Impatient Customers in an M/M/1 Working Vacation Queue with a Waiting Server and Setup Time

ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Impatient Customers in an M/M/1 Working Vacation Queue with a Waiting Server and Setup Time

P. Manoharan* and T. Jeeva Annamalai University, INDIA.

email:[email protected], [email protected]

(Received on: May 14, 2019) ABSTRACT

Using probability generating function method, Impatient customers in an M/M/1 working vacation queue with a waiting server and set-up time is discussed in this paper. Customers impatience is due to the servers vacation. We obtain the probability generating functions of the stationary state probabilities, performance measures, sojourn time of a customer and stochastic decomposition of the queue length and waiting time.

Keywords: M/M/1 queue; Impatient customers; Working vacation; Setup time;

Probability generating function.

1. INTRODUCTION

Server vacations were introduced first by Levy and Yechiali

⁸

. Substantial work in this area has been conducted during the last few decades while Tian and Zhang

¹³

, Doshi

⁶

, Baba

²

and Takagi

¹²

surveyed the vacation models. Recently, Ke, Wu and Zhang

⁷

did a survey about the vacation models. These studies help understanding about a variety of vacation models.

Queueing systems with impatient customers appear in many real life situations such as those involving impatient telephone switchboard customers, hospital emergency rooms handling critical patients, and inventory systems that store perishable goods

^9,1

. There is growing interest in the analysis of queueing systems with impatient customers. This is due to their potential application in communication systems, call centers, Production-inventory systems and many other related areas, see for

^4,3

.

Queueing systems with impatient customers have been studied by a number of

authors. We mention a few of the more significant works below. Palm's pioneering work

¹¹

seems to be the first to analyze queueing systems with impatient customers by considering the

M/M/C queue, where the customers have independent exponentially distributed waiting times.

The situation of impatient customers in a server vacations period was investigated by Altman and Yechialli, Yue et al.

¹⁴

have analyzed an M/M/1 queueing system with working vacation and impatient customers and obtained the pgf of the number of units in the model when the server is in a working vacation period and in a service period. Padmavathy et al.

¹⁰

have studied the steady state behaviour of vacation queues with impatient customers and a waiting server.

In this paper we analyze the impatient customers in an M/M/1 working vacation queue with waiting server and setup times. In this model, if the system becomes empty, then the server waits for a random period of time. This option of the server waits reflects many real life queueing systems. It was first introduced by Boxma et al.

⁵

for the ordinary M/G/1 queue with server vacations.

The rest of the paper is organized as follows. Section 2, gives a description of the Impatient customers in an Markovian working vacation queue with waiting server and setup times. Section 3, gives the analysis of the M/M/1 working vacation model with the stochastic decomposition property. In section 4, gives the numerical results.

2. MODEL DESCRIPTION

We consider a Impatient customers in an M/M/1 working vacation queue with a waiting server and setup time. The assumption of the model are as follows: Customers arrive according to a poisson process with arrival rate 𝜆. The service rate is 𝜇

_𝑏

during a normal service period. When the busy period is ended the server waits a random duration of time before beginning a vacation. This waiting time duration follows the exponential distribution with rate 𝜂 is the waiting rate of a server. The server takes a working vacation as soon as the system becomes empty. During the vacation period, the server serves the customers at an exponential rate 𝜇

_𝑣

, where 𝜇

_𝑣

< 𝜇

_𝑏

. The duration of a vacation is also exponential with rate 𝛾.

The customers in this system are assumed to be impatient as described below.

Whenever arriving customer has to join a queue and finds the system in working vacation, the customer activates an impatient timer 'T', which is exponential with rate 𝛼 and is independent of the number of customers in the system at that moment. If the server ends the vacation before the time T expires, the customer stays in the system till his service is completed, otherwise the customer leaves the system and will never return. When a vacation ends, if there are customers in the queue, the server changes his service rate from 𝜇

_𝑣

to 𝜇

_𝑏

and a regular busy period starts, otherwise the server begins a closed-down period. However, a server needs some setup time to be active so as to serve waiting customers. We assume that the setup time follows the exponential distribution with mean

¹_𝛽

.

