Approximation of Tandem Queues with Blocking

Dug Hee Moon

and Yang Woo Shin

2,∗

School of Industrial Engineering and Naval Architecture, Changwon National University,

Changwon, Gyeongnam 51140, Korea

Department of Statistics, Changwon National University, Changwon, Gyeongnam 51140, Korea

Keywords:

Tandem Queue, Phase Type Distribution, Decomposition, Blocking.

Abstract:

In this paper, we present an approximate analysis for tandem queues with single reliable server at each service

station and a buffer of ﬁnite capacity between service stations. Blocking-After-Service (BAS) rule is adopted.

The effects of the moments of service times to the throughput are investigated numerically and the service

time is ﬁtted with a phase type (PH) distribution by matching the ﬁrst two moments. The system with phase

type service times is approximated based on the decomposition method with two-server-one-buffer subsystem.

Some numerical examples are presented for accuracy of approximation.

1 INTRODUCTION

Tandem queue sometimes called transfer line in man-

ufacturing system with ﬁnite buffers have been widely

used for performance modeling of computer sys-

tems and production systems. e.g. see the mono-

graphs Gershwin (1994), Buzzacott and Shanthiku-

mar (1993), the survey papers Dallery and Gersh-

win (1992), Papadopoulos and Heavey (1996) and the

references therein. Although the system with ﬁnite

buffer is modeled by a Markov chain, the number of

states of the Markov chain increases drastically as the

number of stages increases, which makes analytical or

numerical solutions intractable for the systems with

long line. Approximations of the queueing networks

have been developed in many directions. One is to

overcome the problem of dimension of state space

and another is to reduce the assumption of exponential

service time. The system with phase type service time

or approximate formula of G/G/m/N system have

been used for approximate analysis of the system with

non-exponential service time. One of the most com-

mon method among the approximation techniques

to solve the dimensional problem is decomposition

method developed by Gershwin (1987,1994). The

method decomposes the long line into subsystems

with two service stations and one buffer, and derives

a set of equations that determine the unknown param-

eters of each subsystem, and ﬁnally develops an it-

erative algorithm to solve these equations. There are

some approximations for the system that the service

time distributions are not exponential or geometric (in

discrete time case) based on decomposition method,

see Templemeier and B

urger (2001), Bierbooms et al.

(2011) for system with general service times, Helber

(2005) for the reliable systems with Cox-2 distribu-

tion of service time, and Colledani and Tolio (2011),

Shin and Moon (2018) for discrete time system with

unreliable servers of discrete PH-distribution of geo-

metric or repair time.

In this paper, we present an approximate analysis

for tandem queues with single reliable server at each

service station and a buffer of ﬁnite capacity between

service stations under the blocking-after-service rule.

The approximation is based on the decomposition

method with two-server-one-buffer subsystem. The

system with phase type (PH) distribution of service

time is approximated, where the states of the server

include the state of the number of customers in up-

stream subsystem as well as the states of the server

(blocking, starvation, working) to reﬂect the depen-

dence of consecutive subsystems. In case of general

distribution of service times, approximate the distri-

bution of service times of original system with PH-

distributions and then use the system with PH-service

time as an approximation of original system.

The paper is organized as follows. The effects of

moments of service times to the throughput are nu-

merically investigated in Section 2. The approxima-

tion procedure is described in Section 3. An algo-

rithm for the parameters of subsystems is presented

in Section 4. The effectiveness of the approximation

is investigated numerically in Section 5. Finally, con-

cluding remarks are given in Section 6.

422

Moon, D. and Shin, Y.

Approximation of Tandem Queues with Blocking.

