ith station of the line is denoted by E[HKwj], given that its length (the maximum numi ber of stations, K) is greater than or equal to j. Let ¹ represent the sojourn time, i.e., the time a part spends in the system (except for the time at the first queue). This is a random variable with expected value (see Step 4) K~1 E[¹]"E[S ]# + E[H ], K i i/1 where
(22)
(23) E[S ]"k~1, K K #k )~1], (24) E[HKw3]"k~1 #A[k~1!(k K K~1 K K~1 K~1 2,1 C E[HKw4]"k~1 # + B K~2 K~2 i1 i1/1 K #k )~1] ] + [k~1!(k i K~2 i i/K~1 #B [(k #k )~1 C2,1`1 K~1 K !(k #k #k )~1], (25) K~2 K~1 K C3,1 E[HKw5]"k~1 # + D K~3 i1 K~3 i1/1 K #k )~1] ] + [k~1!(k i K~3 i i/K~2
301
C3,1`C3,2 K # + D + [(k #k )~1 i2 i j i2/C3,1`1 i,j/K~2 jEi !(k #k #k )~1] K~3 i j #D C3,1`C3,2`1 ][(k #k #k )~1 K~2 K~1 K !(k #k #k #k )~1], K~3 K~2 K~1 K (26) C4,1 E[HKw6]"k~1 # + D K~4 K~4 i1 i1/1 K ] + [k~1!(k #k )~1] i K~4 i i/K~3 K C4,1`C4,2 D + [(k #k )~1 # + i2 i j i, j/K~3 i2/C4,1`1 jEi !(k #k #k )~1] K~4 i j C4,1`C4,2`C4,3 # D + i3 i3/C4,1`C4,2`1 K [(k #k #kl)~1 ] + i j l i, j,l /K~3 EjEi !(k #k #k #kl)~1] K~4 i j #D C4,1`C4,2`C4,3`1 [(k #k #k #k )~1 K~3 K~2 K~1 K !(k #k #k K~4 K~3 K~2 #k #k )~1], (27) K~1 K F E[HKw3]"k~1#k~1!(k #k )~1 1 1 2 1 2 K CK~2,1 # + D + [k~1!(k #k )~1 i 1 i i1 i/3 i1/1 !(k #k )~1#(k #k #k )~1] 2 i 1 2 i CK~2,1`CK~2,2 K # D + [(k #k )~1 + i2 i j i, j/3 i2/CK~2,1`1 jEi !(k #k #k )~1!(k #k #k )~1 1 i j 2 i j #(k #k #k #k )~1] 1 2 i j
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CK~2,1`CK~2,2`CK~2,3 D # + i3 i3/CK~2,1`CK~2,2`1 K ] + [(k #k #kl)~1 i j i,l j,l/3 EjEi !(k #k #k #kl)~1 1 i j !(k #k #k #kl)~1 2 i j #(k #k #k #k #kl)~1]# 1 2 i j F #D [(k #2#k )~1 K iK~2 3 !(k #k #2#k )~1 1 3 K !(k #k #2#k )~1 2 3 K #(k #k #k #2#k )~1], 1 2 3 K
A
B
E[HK/3]"3#B(1!2#1), 2 1 1 2 3 9#2B " , 6
(28)
2#B . E[HK/3]" 2 2
(33) (34) (35)
Applying the value 0.82 to B, we get E[HK/3]"1.77, E[HK/3]"1.41 and E[HK/3] 3 2 1 "E[SK/3]"1. Hence, E[¹K/3]"4.18. 3
where i "C #C #2#C K~2 K~2,1 K~2,2 K~2,K~2 "2K~2!1
B
1 # , (31) k #k #k 1 2 3 1 1 1 E[HK/3]" #B ! , (32) 2 k k k #k 2 3 2 3 where B is a real number, between 0 and 1 (0.82 is its value which is obtained from the solution of two non-linear equations). 2. Case K"3 with identical servers (balanced line): If k "k "k "k"1, then Eqs. (31) 1 2 3 and (32) give
(29)
4. Experimentation and numerical results
and, in general, for n"1, 2,2, m,
AB m
C "C(m, n)" . m,n n
(30)
All the coefficients (A, B , C , D ) contained in the i i i above expressions obey to the rule that they make the cumulative distribution functions of the various random variables involved, valid distribution functions (e.g., R`(t) for j"3, 4,2, K). j Example: We show the application of the proposed method to find the mean holding times of a line with 3 stations. More results are available from the author. 1. Case K"3 with non-identical servers (unbalanced line): Applying formulas (28) and (24) we calculate E[HK/3] and E[HK/3], respectively, 1 2 1 1 1 E[HK/3]" # ! 1 k k k #k 1 2 1 2 1 1 1 #B ! ! k k #k k #k 3 1 3 2 3
A
A
B
