Variance of Departure Process in Two-Node Tandem Queue with

Unreliable Servers and Blocking

Yang Woo Shin

and Dug Hee Moon

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

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

Changwon, Gyeongnam 51140, Korea

Keywords:

Variance, Departure Process, Tandem Queue, Finite Buffers, Blocking, Markovian Arrival Process.

Abstract:

This paper provides an effective method for evaluating the second moments such as variance and covariance

for the number of departures in two-node tandem queue with unreliable servers. The behavior of the system

is described by a level dependent quasi-birth-and-death process and the departure process is modeled by a

Markovian arrival process. Algorithms for the transient behavior, the variance and covariance structure for the

output process and the time to the nth departure are developed. We show that the results can be applied to

derive approximate formulae for the due-date performance and the distribution of the number of outputs in a

time interval.

1 INTRODUCTION

There is an extensive literature for the analysis of

manufacturing systems with ﬁnite buffers and unre-

liable servers. Most of the works related to the per-

formance evaluation of manufacturing systems have

been focused on analyzing the ﬁrst order measures

such as average production rates and average buffer

levels in steady-state e.g. see the monographs (Buza-

cott and Shanthikumar, 1993; Gershwin, 1994), the

survey papers (Dallery and Gershwin, 1992; Pa-

padopoulos and Heavey, 1996; Li et al., 2009) and

the references therein. The ﬁrst order measures can be

used to get information about the capabilities of a pro-

duction system in the long run. However, there may

be tremendous variability from a time period to pe-

riod (Gershwin, 1994, Section 3.2; Tan, 1999a). Thus

the second order measures such as the variance of the

number of parts produced in a given time period and

the inter-departure times and covariancebetween con-

secutive inter-departure times are also very useful to

design and control production systems in a more ef-

fective way. The information about the time depen-

dent second order measures can especially be useful

to respond short-term and long-term requirements in

an effective and timely way.

Studies on variance of the output process in a se-

rial production line have been presented during the

last decades, for a review of recent studies on the

variance of the output for production systems, one

can refer to the papers (Tan,2000; Tan, 2013; Lager-

shausena and Tan, 2015). For discrete material ﬂow

production systems with ﬁnite buffers, Tan (1999b,

2000) use a Markov reward model to calculate the

variance of the number of parts produced in a given

time period in a two-station production line with ﬁnite

buffer capacities and deterministic processing times

and geometrically distributed failure and repair times.

Our approach to be developedin this paper is to model

the output process by a Markovian arrival process

(MAP) and to use the closed formulae for the tran-

sient behavior and the variance and covariance struc-

ture for the number of outputs during a period (0, t]

and the nth departure time in the literature.

This paper is aimed on providing an effective

method for evaluating the second moments of the

number of outputs and inter-departure times and in-

vestigating the effects of the system parameters to the

second moments. The results can be applied to the

practical problem such as due-time performance in

manufacturing system and are basis on analyzing the

long line. This paper concerns to the two-station sys-

tem with ﬁnite buffer capacities. A model of a two-

node system is simple, but it helps us to understand

the behavior of the system and gives some insights

of the more complicated system. The approach can

also be used as building block for analyzing the more

complex system with multiple nodes.

258

Shin Y. and Moon D.

Variance of Departure Process in Two-Node Tandem Queue with Unreliable Servers and Blocking.

DOI: 10.5220/0006119302580264

In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 258-264

ISBN: 978-989-758-218-9

This paper is organized as follows. In Section 2,

the model is described in detail. The moment formu-

lae for MAP are reviewed and algorithms for the per-

formance measures are presented in Sections 3 and 4,

respectively. In section 5, numerical results are pre-

sented. Concluding remarks are given in Section 6.

2 MODEL

We consider a tandem queueing network that consists

of two service stations S

and S

and one buffer of

ﬁnite size b between them. Each station S

has an

unreliable server M

, i = 1,2. Assume the following

system characteristics.