DOI: 10.5220/0007469504220428

In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 422-428

ISBN: 978-989-758-352-0

2 SENSITIVITY OF

THROUGHPUT

In this section, the effects of moments of service times

to the throughput are numerically investigated. We

consider the systems with ﬁve servers where the ser-

vice times are identical with common means 1.0 and

the buffer size between servers are identical. The

buffer size c

between servers are chosen as c

1,5,10. We use the Erlang distribution of order k (E

)

for squared coefﬁcient of variation C

< 1, ex-

ponential distribution (Exp) for C

= 1 and hyperex-

ponential distribution of order 2 (H

) with balanced

mean for C

> 1 whose probability density function

f (t) = p

−λ

+ p

−λ

, t ≥ 0,

with λ

= 2p

µ, λ

= 2p

µ and

1 +

− 1

+ 1

, p

= 1 − p

The lognormal distribution (LN), gamma distribu-

tion (GAM) and Weibul distribution (WEIB) are

compared with the PH-distributions whose the ﬁrst

two moments are the same as those of the distribu-

tions LN, GAM and WEIB. Simulation results of the

throughput are listed in Table 1. The deviation (D)

between PH-distribution and the non PH-distribution

(G) is calculated by

D(%) =

PH − G

× 100,

where PH and G are the simulation results for

throughput of PH-distribution and others, respec-

tively. The table shows that the deviation decreases

as the buffer size increases and the deviation is suf-

ﬁciently small in practical sense even for the system

with moderate buffer size (c

= 5) and high variabil-

ity of service time (C

= 5.0). We have observed the

similar phenomena for the system with 10 servers, but

the results are not listed here. Based on the observa-

tions, we approximate the system with general dis-

tribution of service time using the system with PH-

service time.

3 APPROXIMATION OF TANDEM

QUEUE WITH PH-SERVICE

TIME

3.1 Model

We consider a tandem queueing network L with N

∗

N + 1 servers M

, i = 0,1,··· ,N and buffers B

Table 1: Throughput for tandem queues with ﬁve servers.

1 5 10

Serv. sim(D(%)) sim(D(%)) sim(D(%))

0.25 E

0.805 0.929 0.961

LN 0.803(-0.2) 0.928(-0.1) 0.960(+0.0)

WEIB 0.808(+0.4) 0.930(+0.1) 0.961(+0.0)

0.5 E

0.712 0.875 0.927

LN 0.713(+0.2) 0.873(-0.2) 0.926(-0.1)

WEIB 0.713(+0.2) 0.876(+0.1) 0.927(+0.0)

1.0 Exp 0.608 0.793 0.869

LN 0.618(+1.7) 0.795(+0.2) 0.869(-0.1)

2.0 H

0.513 0.690 0.784

LN 0.530(+3.3) 0.701(+1.6) 0.789(+0.7)

GAM 0.508(-1.0) 0.689(+0.0) 0.785(+0.1)

WEIB 0.508(-1.0) 0.689(-0.1) 0.784(+0.0)

5.0 H

0.416 0.553 0.646

LN 0.416(-0.0) 0.553(+0.0) 0.646(+0.0)

GAM 0.408(-1.8) 0.558(+0.9) 0.655(+1.4)

WEIB 0.402(-3.3) 0.549(-0.7) 0.645(-0.1)

capacity c

with 0 ≤ c

< ∞ between M

i−1

and M

i = 1,2,··· ,N. Service time distribution of M

of phase type denoted by PH(α

), where α

(α

(1),··· ,α

)) is a probability distribution and

= (s

(i, j)) is a nonsingular matrix of size h

such

that s

( j, k) ≥ 0, j 6= k, s

( j, j) = −s

( j) < 0, 1 ≤ j ≤

. Let S

= −S

e = (s

(1),··· ,s

))

, where e is

the column vector of appropriate size whose compo-

nents are all 1. See Neuts (1981) for more about phase

type distribution. Transportation times of customers

through buffers and servers are assumed to be negligi-

ble comparing to service time. We assume the block-

ing after service (BAS) rule. That is, when a server

completes its service at a stage, if the buffer of next

stage is full at that time, then the server is forced to

stop its service and the customer is held at the sta-

tion where it just completed its service until the des-

tination can accommodate it. A server M

is said to

be starved if there are no customers to be served on

the server M

. We assume that the initial server M

never starved and it starts new service immediately af-

ter a service completion unless the server is blocked.

The last server M

is never blocked and the customer

at M

leaves the system immediately after completing

its service.

