Computer simulation of a material flow system

Computer simulation of a material flow system

Comput & Indus. Engng Vol. 7, No. 1, pp. 23-32, 1983 Printed in Great Britain. 0360-8352/83/010023-10503.00/0 Pergamon Press Ltd, COMPUTER SIMULATIO...

540KB Sizes 0 Downloads 8 Views

Recommend Documents

No documents
Comput & Indus. Engng Vol. 7, No. 1, pp. 23-32, 1983 Printed in Great Britain.

0360-8352/83/010023-10503.00/0 Pergamon Press Ltd,

COMPUTER SIMULATION OF A MATERIAL FLOW SYSTEM H. PAUL Department of Industrial and Systems Engineering, National University of Singapore, Kent Ridge, Singapore 0511

(Received in revisedform January 1982) Abstract--This paper presents a computer simulation model of a material flow system. The flow system is a conveyor fed system for sorting and packing different types of bundles of cigarettes. Several bundle-types of cigarettes are produced at different rates by different machines and then released onto a common belt conveyor. The conveyor transports the bundles to a marshalling area. Because of the nature of the flow system different types of bundles get mixed up in the conveyor and several workers are stationed in the marshalling area to sort the different bundle-types and then pack into different crates. The model is used to answer questions relating to assignment of workers to particular bundle-types, whether workers should help each other in picking up bundles, and manpower levels in the marshalling area. This paper illustrates the application of computer simulation to a real world problem in a small factory. NOTATION AT(1)t CLAT(I) CLOCK CSLT(I, 1) CSLT(I, 2) IARR(I) IDT(1) IHAND(I) ILOST(I) IMISS(I) IRACK(I) ISAVE(I) ISER(I) ISIZE(I) LAST MISS(I) NSIZE(1) STl(I) ST2(I) ST3(I) ST4(I) TIDT(I) TOL TSERV(I) TST2(I)

arrival time of bundles cumulative arrival time chronological time minimum time after type l or type 3 service maximum time after type 1 or type 3 service cumulative number of arrivals idle time code for determining the status of server No. of bundles lost No. of bundles missed No. of bundles on the rack No. of bundles saved by server I + 1 No. of bundles served No. of bundles in hand No. of servers No. of bundles actually missed No. of bundles packed onto crate each time Type 1 service time Type 2 service time Type 3 service time Type 4 service time Total idle time maximum time allowance total no. of bundles served cumulative service time, type 2 service 1. I N T R O D U C T I O N

A cigarette manufacturing company uses a single belt conveyor consisting of three stages to bring bundles of cigarettes of different brands and sizes from bundling machines to a central marshalling area (Fig. 1). The bundling machines (capacities shown in Table 1) produce bundles of several brands concurrently and these bundles are then discharged onto the conveyor belt. Because of the nature of the building system mixing and clustering of different brands occur. Several workers are stationed in the marshalling area along side the moving belt conveyor to sort the mixed-up bundles and then pack the bundles into different crates (Fig. 2). However, the R._r~.t floor_ Morshalling

I

-.~I~1(---

q

1

room el

(

3

t

Jr~nsf~ _~omt Gr~uo~ f ~ Fig. 1. Three-stage conveyor belt. tI in parenthesis refers to the Ith server, except for ISAVE(I). 23

)

24

H. PAUL

Table 1. Capacitiesof bundlingmachines 2 machines with max capacity of ID bundles/min I machlne

wlth max capaclty of 22 bundles/min

I machine

with max capacity of 26 bundles/min

I machlne

with max capacity of 42 bundles/mln

each

and

7 machines with max capaclty of 2u bundles/min each

I/ / / e==

/ / /wot,// ~

~

ConveYor

Rack

belt ~

Crates for server 2

/ / / / ~

Flow

~

of bundles

C~ules for

server 1

Temporary storage area,

///////////// Fig. 2. Plan of Marshallingroom.

workers experience tot of strain and fatigue, and at times, order in the marshalling room is difficult to maintain. A detailed study of the existing material flow system was made. Three important processes in the material flow system were identified. These are: (1) Flow of bundles from the machines to the central marshalling area. (2) Sorting--identifying and picking--of bundles by workers, and (3) Packing of bundles by workers. It was contemplated that the performance of the system would improve if the above processes could be streamlined. Due to constraints imposed by the structure of the factory building, it was not possible to remove the main conveyor although some modifications to the conveyor (like channelling) could be done. In this paper the material flow system is analysed by the techniques of modelling and simulation.