We assume that inter arrival times, service times, working vacation times, impatient times and setup times are mutually independent. In addition the service order is First in First out(FIFO). At time t, let Q(t) denote the total number of customers in the system and J(t) denotes the state of the server with,

J(t)=

0, at time t the server is busy in working vacation period, 1, at time t the server is free during in setup period, 2, at time t the server is busy in normal service period.

Then the pair {(𝑄(𝑡), 𝐽(𝑡)); 𝑡 ≥ 0} defines a two dimensional continuous time Markov chain with state space 𝐸 = {{(0,0), (0,1), (0,2)} ∪ {𝑖, 𝑗}, 𝑖 = 1,2, . . . , 𝑗 = 0,1,2}. Let 𝑃

_𝑖,𝑗

= lim

𝑡→∞

𝑃{𝑄(𝑡) = 𝑖, 𝐽(𝑡) = 𝑗}, (𝑖, 𝑗) ∈ 𝐸. Let 𝑃

_𝑖,0

, 𝑖 ≥ 0, be the probability that there are i customers in the system when the server is in working vacation period, 𝑃

_𝑖,1

, 𝑖 ≥ 0, be the probability that there are i customers in the system when the server is in setup period and 𝑃

_𝑖,2

, 𝑖 ≥ 0, be the probability that there are i customers in the system when the server is in normal busy period. Therefore, using the theory of Markov process, we obtain the following set of balance equations.

(𝜆 + 𝛾 ) 𝑝

_0,0

= 𝜂𝑝

_0,2

+ 𝜇

_𝑣

𝑝

_{1,0 ,}

(1)

(𝜆 + (𝑛 − 1)𝛼 + 𝜇

_𝑣

+ 𝛾) 𝑝

_𝑛,0

= 𝜆𝑝

_𝑛−1,0

+ (𝜇

_𝑣

+ 𝑛𝛼)𝑝

_𝑛+1,0

, 𝑛 ≥ 1, (2)

𝜆𝑝

_0,1

= 𝛾𝑝

_{0,0 ,}

(3)

(𝜆 + 𝛽)𝑝

_𝑛,1

= 𝜆𝑝

_𝑛−1,1

, 𝑛 ≥ 1, (4)

(𝜆 + 𝜂)𝑝

_0,2

= 𝜇

_𝑏

𝑝

_1,2

+ 𝛽 𝑃

_0,1

, (5)

(𝜆 + 𝜇

_𝑏

)𝑝

_𝑛,2

= 𝜆𝑝

_𝑛−1,2

+ 𝛽𝑝

_𝑛,1

+ 𝛾𝑝

_𝑛,0

+ 𝜇

_𝑏

𝑝

_𝑛+1,2

, 𝑛 ≥ 1 . (6)

Define the Probability generating functions (PGFs) 𝑄

₀

(𝑧) =

₀ 0 , n n n

p z , 𝑄

₁

(𝑧) =

₁ 0 , n n n

p z , 𝑄

₂

(𝑧) =

₂ 0 , n n n

p z (7)

with 𝑄

₀

(1) + 𝑄

₁

(1) + 𝑄

₂

(1) = 1 and 𝑄

₀^′

(𝑧) =

1 1 ,0 n n

np z

n

Multiplying (2) by 𝑧

^𝑛

and summing over n and using (1) , we get

(𝜆 + 𝜇_𝑣+ 𝛾) ₀ 1 , n n n

p z

+ 𝛼 ₀ 1 , n n n

p z

(𝑛 − 1) + (𝜆 + 𝛾)𝑝₀₀= 𝜆𝑧 _,0 1 1 n n

p

n

z

+ 𝜇𝑣 ,0 1 1 n n

p

n

z

+𝛼 _1,0 1 n n

np

n

z

+ 𝜂𝑝0,2+ 𝜇𝑣𝑝1,0 𝛼𝑧(1 − 𝑧)𝑄0′(𝑧) − 𝑄0(𝑧)[(𝜆𝑧 − 𝜇𝑣+ 𝛼)(1 − 𝑧) + 𝛾𝑧] + (𝛼 − 𝜇𝑣)(1 − 𝑧)𝑝0,0+ 𝜂𝑧𝑝0,2= 0