Denote the state of the server M

at time t by

(t) =











j, M

is working and its service

phase is j

b, M

is blocked

s, M

is starved

and let B

(t) be the number of customers waiting in

at time t. By X

(t), denote the number of cus-

tomers waiting in B

and the customers being served

or blocked at M

and the customers blocked at M

i−1

Approximation of Tandem Queues with Blocking

423

at time t, that is,

(t) = B

(t)+1(M

i−1

(t) = b)+1(M

(t) ∈ {w

w(i),b}),

where 1(A) = 0 if A is true and 0, otherwise and

w(i) = {1, 2, · · · ,h

} the set of phases of service time

of M

. Then X

(t) takes values on {0,1,··· , K

where K

= c

+ 2.

In this paper, we propose an approximate analy-

sis of this system based on decomposition approach.

The ﬁrst step is to decompose the N + 1 server sys-

tem into a set of N two-server-one-buffer subsystems

, i = 1, 2, · · · ,N. Each subsystem L

is consists of

upstream server M

i−1

, downstream server M

and a

buffer B

between them. The subsystems are modeled

with a quasi-birth-and-death process. The transition

rates among the states of servers in L

are presented in

terms of the stationary distributions of adjacent sub-

systems L

i−1

and L

i+1

. Unknown parameters are cal-

culated by an iterative scheme.

In what follows, we use the following conven-

tions. Denote the identity matrix by I the identity

matrix of appropriate size and denote I

if it is nec-

essary to designate the size n of the matrix. Similarly,

means the vector e of size n.

3.2 Subsystems

Subsystem L

consists of two servers M

i−1

and M

and a buffer B

between them. Since M

is a down-

stream server in L

and upstream server in L

i+1

, de-

note the downstream server in L

by M

and the up-

stream server in L

i+1

by M

, if necessary to distin-

guish them. Denote the states of M

and M

at time t

by M

(t) and M

(t), respectively. Deﬁne the the state

i−1

(t) of the upstream server M

i−1

with X

i−1

(t) in

as follows:

i−1

(t) =











( j), if M

i−1

(t) = j, X

i−1

(t) = 1

( j), if M

i−1

(t) = j, X

i−1

(t) ≥ 2

, if M

i−1

(t) = b, X

i−1

(t) = 1

, if M

i−1

= b, X

i−1

(t) ≥ 2

s, if M

i−1

(t) = s

and the state of M

(t) of the downstream server M

(t) =







j, if M

(t) = j

b, if M

(t) = b

s, if M

(t) = s.

Let w

(i) = {w

(1),··· ,w

)}, k = 1, 2, b

b =

}, w

(i) = w

(i) ∪w

(i). Let

i−1

(t) = j} = {M

i−1

(t) ∈ {w

( j), w

( j)}},

i−1

(t) = b} = {M

i−1

(t) ∈ b

b}.

Let Z

(t) = (X

(t),M

i−1

(t),M

(t)). The state space

of Z

(t) can be easily obtained. For example, when

(t) = n, 1 ≤ n < K

, the set on which Z

(t) can attain

the values is

(n) = {(n, x, y) : x ∈ {w

(i − 1), s}, y ∈ {w

w(i),b}}.

The stochastic process Z

is modeled by a Markov

chain on S

with generator of the form







(0)

(1)

−1)

)







where B

(n)

is the square matrix of size m

(n) whose

((x,y),(x

))-component [B

(n)

]

(x,y),(x

)

is the tran-

sition rate of Z

(t) from the state (n,x,y) to the state

(n,x

) without change of X

(t). Similarly, the

((x,y),(x

))-component of A

(n)

is corresponding to

the transition rate from (n,x, y) to (n + 1, x

) and

(n)

]