2. SIMULATION MODEL

2.1 Basic features Several authors (Disney f 1], Pritsker[2], Phillips & Skeith[3] and E1 Sayed et al.[4]) have studied conveyor systems using queueing theory. However, most authors studied systems with many servers but only one product although the workers may perform different types of operation. The problem presented in this paper involves several products and several workers, each of the workers being assigned some specific product(s). Computer simulation is applied to this conveyor fed sorting and packing system because the model can include many of the parameters that are present in the real-life situation. Some of the features of the simulation model are described below: 2.1.1 Distribution of arrivals of bundles. By timing a considerable number of arrivals and statistical curve fitting it has been determined that the distribution that describe the interarrival times of bundles are truncated normal distributions at confidence levels ranging from 95% to 99%. 2.1.2 Service time distribution. Four different service times are identified in the simulation model. These are: STI(I): Service time of the Ith server to identify and pick a bundle of the type assigned to him (type 1 service).

Computersimulationof a materialflowsystem

25

ST2(I): Service time of the Ith server to turn around and put NSIZE(I) bundles into crates (type 2 service). ST3(I): Service time of the Ith server is to pick a bundle from the rack (type 3 service). ST4(I): Service time of the (I + 1)th server for picking and putting onto a rack a bundle assigned to server I (type 4 service). It has been determined that each of the above service times has exponential distribution. The parameters of the distribution functions have been obtained from observed values of service times by statistical curve fitting and appropriate tests for goodness of fit (chi-square). 2.1.3 Status of the servers. The following different status of servers are defined: (1) Idle--not performing any service and both hands are free. (2) Performing type 1 service with one hand, the other hand is free. (3) Performing type 1 service with one hand and type 3 service with the other hand. (4) Performing type 1 service with both hands. (5) Performing type 2 service. To identify the status of a server a code IHAND(I) is used. If IHAND(I) = 3 the server is performing type 2 service. If IHAND(I) = 0 the server is in any of the states defined by (1), (2), (3) or (4) above. 2.1.4 Range of reach and speed of conveyor. If a bundle arrives when the Ith server is in a state defined by (3), (4) or (5) above he is unable to service this bundle immediately. Once the server has a reach of 0.5-1 m along the length of the conveyor, he may be able to service this bundle at a later point in time. The maximum time allowance available to the server to service this bundle is given by TOL --

maximum range of reach minimum speed of conveyor

If the server is unable to service the bundle at the end of the time allowance then he misses the bundle. The bundle will be lost if the next server is also unable to pick up the bundle and place it on a rack so that the server assigned to it may be able to collect it later. 2.2 Computer model The use of the basic features in the simulation model is shown in the flow chart in Fig. 3. A computer program is written based on the flow chart using Fortran IV. The initial conditions of the model are set in Block 1 and data on input parameters are read in the same block. The input information required include (1) The number of servers, LAST. (2) The assignment of servers to bundle types, (M(I), I = 1, LAST). (3) The standard deviation of the interarrival time, VAR. (4) Mean ST1, ST2, ST3 and ST4 in centiminutes/bundle. (5) The overall efficiency of the bundling machines, EFF in percentage. (6) Number of bundles put into the crates at any one time, NSIZE(I). NSIZE(I) is assumed to have a distribution as follows: NSIZE(I) = N - 1 (25% of the time) = (50% of the time) = N + 1 (25% of the time) The value of N is 5 or 6. (7) Range of reach of the servers, DIST. The initial sequence of operation of the bundling machines is also determined in Block 1. Simulation starts by determining which machine is due to discharge a bundle (Block 2a). The time at which this bundle arrives at the corresponding server is determined by updating CLAT(I) in Block 2b. The rest of the simulation program determines what will happen to this bundle when it arrives at server I. If I -- 1 then the program proceeds to determine the status of server 1. Otherwise, the program checks if the previous server, I - l, has missed any bundle. If server I - 1 has not missed any bundle, i.e. IMISS(I- 1) -- 0, the program proceeds to check the status of server I. This is done in Block 8.