BAS blocking mechanism : Blocking after service

(BAS) rule is adopted, that is, if the buffer is full upon

a completion of service at the ﬁrst station, the server

is blocked and the customer is held at the station

where it just completed its service until the station S

can accommodate it.

Open and saturated system: In many manufac-

turing system, it has been assumed that the ﬁrst sta-

tion the ﬁrst station is never starved and the last sta-

tion is never blocked. For potential applications of

the method and results to developing approximation

method of more complicated system, we assume that

the server M

in the ﬁrst station is never starved and it

starts newservice immediately after a service comple-

tion unless the server is blocked and the server M

the second station is never blocked and the customer

at M

leaves the system immediately after completing

its service.

ODF rule : Each server is either up (operational)

or under repair (broken-down) at any time. Operation

dependent failure (ODF) is assumed. That is, a server

can fail only while the server is working and a server

never fails while the server is blocked or starved.

Exponential distributions of service time, failure

time and repair time: We deﬁne the failure time by

the operation time in units between two successive

failures (from a repair to a failure). The failure time

does not contain the time period while the server is

being blocked, starved or repaired. Service time, fail-

ure time and repair time of M

are assumed to be of

exponential with rates µ

, ν

and η

, respectively.

Let X(t) be the number of customers in the buffer

and at station S

and the customer blocked at station

. The state space of X(t) is {0, 1,···,K}, where

K = b + 2. Let J

(t) be service phase of the server M

at time t denote the states of J(t) by

(t) =











w, M

is working

s, M

is starved

b, M

is blocked

f, M

is failed.

The state space of the stochastic process Z

Z =

{Z(t), t ≥ 0} with Z(t) = (X(t),J

(t), J

(t)) is

S = ∪

n=0

where

= {(0, w, s),(0, f,s)},

= {(n, j

, j

) : j

, j

∈ {w, f}}, 1 ≤n ≤K −1,

= {(K, b,w),(K,b, f)}.

The stochastic process Z

Z = {Z(t), t ≥ 0} forms a

Markov chain with generator of the form

Q =







K−1







The matrices B

, A

, C

are as follows:







∗ ν

∗ 0 ν

0 ∗ ν

0 η

∗







, 1 ≤n ≤K −1,



∗ ν

∗



, B

∗



∗ ν

∗









0 0 0

0 µ

0 0

0 0 0 0







, 1 ≤n ≤K −2,



0 0 0

0 0 0 0



K−1







0 µ

0 0













0 0 0

0 0 0 0

0 0 µ

0 0 0 0







, 2 ≤n ≤K −1,







0 0

0 µ

0 0







, C



0 0 0

0 0 0 0



where the diagonal entries of B

are determined by

Qe = 0 and e is a column vector of appropriate size

whose elements are all 1.

Variance of Departure Process in Two-Node Tandem Queue with Unreliable Servers and Blocking

259

3 DEPARTURE PROCESS

Let T be the ﬁrst time until a customer leaves the sys-

tem and

′

(t) = P(Z(T) = z

′

,T ≤ t |Z(0) = z), z,z

′

∈ S .

Then T is the same as the absorbing time of a Markov

chain with rate matrix of the form



0 0



where







K−1



















The matrix F(t) = (F

′

(t)) is given by

F(t) =

exp(D

u)duD

, t ≥ 0

which is the inter arrival time of a Markovian arrival

process (MAP) with representation MAP(D

), see

Lucantoni et al. (1990).

Let N(t) be the number of customers that leave

the system during an interval (0, t] and P(n,t) =

′

(n,t)) be the square matrix of size |S | whose

(z,z

′

)-component is

′

(n,t) = P(N(t) = n, Z(t) = z

′

|Z(0) = z).

It follows from the Kolomogorov equations that

P(n,t) = P(n,t)D

+ P(n−1,t)D

, n ≥1, t ≥ 0

(1)

and P(0,0) = I the identity matrix. The matrix gener-

ating function P

∗

(w,t) =

∑

∞

n=0

P(n,t) is given by

∗

(w,t) = exp[(D

+ wD

)t] , |w| ≤ 1, t ≥ 0.