(x,y),(x

)

is corresponding to the the transition

rate from (n,x,y) to (n − 1,x

3.3 Transition Rates

Note that when X

(t) = n with 1 ≤ n ≤ K

− 2, the be-

havior of M

i−1

does not depend on the state of M

and it depends only on the state of X

(t) and M

j = 0, 1, · · · ,i−2. Similarly, if X

(t) = 0, then the state

of M

is changed by the service completion of M

i−1

and if X

(t) = K

, then M

i−1

(t) ∈ b

b and the state transi-

tion of M

i−1

occurs by a departure from M

. Further-

more, if M

i−1

(t) = b

, then the state transition of M

i−1

occurs by an arrival from the M

i−2

as well as a depar-

ture from M

. For the derivation of the rates, we adopt

the approximation assumption that given M

(t) = x,

i−1

(t) and X

(t) are conditionally independent. Here,

we omit details and just describe the formulae of the

matrices A

(n)

, B

(n)

and C

(n)

Formula of A

(n)

. The approximate formula for

(n)

is given as follows: for 0 ≤ n ≤ K

−1, 1 ≤ i ≤ N,

(n)

= A

i−1

(n) ⊗ A

(n),

where A ⊗ B denotes the Kronecker product of the

matrices A and B. The matrices A

(n) and A

(n) are

given as follows: for 0 ≤ n ≤ K

i+1

−2, A

(n) = S

(n) =





)α

0 0 S

s 0 0 0





ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

424

and A

− 1) = S

i+1

− 1) =





0 S

s 0 0





where q

) and q

) are the column vec-

tors of size h

whose jth-entries are as follows: for

1 ≤ j ≤ h

( j, w

) = P



(t) = 2



(t) = j,

(t) ≥ 2



( j),

( j, w

) = s

( j) − q

( j, w

and for 1 ≤ i ≤ N − 1,

(0) =



s α



, A

(0) = α

(n) = I

, 1 ≤ n ≤ K

− 1, 1 ≤ i ≤ N.

Formula of C

(n)

. The matrices C

(n)

, 1 ≤ n ≤ K

0 ≤ i ≤ N − 1 are given by

(n)

= C

i−1

(n) ⊗C

(n).

The matrices C

(n) and C

(n) are as follows:

) = α

, C

(n) = I

, 1 ≤ n ≤ K

i+1

− 1, and

i+1

) =



(1 − p

(b,s))α

(b,s)α

0 0 1



where

(b,s) = P(X

(t) = 2|M

(t) = b, X

(t) ≥ 2)

and

(n) =



, n = 1,

, 2 ≤ n ≤ K

and for 1 ≤ i ≤ N − 1,

(1) =



w δ

b δ

(b)



(n) =



w b

w δ

w)α

b δ

(b)α



, 2 ≤ n ≤ K

where δ

w) is the column vector of size h

whose jth

component (1 ≤ j ≤ h

) is

( j) = P(X

i+1

(t) ≤ K

i+1

− 2|M

(t) = j)s

( j),

(b) =

∑

y∈M

i+1

P(M

i+1

(t) = y|M

(t) = b)δ

i+1

(y).

Formula of B

(n)

. The matrices B

(n)

are given as

follows

(n)







i−1

− ∆

(0), n = 0,

i−1

⊕ B

− ∆

(n), 2 ≤ n ≤ K

− 1,

i−1

) ⊕ B

− ∆

), n = K

where A ⊕ B = A ⊗ I + I ⊗ B denotes the Kronecker

sum of the matrices A and B and ∆

(n) is the diagonal

matrix that makes Q

e = 0. Denote by S

∗

the square

matrix of size h

whose diagonal elements are all zero

and off diagonal elements are the same as those of S

The matrices B

and B

are as follows:

= S

∗

, B

= S

∗

and for 1 ≤ i ≤ N − 1,





∗

0 0

) S

∗

s q

(s,w

)α





i+1

) =



∗

) 0





w b

w S

∗

w,b)

b 0 0



The q

) is the diagonal matrix of size h

whose

jth diagonal element is

)]

i−1

∑

k=1



i−1

(t) = k



(t) = j

(t) = 1



i−1

(k)

and q

w,b) = (q

( j, b), j = 1,··· , h

) is the column

vector os size h

with

) =

i−1

∑

k=1

P(M

i−1

(t) = k|M

(t) = b

i−1

(k),

(s,w

) =

i−1

∑

k=1

P(M

i−1

(t) = k|M

(t) = s)s

i−1

(k),

( j, b) = P(X

i+1

(t) = K

i+1

− 1|M

(t) = j)s

( j).