~

__

li.OSlll-ll. iMISSli-ll. I~ISSIII • i,tlS5 ['1 •

Gtllll

I,,E.,

|,

I11 ILGSIlI-li.I IXlSSli-II-1 I~I~SIII.I MISSIII* 1 I

| I I |

NO



1

I I I I /"

lsttiill I

1

I S E R I I I . ISER Ill ,I I ISlZE ill • ISiZE til "l - -

i

m

, IMISSII-1) - 1

ilR.JC.Kll-ti.li~Cml-ti.,

I IMISS l - l l

I

'

I

t~?

II

I

I

I

.!)

,/. |

NSIZE ill

'

li"

1

I

~ l l ~ l Generllie ST 2 III =,

J. ~ ,~, ~ "V [i~IZIEnI>ISIZ£11I?j

Generoil

"i.STil

V

el

.

• CLAI' Ill- IOl. i

I

](llll

~

I

I

{

/ | l I w

|

[

I il41~ill.iMi~lil.i

|

Reoion a

I

.

I ClSi, . . . . . . . . . . .

II

C~rme

l

iTTi

1151

u

~tUll

i $tl III

INO

i t

I

u'li:l"l'~Ulli'm°xlS"illlll l . • / IR.I~KIII, IRACKIII- I q ISERIll = ISERIII* 1 [ ISI~II) . Ic.lZill) • 2

1

I

I

I

lum,ll.timl,l.im

i ,

I

I

G~nero,. ST, II,

"

ll.OStlli. 1 I

I

"I

J

.

t llT.~,,,,.~,,,,.,l'--

CLSIII.11J

-

I ILl.till.

I'e--~LCLOCK

mill:Ill I I

.

il~i

()

I ~

,

i~A~..---i--.-.......,

.o. :

I I

I I_ i ! serapes clo not t llilli e(~ch oiler.

121

Fig. 3. Basic model flow chart.

I

I

I

t

I

,~em.,~m.,

ISEItli). 15EIIII)

i CLSTIIIi

I

I

l

~/llll

. i ......

tlinemle

_

! _ I1

Obloin mirl ICLAI|I) Generate AT (I) CLATII) = CLATI]). ATII| IAII~II)= Isl,l t ~ l l l ) , l

CLST(I l)=mox {CLST IX 11

.-

' ICLAIIII I IS141111 [ ClLST~21=I II. -mcix I II

I

I

I I l~ll~SIl-ll.lt41SSll-l)-I

I

,

ST/,III

iill~ ILII_

Iclsm,.,p.smll..i,

'

I

~~

111 ,,¢llt iniliol conditioi'~$. A(:cepl system porameters : PrH",I title I~ge. C.erieroie inilioI sequence.

l i "~'"""<~T"-"'"~' I

I NG

I 1

llalllll

C'- i. LASt ,-"l.-ll--.~

i

YES

~

r-

Computer simulationof a materialflowsystem

l t =0

CLST11,11

time

27

B

CLST(1,21

Fig. 4. Rules for serverstatus.

If server I - 1 has missed a bundle, i.e. I M I S S ( I - 1 ) = 1, the program uses the code IHAND(I) to determine if server I is packing his bundles into crates. If he is packing his bundles, the bundle missed by server I - 1 is lost to the system. At the same time server I misses a bundle assigned to him. If the last server I = LAST misses a bundle, the bundle is immediately lost to the system because there will not be any server to pick up this missed bundle (Block 4). If server I is not packing his bundles into crates then he must be performing type 1 and type 4 service simultaneously. Block 5 generates the service times ST1 and ST4. Block 6 then updates clocks CLST(I, 1) and CLST(I, 2). Block 7 updates the number of bundles served and the number of bundles on the rack. If a bundle arrives when both hands of the server are busy (Region A in Fig. 4), then the maximum time allowance TOL is added to clock CLAT(I) (Block 9). If the server is still busy at the end of this time, then the server misses the bundle. If the serve has at least one hand free at the end of this time, the bundle will be picked up once the previous service is completed. If a bundle arrives when one hand of the server is free (Region B in Fig. 4), Block 11 then generates the time for this service. Block 12 then updates CLST(I, 1) and ISIZE(I), the number of bundles in his hand and Block 13 resets the clocks CLST(I, 1) and CLST(I, 2), If a bundle arrives when both hands of the server are free (Region C in Fig. 4) the program calculates the idle time of the server (Block 14). Block 15 then sets IHAND(I)= 0 because when the server returns to the conveyor after a type 3 service both of his hands are free, and the server is idle until the next bundle of the type assigned to him arrives. The progran then checks if there are any bundles on the rack (Block 6). If there are none, then the server picks up the bundle from the conveyor. If there are bundles on the rack then the server uses one hand to pick up a bundle from the rack while he uses the other hand to pick up a bundle from the conveyor. Block 17 generates the times required for these services. Block 18 determines the number of bundles the server accumulates before he turns around and packs the bundles into crates. If there are insufficient number of bundles in his hands the simulation proceeds to Block 21. If there are sufficient number of bundles in his hands then Block 19 generates the time required for him to place the bundles in the crates and to return to the conveyor. Block 20 sets the code IHAND(I) -- 3 and ISIZE -- 0. Block 21 determines whether the simulation has been completed. If simulation is not completed, the program returns to Block 2 and generates another arrival. Otherwise the program proceeds to print out all the relevant results. Most of the results are obtained by continual updating throughout the simulation. However, the total service time is obtained from the following equation: ISERV(I) = CLAT(I) - TIDT(I). This indirect approach is necessary because ST1, ST3 and ST4 are sometimes overlapping because of use of the two hands for different types of service (type 1, 3 or 4). Another variable ISAVE(I) has been defined as follows: ISAVE(I) = MISS(I) - ILOST(I). ISAVE(I) gives the number of bundles saved by server I + 1 but missed by server I. 2.3 Modilications o/ the basic model Two modifications of the above model have been made. The above basic model assumes