To calculate the mean sojourn time of a K-station line with exponential service times and no intermediate buffers, we have to apply Eq. (22) in conjunction with Eqs. (23)—(28). However, to determine the coefficients involved in E[H ]’s, numerical i computations are involved. More specifically, we applied the Newton—Raphson method to solve the non-linear equations to which the integral equations of Step 2 are transformed. A S/W programme was written in Fortran, for this particular problem. By applying the HTM approximate method, the following numerical results were obtained for the mean holding times, E[H ]’s and the mean sojourn i time spent in the line by a part (job), E[¹ ]HTM (see K Table 1), as well as for the variances of the respective holding times (Var(H )’s, see Table 2), for lines i with K"2, 3, 4, 5, and 6 stations. For comparison reasons, the balanced line cases were treated (with k "k"1, for all i). A simulation programme was i written in Simscript to derive numerical results for the mean sojourn time (this is denoted by E[¹ ]SIM) K as opposed to the same result derived by the proposed method (E[¹ ]HTM). The last column of K
303
H.T. Papadopoulos/Int. J. Production Economics 54 (1998) 297—305 Table 1 Mean holding times and mean sojourn time of balanced lines K
E[H ] 1
E[H ] 2
E[H ] 3
2 3 4 5 6
1.50 1.77 1.95 2.08 2.18
1.00 1.41 1.65 1.82 1.94
1.00 1.39 1.62 1.78
E[H ] 4
1.00 1.38 1.61
Table 2 Variance of the holding times of balanced lines Var(H ) Var(H ) Var(H ) Var(H ) Var(H ) Var(H ) 1 2 3 4 5 6
2 3 4 5 6
1.25 1.37 1.42 1.47 1.52
1.00 1.23 1.37 1.45
1.00 1.24 1.37
1.00 1.24
1.00 1.38
E[H ] 6
E[¹ ]HTM K
E[¹ ]SIM K
PD
1.00
2.50 4.18 5.99 7.90 9.89
2.56 4.38 6.33 8.28 10.31
2% 5% 5% 5% 4%
Table 3 Mean holding times and mean sojourn time of longer balanced lines
K
1.00 1.24 1.33 1.44 1.52
E[H ] 5
1.00
Table 1, gives the percentage difference (denoted by PD"(E[¹ ]SIM!E[¹ ]HTM)/E[¹ ]SIM) between K K K these two results. This difference lies between 2% and 5% and it is caused by the omission of the realisations of certain events (that two adjacent stations cannot be starved simultaneously). Taking into account these probabilities would increase the complexity of the mathematics involved 100%, something that cannot be justified from the slight improvement in the performance evaluation. We also ran a heuristic algorithm developed by King [13], the so-called DPL (enhanced deterministic simulation heuristic). Comparing King’s results (these are not tabulated) against ours and simulation, we observed that our method gives better results for the mean sojourn time. Finally, to investigate the applicability of the proposed approximation to longer production lines, we performed simulation runs for 5000—10000 time units each, for systems with 7—20 stations with exponential servers and no intermediate buffers. For the termination of the simulation runs we used as a rule of thumb the exact numerical result for the mean throughput of the lines as we know it from
K
E[H ]SIM K
E[H ]HTM K
PD
7 8 9 10 11 12 13 14 15 16 17 18 19 20
12.34 14.52 16.72 18.75 20.86 22.83 25.24 27.52 29.47 31.75 33.61 35.79 38.13 40.10
11.85 13.79 15.88 17.81 19.85 21.70 23.99 26.13 28.01 30.15 31.58 34.01 35.83 38.11
4% 5% 5% 5% 5% 5% 5% 5% 5% 5% 6% 5% 6% 5%
the Markovian state model for lines with up to 14 stations. The simulated results matched perfectly well this exact result. As it may be seen from Table 3, these results deviate from those derived from the HTM method from 5% to 6%. We also observed that the average delay due to blocking, at the workstations of these balanced lines (with k "k"1, for all is) started exceeding the average i processing time for lines with more than 12 stations. This delay for K"2, 6, 12, 20 accounts for 55%, 86%, 99% and 106% of the mean processing time, respectively, of each station of the balanced line. Concerning the variability of the output process of serial production lines, the interested reader is referred to Hendricks [18] and Hendricks and McClain [19] who dealt with this important issue.