3.4 Approximation of Parameters

Now we assume that the system is in stationary

state and let π

be the stationary distribution of Q

We can express the parameters q

(x,x

), p

(b,x) for

(n), B

, C

(n) of the subsystem L

i+1

and δ

i−1

(y),

Approximation of Tandem Queues with Blocking

425

i−1

(w,b) of the subsystem L

i−1

in terms of π

by us-

ing the approximation

P(X

(t) = n, M

i−1

(t) = j,M

(t) = y)

= π

(n,w

( j), y) + π

(n,w

( j), y),

P(X

(t) = K

i−1

(t) = b, M

(t) = y)

= π

,y) + π

,y).

For example,

( j, w

) =

(2, j)s

( j)

( j))

i−1

( j, b) =

i−1

− 1, j)s

i−1

( j)

i−1

( j)

where

(n,y) = P(X

(t) = n, M

(t) = j),

( j)) =

∑

n=2

(n, j),

i−1

( j) = P(M

i−1

(t) = j),

i−1

(n, j) = P(X

(t) = n, M

i−1

(t) = j).

Performance measures. Once the stationary distribu-

tion π

of Z

is obtained, the performance measures

such as the throughput (E) and mean number

customers in B

can be obtained as follows:

E =

∑

j=1

P(M

(t) = j)s

( j),

−1

∑

n=1

(n − 1)π

(n)e +(K

− 2)π

)e.

4 ALGORITHM

In this section, an iterative algorithm for solving the

proposed decomposition equations is presented.

Initial Step: Initially the upstream servers are as-

sumed to be never starved and A

(n), B

(n) and

(n) are as follows: for 0 ≤ i ≤ N − 1,

(n) =



, 0 ≤ n ≤ K

i+1

− 2,

, n = K

i+1

− 1,

= S

∗

(n) =



I, 1 ≤ n ≤ K

i+1

− 1,

, n = K

i+1

Compute the stationary distribution π

and de-

termine the parameters for A

N−1

(n), B

N−1

and

N−1

(n)

Iteration Step:

Step 1. Backward iteration. For i = N − 1, N −

2,··· ,1, compute the stationary distribution π

. If

i ≥ 2, then update A

i−1

(n), B

i−1

and C

i−1

(n), oth-

erwise (i = 1) update A

(n), B

and C

(n).

Step 2. Forward iteration. For i = 2, 3, · · · ,N, com-

pute the stationary distribution π

. If i ≤ N − 1,

then update A

(n), B

and C

(n), otherwise (i =

N) GO TO next step.

Step 3. Calculate throughput and check the toler-

ance. Once the stationary distribution π

of Z

obtained, compute the throughput and check the

stopping criterion

TOL = |E

(m)

− E

(m−1)

| < ε, (1)

where E

(m)

is the throughput obtained in the mth

iteration and ε > 0 is the tolerance predetermined.

If the stopping criterion is not satisﬁed, update

N−1

(n), B

N−1

and C

N−1

(n) and GO TO Step 1

and repeat the backward and forward iteration un-

til the stopping criterion is satisﬁed.

The stationary distribution π

can be computed by

well known matrix analytic method that uses the in-

versions of matrices whose sizes are the same as those

of the block matrices B

(n) in Q

, see e.g. Latouche

and Ramaswami (1999), Shin and Moon (2017). The

complexity of the algorithm in an iteration is O(Nm

where m

is the maximal size of the matrices B

(n).

5 NUMERICAL RESULTS

The effectiveness of the method is investigated nu-

merically in this section. Approximations (App) are

compared with the simulations (Sim). Simulation

models for the systems in the tables are developed

with ARENA. Simulation run time is set to 550,000

unit times including 50,000 unit times of warm-up pe-

riod. Twenty replications are conducted for each case

and the maximum half width of 95% conﬁdence in-

tervals (c.i.) is less than 0.001 and conﬁdence inter-

vals are omitted in the following tables. Tolerance

ε = 10

−5

is used for stopping criterion (1). The de-

viation (D) between approximation and simulation is

calculated by

D(%) =

App − Sim

Sim

× 100,

where Sim and App are the simulation results and ap-

proximation results for throughput.