(8) Putting 𝑧 = 1 in (8) we get

𝛾𝑄

₀

(1) = 𝜂𝑝

_0,2

−𝑄

₀

(𝑧)[(𝜆𝑧 − 𝜇

_𝑣

)(1 − 𝑧) + 𝛾𝑧] − 𝜇

_𝑣

(1 − 𝑧)𝑝

_0,0

+ 𝜂𝑧𝑝

_0,2

= 0 𝑄

₀

(𝑧) = 𝜇

_𝑣

(1 − 𝑧)𝑝

_0,0

+ 𝜂𝑧𝑝

_0,2

𝜆𝑧 − 𝜆

²

− 𝜇

_𝑣

+ 𝜇

_𝑣

𝑧 + 𝛾𝑧 Equation (8) can be written as 𝑄

₀

′(𝑧) − (

^{𝜆𝑧−𝜇𝑣+𝛼}

𝛼𝑧

+

^𝛾

𝛼(1−𝑧)

)𝑄

₀

(𝑧) +

^{(𝛼−𝜇𝑣)}

𝛼𝑧

𝑝

_0,0

+

^𝜂

𝛼(1−𝑧)

𝑝

_0,2

= 0

Integrating Factor

𝑒

^{− ∫ (𝜆𝑧−𝜇𝑣+𝛼𝛼𝑧 +𝛼(1−𝑧)𝛾 )𝑑𝑧}

= 𝑒

^{−𝜆𝛼𝑧}

𝑧

^{𝜇𝑣𝛼−1}

(1 − 𝑧)

^𝛾𝛼

The general solution to the differential equation is given by

𝑄0(𝑧) = 𝑒^𝛼𝜆𝑧𝑧^{−𝜇𝑣𝛼−1}(1 − 𝑧)^−𝛾𝛼((𝜇𝑣

𝛼 − 1)𝑝00∫

𝑧 0

𝑒^{−𝜆𝛼𝑥}𝑥^{𝜇𝑣𝛼−2}(1 − 𝑥)^𝛾𝛼𝑑𝑥 −𝜂

𝛼𝑝_0,2∫^𝑧

0

𝑒^{−𝜆𝛼𝑥}𝑥^{𝜇𝑣𝛼−1}(1 − 𝑥)^𝛾𝛼−1𝑑𝑥)

𝑄

₀

(𝑧) = 𝑧

^{−𝜇𝑣𝛼−1}

(1 − 𝑧)

^−𝛼𝛾

((

^𝜇_𝛼^𝑣

− 1)𝑝

_0,0

𝑘

₁

(𝑧) −

^𝜂_𝛼

𝑝

_0,2

𝑘

₂

(𝑧)) (9) 𝑘

₁

(𝑧) = ∫

₀^𝑧

𝑒

^{𝜆𝛼(𝑧−𝑥)}

𝑥

^{−𝜇𝑣𝛼−2}

(1 − 𝑥)

^𝛾𝛼

𝑑𝑥

𝑘

₂

(𝑧) = ∫

₀^𝑧

𝑒

^{𝛼𝜆(𝑧−𝑥)}

𝑥

^{𝜇𝑣𝛼−1}

(1 − 𝑥)

^𝛾𝛼−1

𝑑𝑥 Taking

1

lim

z

in equation (8) we get

1

lim

z

𝑄

₀

(𝑧) = 𝑄

₀

(1) = [( 𝜇

_𝑣

𝛼 − 1)𝑝

_0,0

𝑘

₁

(𝑧) − 𝜂

𝛼 𝑝

_0,2

𝑘

₂

(𝑧)]