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They proved that this process has a negative correlation, which means that an especially long sojourn time with respect to the average at one station leads to shorter times at the succeeding station, while a short sojourn time leads to a longer stay at the succeeding station. This negative correlation causes the variance of the overall sojourn time to obviously be less than the sum of the individual station sojourn variances.
5. Conclusion and further research Exploiting the findings of the holding time model method, a general algorithm was developed for calculating the mean sojourn time of a K-station production line with exponential service times, no intermediate buffers between any two successive stations and manufacturing blocking (blocking after service). Comparing the numerical results for K"2, 3, 4, 5 and 6 stations against simulation we observed that the proposed approximation method gives numerical results for the mean sojourn time with a good accuracy. Their deviation from the simulation estimates lies between 2% and 5%. Finally, to investigate the applicability of the approximation method to longer production lines, we conducted simulation runs for 5000—10000 min for systems with 7—20 stations and we verified that the approximate results underestimate the mean sojourn time by 6%, the maximum. We also observed that the average delay due to blocking, at each workstation of these balanced lines (with k "k"1, for all is) started exceeding unity (the i average processing time), for lines with more than 12 stations. This delay for K"2, 6, 12, 20 accounts for 55%, 86%, 99% and 106%, of the mean processing time, respectively. As areas for further research we recommend the use of the exact HTM method in conjunction with curve fitting, in order to determine easily the open coefficients involved in the expected holding times at the various stations of the production line. That would facilitate a lot, making the proposed algorithm quite attractive, especially if this could be extended to handle production lines with intermediate buffers of non-zero finite capacity.
Acknowledgements The author would like to express his sincere thanks to the anonymous referees for their valuable comments which improved significantly the appearance of the paper. References [1] H.T. Papadopoulos, C. Heavey, J. Browne, Queueing Theory in Manufacturing Systems Analysis and Design, Chapman Hall, London, 1993. [2] J.A. Buzacott, J.G. Shanthikumar, Stochastic Models of Manufacturing Systems, Prentice Hall, Englewood Cliffs, NJ, 1993. [3] E. Reich, Waiting times when queues are in tandem, Annales of Mathematical Statistics 28 (1957) 768—773. [4] E. Reich, Note on queueing in tandem, Annales of Mathematical Statistics 34 (1963) 338—341. [5] J. Walrand, P. Varaija, Sojourn times and the overtaking condition in Jacksonian networks, Advances in Applied Probability 12 (1980) 1000—1018. [6] B. Melamed, Sojourn times in queueing networks, Mathematics of Operations Research 7 (2) (1982) 223—244. [7] B. Melamed, M. Yadin, Randomization procedures in the computation of cumulative time distributions over discrete state Markov processes, Operations Research 32 (1984) 926—944. [8] I. Mitrani, Response time problems in communication networks, Journal of the Royal Statistical Society, Series B 47 (3) (1985) 396—406. [9] C. Knessl, On the sojourn time distribution in a finite capacity processor shared queue, Journal of the Association for Computing Machinery 40 (5) (1993) 1238—1301. [10] B. Simon, R.D. Foley, Some results on sojourn times in acyclic Jackson networks, Management Science 25 (1979) 1027—1034. [11] I. Mitrani, A critical note on a result by Lemoine, Management Science 25 (1979) 1026—1027. [12] G. Fayolle, R. Iasnogorodski, I. Mitrani, The distribution of sojourn times in a queueing network with overtaking: reduction to a boundary value problem, in: A.K. Agrawal, S.K. Tripathi (Eds.), Performance ’83, North-Holland, Amsterdam, 1983, pp. 477—486. [13] R.E. King, Sojourn distributions for particular customers in networks of queues, Ph.D. Dissertation, Department of Industrial and Systems Engineering, University of Florida, USA, 1986. [14] E.J. Muth, Stochastic processes and their network representations associated with a production line queueing model, European Journal of Operational Research 15 (1984) 63—83. [15] A. Alkaff, E.J. Muth, The throughput rate of multistation production lines with stochastic servers, Probability in the Engineering and Informational Sciences 1 (1987) 309—326.
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[18] K.B. Hendricks, The output processes of serial production lines of exponential machines with finite buffers, Operations Research 40 (6) (1992) 1139—1147. [19] K.B. Hendricks, J.O. McClain, The output processes of serial production lines of general machines with finite buffers, Management Science 39 (10) (1993) 1194—1201.