We consider the systems with 5 servers and 10

servers where the service times are identical with

common means 1.0 and the buffer size between

servers are identical. The buffer size c

between

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

426

Table 2: Throughput for tandem queues with 5 servers.

1 5 10

Sim Sim Sim

App(D(%)) App(D(%)) App(D(%))

0.25 0.805 0.929 0.961

0.790(-1.8) 0.927(-0.2) 0.961(-0.0)

0.5 0.712 0.875 0.927

0.696(-2.3) 0.871(-0.5) 0.926(-0.1)

1.0 0.608 0.793 0.869

0.594(-2.3) 0.786(-0.9) 0.867(-0.2)

2.0 0.513 0.690 0.784

0.512(-0.2) 0.687(-0.4) 0.785(+0.2)

5.0 0.416 0.553 0.646

0.419(+0.7) 0.552(-0.2) 0.650(+0.7)

servers are chosen as c

= 1,5,10. We use the Er-

lang distribution of order k (E

) for C

< 1, ex-

ponential distribution (Exp) for C

= 1 and hyperex-

ponential distribution of order 2 (H

) with balanced

mean for C

> 1. The throughput for the system with

5 servers and 10 servers are presented in Tables 2 and

3. The deviations of approximations from simulation

in Tables 2 and 3 are less than 1% for short length sys-

tem (N

∗

= 5) and are less than 5% even for moderate

size of system with small buffer size (N

∗

= 10,c

= 1)

which may be acceptable in practicable situation. Fur-

thermore, the accuracy of approximations increases as

buffer size increases and the deviations are less than

1% for c

= 10. The throughput for the systems with

10 identical servers of Lognormal (LN) service times

(Sim) and PH-service times (App) is listed in Table

4, where the deviation (D) is the relative deviation

between LN service system and PH service system

with respect to LN service system. Table 4 shows that

the deviation increases as the variation (C

) of service

time increases or the buffer size (c

) decreases which

can be expected from Section 2. It can be seen from

Table 4 that the approximation performs well for the

system in which the coefﬁcient of variation of service

time is not large and buffer size is not so small.

The current algorithm was run on a laptop

computer at 2.70 GHz with 16.0 GB RAM using

Mathematica

11. The maximum CPU time for the

results in Table 3 is 1.25 seconds with 15 iterations

for E

service time and others are less than 0.6 sec-

onds.

6 CONCLUSIONS

In this paper, an approximate analysis for tandem

queues with ﬁnite buffer has been presented. The ser-

vice time is ﬁtted with a phase type (PH) distribution

by matching the ﬁrst two moments of service times

and the system with PH-service times is approximated

Table 3: Throughput for tandem queues with 10 servers.

1 5 10

Sim Sim Sim

App(D(%)) App(D(%)) App(D(%))

0.25 0.776 0.917 0.954

0.755(-2.8) 0.917(-0.1) 0.955(+0.2)

0.5 0.673 0.855 0.915

0.649(-3.7) 0.851(-0.5) 0.916(+0.1)

1.0 0.560 0.763 0.849

0.537(-4.1) 0.752(-1.4) 0.848(-0.1)

2.0 0.449 0.643 0.749

0.445(-0.9) 0.634(-1.4) 0.750(+0.1)

5.0 0.334 0.487 0.591

0.338(+1.2) 0.474(-2.7) 0.589(-0.4)

Table 4: Throughput for the systems with 10 servers of Log-

normal (LN) service times and PH-service times.

1 5 10

LN LN LN

PH (D(%)) PH(D(%)) PH(D(%))

0.25 0.770 0.916 0.953

0.755(-2.0) 0.917(+0.1) 0.955(+0.2)

0.5 0.666 0.851 0.912

0.649(-2.8) 0.851(+0.0) 0.916(+0.4)

1.0 0.558 0.757 0.844

0.537(-3.9) 0.753(-0.6) 0.848(+0.4)

2.0 0.459 0.646 0.748

0.445(-3.1) 0.634(-1.9) 0.750(+0.2)

5.0 0.355 0.507 0.604

0.338(-5.1) 0.474(-6.9) 0.589(-2.6)

based on the decomposition method. To reﬂect the

dependence between consecutive stages, the states of

the servers in subsystems are indicated by the state

of the number of customers in upstream subsystem

as well as the states of the server (blocking, starva-

tion, working), and the transitions among the states

are considered. Numerical experiments indicated that

the method works reasonably.