28

H. PAUL Ill Accept system i:x:smmete~: Print title poge. Generate ~itiat sequence.

t

12} I

I C] LfiOl~ain eA ntlccMt I I I ) Im~n _AT(I) _

I I CLAT=CLAT" AT(I'

IARR(1) = IARR(I) * NSIZE

STIll

31

C~emte IOT(1) • CLATll) - CLST I l l

IS£R{ll • ISERII) • NSIZE TS1"II), TST(I) • ST(1) CLST(1)• (:l_STIl) • STIll TIOTll) = TIDT(1). IDTIII

)

I

I Cl.O0(_-CLATII), TOL I }

110) Ger~ot~ Slft) TSTIll = ~ST111• ST(l) c ~ ( z ) : c t s r ( l ) , sT (I) ZSER(II = ~ ( Z )

I ILOSTII)= IL0ST(l)

-

NSIZE

,

I

• NSZZE

I

I ReSults. I Fig. 5. Model flow chart for group discharge.

that if a server misses a bundle of a type assigned to him, the next server along the conveyor belt will help to retrieve the missed bundle if he is free. The first modification (MOD 1) assumes that if a server misses a bundle assigned to him the bundle is lost. In this modification Blocks 3-7 in the basic model are excluded. The second modification (MOD 2) assumes that bundles arrive in groups of six bundles of each type. The flow chart for this modification is given in Fig. 5. This modification is done to simulate the effect of releasing six bundles at a time from the bundling machines instead of one bundle at a time. The bundling machine release mechanism can be modified to accomplish this. 3. S I MU L A T I ON E X P E R I M E N T S

Simulation experiments have been performed both on the basic model and the modifications to analyse the effects of varying assignments of servers to bundle types and variations in other factors such as number of servers, standard deviation of the interarrival times of the bundles, bundling machine efficiency, service times for type 1, type 2, type 3 and type 4 service, number of bundles packed into crates each time, and the range of reach of servers. The effectiveness of the sorting and packing system is measured by percentage of bundles served and percentage utilization of the servers. The first set of simulation experiments reveal that the percent of bundles served is maximum when the servers are assigned bundle types in the increasing order of production rates of the bundles (the numbering of servers is as shown in Fig. 2). This result has been obtained for both arbitrary and actual bundle production rates. The assignments of bundle types to workers examined are shown in Table 2. The assignment shows how servers are allocated duties to serve bundles from various bundling machines with varying outputs. For example, in Assignment No. 1 the first server is allocated to service bundles from another machine of 15 bundles/minute, the second server is assigned to service bundles from another machine with output rate of 15 bundles/minute, the third server is assigned to service bundles from a machine with output rate of 20 bundles/minute, and so on. In Assignment No. 5 the first server is assigned to service bundles from a machine with output rate of 20 bundles/minute, the second server is assigned to service bundles from a machine with output rate of 22 bundles/minute, the third server is assigned to service bundles from a machine with output rate of 28 bundle/minute, the fourth server is assigned to service bundles from two machines with output rates of 20 bundles/minute and 15 bundles/minute, and so on. In all the assignments the servers are assigned to service bundle types in the ascending order of bundling rates. The total

Computer simulation of a material flow system

29

Table 2. Assignment of workers to bundle types As s ignment No.