For later use, deﬁne the following notation. Let π

π =

(π(x),x ∈ S ) be the stationary distribution of Q and

Π = eπ

π, Ψ = (eπ

π−Q)

−1

, λ = π

πD

c = π

πD

Ψ, d = ΨD

It can be easily seen that π

πΨ = π

π, Ψe = e and ce =

λ = π

πd.

The following theorem can be found in (Neuts,

1989, Theorems 5.4.1 and 5.4.2; Artalejo et al, 2010).

Theorem 3.1. In stationary state, that is, π(x) =

P(Z(0) = x), mean µ(t) = E[N(t)], variance

(t) = Var[N(t)] and the covariance Cov(t,u,v) =

Cov[N(t),N(v) −N(u)] (0 < t ≤ u < v) are given as

follows:

µ(t) = λt,

(t) =

(t) + 2c[exp(Qt) −Π

Π]d,

Cov(t,u,v) = π

πD

[I −exp(Qt)] exp[Q(u−t)]

×[I −exp(Q(v−u))]Ψd.

where

(t) = 2(λ

−cd) + (λ−2λ

+ 2cD

e)t. (2)

Remark 1. It is well known that as t → ∞

exp(Qt) = Π

Π+ O(t

r−1

−ηt

), (3)

where −η is the real part of η

∗

, the non-zero eigen

value of Q with maximum real part, and r is the mul-

tiplicity of η

∗

, see e.g. (Narayana and Neuts, 1992).

It can be easily seen from Theorem 3.1 and (3) that

Cov(t,u,v) → 0 as u−t → ∞.

Remark 2. It can be seen from Theorem 3.1 that

the variance rate is given by the closed formula

V = lim

t→∞

Var[N(t)]

= (λ −2λ

+ 2cD

e).

Tan (1999b) use numerical result of the asymptotic

variance rate V to determine the variance σ

(t) ≈Vt

for large t. We can see that

(t) provides more ac-

curate approximation of σ

(t) than that of Vt and it is

easy to compute

(t).

Let ξ

, n = 0,1,2, ··· be the nth transition time

of N

N with ξ

= 0 and set τ

= ξ

−ξ

n−1

and Z

Z(ξ

+ 0), n = 1,2,··· with Z

= Z(0). The tran-

sition probability matrix of {Z

,n = 0, 1,2, ···} is

P = (−D

)

−1

and the stationary distribution p

p of

P is given by

p =

πD

The following theorem can be found in Artalejo et al.

(2010).

Theorem 3.2. Assume that Z(0) has a distribution

a = (a(x),x ∈ S ) with a(x) = P(Z(0) = x). The mean,

variance and covariance of τ

are given as follows:

E[τ

] = aP

n−1

Var[τ

] = 2aP

n−1

(−D

)

−2

e−



n−1



Cov(τ

,τ

) = aX

k,n

e−(E[τ

])(E[τ

]), 1 ≤ k < n,

where

= (−D

)

−1

k,n

= P

k−1

(−D

)

−1

n−k

(−D

)

−1

P, 1 ≤k < n.

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

260

Remark 3. Since lim

n→∞

= ep

p, it can be easily

seen that for each k = 1, 2,···,

lim

n→∞

Cov(τ

,τ

) = 0.

Remark 4. The mean and variance of ξ

are as

follows

E[ξ

] =

∑

i=1

E[τ

Var[ξ

] =

∑

i=1

Var[τ

] + 2

n−1

∑

i=1

∑

j=i+1

Cov(τ

,τ

Remark 5. Assuming a = p

p, the mean, variance

and covariance of τ

are as follows:

E[τ

] =

Var[τ

] =

πd

−

Cov(τ

,τ

) =

πP

−

,n = 1,2,···.