We have used the Erlang distribution and hyperex-

ponential distribution for ﬁtting the ﬁrst two moments

of service time. There are several ways to use PH dis-

tribution matching the ﬁrst two or three moments of a

nonnegative random variable, see e.g. Bobbio et al.

(2005), Osogami and Harchol-Balter (2006), Tijms

(2003) and references therein. However, it is not easy

to choose the best one among the candidates of PH

distributions. The choice may depend on the model

and it requires preliminary experiment.

ACKNOWLEDGEMENTS

The ﬁrst author and the second author

∗

were sup-

ported by Basic Research Program through the

∗

Corresponding author

Approximation of Tandem Queues with Blocking

427

National Research Foundation of Korea (NRF)

funded by the Ministry of Education, Grant

Numbers NRF-2016R1D1A1A09917954 and NRF-

2018R1D1A1A09083352, respectively.

REFERENCES

Bierbooms, R., Adan, I.J.B.F., van Vuuren, M. (2011).

Approximate analysis of single-server tandem queues

with ﬁnite buffers. Annals of Operations Research,

DOI 10.1007/s10479-011-1021-1.

Bobbio, A., Horvath, A., Scarpa, M., Telek, M. (2005).

Matching three moments with minimal acyclic

phase-type distributions, Stochastic Models 21,

303-326.

Buzzacott, J. A., Shanthikumar, J. G. (1993). Stochastic

Models of Manufacturing Systems, Prentice-Hall.

Colledani, M., Tolio, T. (2011). Performance evaluation of

transfer lines with general repair times and multiple

failure modes, Annals of Operations Research 182,

31-65.

Dallery, Y., Gershwin, B. (1992). Manufacturing ﬂow line

systems: a review of models and analytical results.

Queueing Systems 12, 3-94.

Gershwin, S. B. (1987). An effﬁcient decomposition al-

gorithm for the approximate evaluation of tandem

queues with ﬁnite sotrage space and blocking, Opera-

tions Research 35, 291-305.

Gershwin, S. B. (1994). Manufacturing systems engineer-

ing. Prentice-Hall, Englewood Cliffs.

Helber, S. (2000). Approximate analysis of unreliable

transfer lines with splits in the ﬂow of material, An-

nals of Operations Research 93, 217-243.

Latouche, G., Ramaswami, V. (1999). Introduction to Ma-

trix Analytic Methods in Stochastic Modeling. Siam.

Neuts, M. F., (1981). Matrix Geometric Solutions in

Stochastic Models. Dover Publishing Co., New York.

Osogami, T., Harchol-Balter, M. (2006). Closed form so-

lutions for mapping general distributions to quasi-

minimal PH distributions, Performnce Evaluation 63,

524-552.

Papadopoulos, H. T., Heavey, C. (1996). Queueing theory

in manufacturing systems analysis and design: a clas-

siﬁcation of models for production and transfer lines,

European Journal of Operational Research 92, 1-27.

Shin, Y. W., Moon, D. H. (2017). Approximation of dis-

crete time tandem queueing networks with unreliable

servers and blocking, Journal of Industrial and Man-

agement Optimization 13(2), 901-916.

Shin, Y. W., Moon, D. H. (2018). Approximation of dis-

crete time tandem queueing networks with unreliable

servers and blocking, Performance Evaluation 120,

49-74.

Templemeier, H., B

urger, M. (2001). Performance evalua-

tion of unbalanced ﬂow lines with general distributed

processing times, failures and imperfect production,

IIE Transactions 33, 293-302.

Tijms, H. (2003). A First Course in Stochastic Models, Wi-

ley.

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

428