Server

I

2

3

4

5

15

15

20

20

20

20 20 20 20 22 28 42

20

20

20

20

20

20

20

20

20

20

20

20

22

2 8 } }20 4 2 20 15 15

6

7

8

22

9

28

10 11 12

15~ 42

15J

20 20 20 22 28 20 20} 20}42 15J 15 2 20

22

28

20, 20~ \20 20~ 42

15] ~ 1 5 j2o 2o~

28

20\ 20[ 15J 15 20~ 2

22

42

20 20 20} 42 20} 15} 20} 20} 22 28 15 20 20} 20, 20} 15~ 20 20 ~* 20 15~

20 22 j 28 20t 20J

number of alternative assignments considered is 8 and the number of servers is progressively reduced from 12 to 4. Analyses of results of some of the experiments are presented below:

3.1 Varying number of servers The summary of the results of this simulation experiment is presented in Table 3. The number of servers and the assignment of servers have been varied as indicated. The other system parameters have been kept constant. The results indicate that the percent of bundles lost increases rapidly when the number of servers are reduced from 12 to 4. In the present system the percent of bundles lost is about 10 with 12 servers; but there is no systematic assignment of servers to bundle types. Therefore the number of servers cannot be reduced without affecting adversely the present effectiveness of the system. 3.2 Varying service times Simulation experiments conducted to examine the effect of changes in service times reveal that the service times for type 1, type 3 and type 4 service have marginal effects on percent of bundles served. However, the effect on percent utilization of servers was quite substantial (increased from 46 to 52.4%). On the other hand, the effect of changes in time for type 2 service on percent of bundles lost was an increase from 6 to 13% as the service time was increased from 0.08 to 0.115 min per bundle. Idle time of the servers decreased from 57 to 50%.

Table 3. Effect of varying number of servers No. of servers

% served

% lost % saved

% idle % u t i l .

12

82.11

9-77

8.12

52.46

47.54

11

80.44

10.82

8.75

49.31

50.69

10

77.95

13.38

8.67

47.30

52.70

9

75.35

15.87

8.78

44.51

55-49

8

71.56

19.14

9.30

40.72

59.2~

7

65.87

24.38

9.76

36.36

63.64

6

60.2?

29.89

9.89

32.48

67.52

30

H. PAUl.

Table 4. Effectof changes in N N

% served

% lost

¢ saved

% idle

% util. 52.84

4

72.40

18.69

8.91

47.16

5

78.26

13.19

8.55

49.98

5o.o2

6

82.11

9.77

8.12

52.46

47.54

7

85.19

7.32

7.49

55.01

44.99

8

87.20

6.13

6.67

57.08

42.92

9

88.70

4.80

6.50

58.03

41.97

10

90.53

3.80

5.68

60.26

39.74

3.3 Number of bundles packed into crates The effect of changing the average number of bundles packed into crates each time by each server is shown in Table 4. As this is varied from 4 to l0 the per cent of bundles lost reduces from about 19% to about 4%. However, the idle time of the servers increases from 47% to about 60%. 3.4 Range of reach o.f servers The results of simulation experiments by changing the range of reach of servers from 0 to 3.5 m in steps of 0.5 m are shown in Table 5. Although increasing the range has similar effects as increasing NSIZE(I) in terms of per cent of bundles lost, its effect on idle time is negligible. Also, each server serves more of his own bundle type and requires less help from the next server. However, when a server does not help in putting on the rack bundles of type assigned to the previous server the number of bundles lost is about 4% higher and the utilization of servers in lower. 3.5 Discharging bundles in groups The results for discharging bundles in groups of six are shown in Table 6. In this simulation experiment the number of servers and assignments have been varied as indicated. The range of reach of servers has been assumed as zero, and the servers work independently. The results indicate marked improvement in per cent of bundles served when the number of servers are kept constant at 12. However, as the number of servers is reduced, the per cent of bundles served decreases. The per cent of bundles served is about 90% when the number of servers is 8 and utilization of servers is about 44%. These figures are about the same as in the present system with 12 servers. 4.1 Discussion The results of the basic model and modification 1 where servers do not help each other

Table 5. Effectof changes in rangeof reach R~ige of Reach DIST

% served

% lost

¢ saved

~ idle

¢ u~ik.