4 ALGORITHMS

It is necessary to π

π, Ψ = (eπ

π−Q)

−1

and exp(Qt) for

the variance, covariance of N(t) and τ

. In this sec-

tion, some algorithms for computing π

π, Ψ = (eπ

π −

−1

and exp(Qt) are presented.

1. Algorithm for stationary distribution π

π of Q.

Here, we present an algorithm for stationary distribu-

tion π

π of Q. Write π

π = (π

,π

,··· ,π

), where π

the vector of size l

, 0 ≤ i ≤ K. Let R

,··· , R

be the

matrices satisﬁes the following matrix equations

n−1

+ R

n+1

= 0, 1 ≤n ≤K −1,

K−1

+ R

= 0.

The solutions of the equation are given as follows:

= A

K−1

(−B

)

−1

= A

n−1

[−(B

+ R

n+1

)]

−1

, n = K −1,···,1.

Then the stationary distributionπ

π of Q is given as fol-

lows

= π

···R

, n = 1, 2,···,K

with

+ R

] = 0

and normalizing condition

∑

n=1

···R

= 1.

Once π

π is obtained, p

p = (p

, p

,··· , p

) can be cal-

culated by



k+1

, k = 0, 1,··· , K −1,

0, k = K.

2. Algorithm for (−D

)

−1

. It can be seen from the

structure of D

that (−D

)

−1

is of the form

(−D

)

−1







X(0,0) X(0, 1) ··· X(0, K)

X(1,1) ··· X(1,K)

X(K,K)







The block components X(i, j), 0 ≤ i ≤ j ≤ K are cal-

culated following the algorithm in (Shin, 2009) as fol-

lows :

(1) Compute

= A

n−1

(−B

)

−1

, n = K,K −1,N −2, ··· ,1

and G

= (−B

)

−1

(2) Compute X(n, k), 0 ≤ n ≤ K, k = n,n + 1, ··· , K

as follows: For n = 0,1,··· ,K, set X(n,n) =

(−B

)

−1

and

X(n,k) = X(n,k−1)G

, k = n+ 1,n+ 2,···,K.

3. Algorithm for Ψ = (eπ

π − Q)

−1

. Let E

(1,··· ,1)

(0 ≤ n ≤ K) be the l

-dimensional col-

umn vector whose components are all one and E

∗

(1,··· ,1)

be the (

∑

i=1

)-dimensional column vec-

tor, where l

is the number of elements of S

. Let

∗

= (π

,··· ,π

) and Π[i, j] = E

, 0 ≤i, j ≤K. De-

note the (i, j) block matrix of a matrix A correspond-

ing to (i, j) block of Q by A[i, j], 0 ≤ i, j ≤ K. Write

the matrix Q in the block form

Q =





, eπ

π−Q =





where

= E

−B

, A

= E

∗

−Q

= E

∗

−Q

, A

= E

∗

−Q

Then the block matrix form of Ψ is given by (e.g.

(Horn and Johnson, 1985, page 18))

Ψ =



∗−1

−A

−1

∗−1

−A

∗−1

−1

∗−1



where

∗

= A

−A

−1

∗

= A

−A

−1

The matrix Ψ is calculated by the following step:

(1) Calculate (−B

)

−1

using the ordinary algorithm.

(2) Since Q

is block tridiagonal matrix, one can use

the algorithm in (Shin, 2009) for (−Q

)

−1

(3) For A

−1

and A

−1

, one can use the following for-

mula (see Horn and Johnson(1985, page 19))

−1

= (−Q

)

−1

−

1+π

∗

(−Q

)

−1

where q

= (−Q

)

−1

∗

. The inverse matrix

−1

is calculated by usual method.

Variance of Departure Process in Two-Node Tandem Queue with Unreliable Servers and Blocking

261

(4) Calculate A

∗−1

by the formula,

∗−1

= (A

+ A

(−A

−1

)

−1

= A

−1

+ A

−1

∗−1

−1

where A

∗−1

is calculated by usual method.

(5) The (i, j) block Ψ[i, j] of Ψ is

Ψ[i, j] =











∗−1

, i = 0, j = 0,

−A

−1

∗−1

)[ j], i = 0, 1 ≤ j ≤ K,

−(A

∗−1

)[i]A

−1

, 1 ≤ i ≤ K, j = 0,

∗−1

[i, j], 1 ≤ i, j ≤ K.

4. Calculation of exp(Qt) =

∑

∞

n=0

. We use

the uniformization technique. Let

q = max

z∈S

([−Q]

)

and Θ = I +

Q. Then

exp(Qt) = Q

(t) + E

(M)

(t),

where

(t) =

∑

n=0

−qt

(qt)

(M)

(t) =

∞

∑

n=M+1

−qt

(qt)

For given ε > 0, let M(ε) be the positive integer such

that

1−

M(ε)

∑

n=0

−qt

(qt)

< ε.

For large t, the following addition formula is useful.

First, take an integer n

such that t

is moderate

with E

(M)

)e < ε

e. Note that

exp(Qt) = [exp(Qt

)]

) + E

(M)

)

= [Q

)]

+ E

(M)

where

(M)

) =

∑

k=1





(M)

)

−k

)]

Since Q

)e < (1−ε

)e and E

(M)

)e < ε

e, it can

be seen that

(M)

)e < (1−(1−ε

)

)e.

5. Calculation of P(n,t). Let Θ

= I +

. Ap-

plying the uniformization technique to P(n,t) and us-

ing the the Kolmogorov equation (1), it can be seen

that

P(k,t) =

∞

∑

n=0

−qt

(qt)

(n)

, k ≥ 1,

where {K

(n)

}satisﬁes the followings: for k = 1,2,···,

(n+1)

(n)

k−1

+ K

(n)

, n = 0, 1,2, ··· (4)

with K

(0)

= 0, k ≥1 and K

(0)

= I and

(n+1)

= K

(n)

, n ≥ 0.

The recursive formula (4) is also given in (Lucantoni,

1991). Let

(M)

= P(k,t) −

∑

n=0

−qt

(qt)

(n)

, k ≥ 0.

Note that

exp(Qt) =

∞

∑

k=0

P(k,t) =

∞

∑

n=0

−qt

(qt)

∞

∑

k=0

(n)

It can be seen from

∞

∑

k=0

(n)

, n ≥ 0

that K

(n)

e < e and hence

(M(ε))

e < εe, k ≥ 0.

5 NUMERICAL RESULTS

We apply the algorithms in section 4 to the system

with two-node tandem queue with a ﬁnite buffer and

server breakdown. We consider the system with ser-

vice rates µ

= µ

= 1.0, failure rates ν

= 0.1, ν

0.04 and repair rates η

= 0.5, η

= 0.2. The isolated

efﬁciency of each server is the same as

+η

= 0.833.

In this section, we assume that the system is in sta-

tionary state.

1. Speed of convergence to stationary state. We

investigate how fast the distribution of the process

Z converges to the stationary distribution π

π. It can

be seen from (3) that ∆

(t) = log||exp(Qt) − Π||

is almost linear for large t. We also have seen that

the speed of convergence of exp(Qt) decreases as

buffer size increases. For example, the time t

(b) :=

min{t > 0 : ∆

(t) < −5} are t

(3) = 74, t

(5) = 99,

(7) = 134.

2. Variance of N(t). Figures 1 and 2 show vari-

ance σ

(t) and the difference ∆

(t) = σ

(t) −

(t).

Figure 1 exhibits that σ

(t) increases almost linearly

as t increases as expected in the formula in Theorem

3.1. It can be seen from ﬁg. 2 that

(t) can be used

instead of σ

(t) for t ≥ t

. In fact, it follows from

Theorem 3.1 and (3) that for t

with ∆

) < log

ε,

|∆

(t)| < 2|cd|ε, t ≥ t

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

262

Figure 1: The variance σ

(t).

Figure 2: Differences ∆

(t) = σ

(t) −

(t).