0.0 m

76.17

15.10

10.74

51.84

d8.16

0.5 m

82.11

9.77

8.12

52.46

47.54

1.0

m

86.85

7.15

6.00

53.15

46.85

1.5 m

91.00

4.94

4.36

54.28

45.72

2.0 m

93.79

3.55

2.66

54.64

45.36

2.5 m

94.98

2.18

1.84

55.45

44.55

3.0 m

97.57

1.33

1.1C

55,4~

44,52

3.5 m

98.42

0.88

0.70

55.73

44,~7

Computer simulation of a material flow system Table 6. Effect of discharging bundles in groups No. of servers

~ served

)~ lost

~ idle

~ util,

12

96.33

3.67

68.14

31.53

11

95.42

4.58

65.55

34.12

10

93.76

6.24

62.93

36.74

9

92.13

7.87

59-55

40.14

8

89.64

IC.36

55.71

43.97

7

86.74

13.26

50.65

49.02

6

82.39

17.61

45.21

54.49

4

69-59

30.41

30.74

68.94

differ very little. This is due to the fact that in the basic model the use of IHAND(I) to set up link between Server I and Server I + 1 is very simplified. In the actual situation priority is given to the "missed" bundles. Furthermore, the servers change their pace of work in accordance with changing rates of arrivals of bundles. From the analysis of results of the simulation experiments it is apparent that the major parameters affecting the percent of bundles lost are NSIZE(I) (the number of bundles the servers collect before they perform type 2 service), the duration of type 2 service and the distance within which a server works. Numerical results from the simulation experiments have been analysed and two general relationships relating the above parameters with percent of bundles lost and percent idle time of servers have been obtained by the method of multiple regression. These relationships are: Per cent of bundles lost = 83.667- 11.314 LAST + 0.44354 LAST 2 + 0.22650 E F F + 1.8452 S T - 8.2360 NSIZE + 0.4224 Per cent of idle time = 44.783 + 7.029 L A S T - 0.20713 LAST 2 - 0.44207 EFF - 8.9226 ST1 - 2.0695 ST2 + 4.0701 NSIZE 0.13761 NSIZE 2 + 1.9699 DIST - 0.22547 DIST 2 The above relationships have been found to be significant at 95% confidence level for both overall relationship (F-test) and for individual coefficients (t-test). It may be served that the per cent of idle time is significantly affected by type 1 service time (ST1). However, ST1 does not affect the per cent of bundles lost. This is due to the fact that ST1 is only a small fraction of ST2 and bundles are mostly lost when the server is performing type 2 service. Furthermore, because the maximum time allowance (TOL) is substantially higher than ST1 the server has more than sufficient time to identify and pick up bundles if he maintains a minimum range of reach of 0.5 m. 4.2 Conclusions The following conclusions can be drawn from the study: (1) It is more effective for servers to collect more bundles before turning around to pack the bundles into crates. (2) System performance (as measured by per cent of bundles lost and per cent of idle time) will improve if type 2 service time (ST2) is reduced by bringing the crates nearer to the servers. (3) Although increasing the distance within which each server works will improve performance, this may not be practicable. (4) The number of servers may be reduced to eight from twelve at present without adversely affecting present performance if bundles are released in groups of six or more. If bundles do not arrive in groups, reducing the number of servers will increase per cent of bundles lost and will reduce the per cent idle time of servers. CAIE VoL 7, No. I--C

32

H. PAUL REFERENCES

1. R. L. Disney, Some results of multichannel queueing problems with ordered entry--an application to conveyor theory, J. Indus. Engng 14(2), 105-108 (1963). 2. E. A. El Sayed, C. L. Proctor & H. Elayat, Analysis of closed loop conveyor systems with multiple Poisson inputs and outputs, Int. J. Production Res. 14(1), 99-109 (1976). 3. D. T. Phillips & R. W. Skeith, Ordered entry queueing networks with multiple servers and multiple queues. AIIE Trans. 1, 333-342 (1969). 4. A. A. B. Pritaker, Application of multichannel queueing results to the analysis of conveyor systems. J. Indus. Engng 17(1), 14-21 (1966).