Indeed, for b = 5, ∆

(64) < 10

−3

, ∆

(81) < 10

−4

and ∆

(99) < 10

−5

3. Covariance of N(t). Figure 3 de-

picts the Cov[t] = Cov[N(t), N(2t) − N(t)] and

lim

t→∞

Cov[t] = π

πd−λ

. Figure 3 shows that Cov[t]

is positive for b = 3 and negative for b = 5.

Figure 3: Covariance Cov[N(t),N(2t) −N(t)] for b = 3, 5.

5. Distribution of N(t). The distribution of N(t)

is depicted in ﬁgure 4 for t = 30, t = 50, t = 70.

The ﬁgures show that the distribution of N(t) visu-

ally resembles the normal distribution. The the pair

(skewness, kurtosis) of N(t) are (−0.2599,0.5054),

(−0.2393,0.1739) and (−0.2141,0.1486) for t = 30,

t = 50 andt = 70, respectively. Here, we approximate

the distribution of N(t) in stationary state with the

normal distribution N(µ(t),σ

(t)) with mean µ(t) =

λt and variance σ

(t) as Tan (1999b), that is,

P(N(t) ≥ n) ≈ 1−Φ

n−0.5−λt

(t)

, (5)

where Φ(x) =

−∞

√

2π

exp(−y

/2)dy is the distribu-

tion function of the standard normal distribution and

n −0.5 is used for correction of the approximation

of discrete random variable using continuous distri-

bution and

(t) can be used as an approximation of

(t) for large t.

Figure 4: Plot of p(n,t) = P(N(t) = n)] for b = 5.

The approximation errors ∆

(t) between

P(N(t) ≥ n) and normal approximation are depicted

in ﬁgure 5 for t = 30, t = 50, t = 70 and b = 5. The

maximal error of approximation occurs at the mean

λt for each case. Figure shows that the accuracy

increases as t increases.

Figure 5: Error of normal approximation for P(N(t) ≥ n).

6. Due time performance. The due-time perfor-

mance of a production line can be measured by a

probability

p = P(N(t

∗

) ≥n

∗

)

of meeting a customer’s order n

∗

on time t

∗

. Some

numerical results for t

∗

for given n

∗

and p are listed

in Table 1.

Table 1: Due time t

∗

0.5 0.6 0.7 0.8 0.9 0.99

50 72 76 79 84 91 109

70 101 105 110 115 123 143

100 145 150 155 161 170 193

Variance of Departure Process in Two-Node Tandem Queue with Unreliable Servers and Blocking

263

6 CONCLUSIONS

We have provided an algorithm for the transient be-

havior, the variance and covariance structure for the

output process and inter-departure time in two-node

tandem queue. Some numerical results are presented.

We also showed that the results can be applied to

derive approximate formulae for the due-date perfor-

mance and the distribution of the number of outputs

in a time interval.

The algorithm is based on the Markovian arrival

process (MAP) which gives closed formula for vari-

ance and asymptotic variance. This is a different point

from the other methods in the literature for variance

of departure process. The algorithm requires only the

inversions of the block matrices of size 4 in the com-

puting process. Thus the computational complexity of

the algorithm dose not severely depend on the buffer

size of the system. The approach using MAP can be

easily applied to the system with more general ser-

vice, failure and repair time than exponential case.

Although the method developed in this paper is

quite efﬁciently, it will be limited to apply the method

to the system with multiple nodes due to the rapid in-

crease of the number of states when the number of

stations and the buffer capacities increase. There-

fore developing approximation methods to estimate

the second moment measures in multiple node system

are required. There are many approximation methods

for throughput in a complicated system, for exam-

ple, decomposition method and aggregation method

(Dallery and Gershwin, 1992; Li et al., 2009) that use

the the two-node system. The method of analyzing

the two-node system can be used as a building block

of analyzing the more complex system.

ACKNOWLEDGEMENTS

The authors are very thankful to three anonymous re-

viewers for valuable comments and suggestions.

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