1

lim

z

(1 − 𝑧)

^−𝛾𝛼

(

^𝜇_𝛼^𝑣

− 1) 𝑝

_0,0

𝑘

₁

(1) =

_𝛼^𝜂

𝑝

_0,2

𝑘

₂

(1)

𝑝

₀₂

= (

^𝜇𝑣_𝛼^−𝛼𝛼_𝜂

𝑝

_0,0^𝑘_𝑘¹⁽¹⁾

2(1)

) 𝑝

₀₂

= (

^𝜇𝑣_𝜂^−𝛼

𝑝

_0,0^𝑘_𝑘¹⁽¹⁾

2(1)

)

(10)

Multiplying (3)and (4) by 𝑧

^𝑛

and summing over n, we get

(𝜆 + 𝛽)

₀

1 ,

n n n

p z = 𝜆

_,1

1 1 n

n

p

n

z (𝜆 + 𝛽)𝑄

₁

(𝑧) − (𝜆 + 𝛽)𝑝

_0,1

= 𝜆𝑧𝑄

₁

(𝑧)

(𝜆 + 𝛽 − 𝜆𝑧)𝑄

₁

(𝑧) = (𝜆 + 𝛽)𝑝

_0,1

(11)

Similar way (5) and (6) with power of 𝑧

^𝑛

and summing over all possible values of n, we get

(𝜆 + 𝜇_𝑏)(𝑄₂(𝑧) − 𝑝_0,2) + (𝜆 + 𝜂)𝑝_0,2 = 𝜆𝑧𝑄₂(𝑧) + 𝛽(𝑄₁(𝑧) − 𝑝_0,1) +𝜇𝑏

𝑧 (𝑄₂(𝑧) − 𝑃_1,2𝑧 −𝑃0,2) + 𝛾(𝑄0(𝑧) − 𝑃0,0) + 𝜇𝑏𝑃1,2+ 𝛽𝑃0,1

(1 − 𝑧)(𝜆𝑧 − 𝜇

_𝑏

)𝑄

₂

(𝑧) = 𝛾𝑧𝑄

₀

(𝑧) + 𝛽𝑧𝑄

₁

(𝑧) − 𝜇

_𝑏

(1 − 𝑧)𝑃

_0,2

− (𝜂𝑝

₀₂

+ 𝛾𝑝

_0,0

)𝑧 𝑄

₂

(𝑧) =

^{𝛽𝑧𝑄1(𝑧)+𝛾𝑧𝑄0(𝑧)−(𝜂𝑝}_{(𝜆𝑧−𝜇}^{02+𝛾𝑝0,0)𝑧−𝜇𝑏(1−𝑧)𝑝0,2}

𝑏)(1−𝑧)

(12) Adding (5) and (6) over all possible values of n, we get

𝛽𝑄

₁

(1) + 𝛾𝑄

₀

(1) = 𝜂𝑝

_0,2

+ 𝛾𝑝

_0,0

(13) Equation (12) can be written as,

𝑄

₂

(1) =

^{𝛽𝑄1′(1)+𝛾𝑄}_𝜇 ^0′(1)

𝑏−𝜆

+

^𝜇_𝜇^{𝑏𝑝0 2}

𝑏−𝜆

(14)

where, 𝑄

₀

′(1) = 𝐸[𝐿

₀

],𝑄

₂

(1) = 1 − 𝑄

₀

(1) − 𝑄

₁

(1) substituting in(13), we gives

1 − 𝑄0(1) − 𝑄1(1) = ^{𝛽𝐸[𝐿}_𝜇^{1]+𝛾𝐸[𝐿0]}

𝑏−𝜆 +^𝜇_𝜇^𝑏𝑝0,2

𝑏−𝜆

𝐸[𝐿₀] =^(𝜇𝑏_𝛾^−𝜆)(1 − 𝑄₀(1) − 𝑄₁(1)) −^𝛽_𝛾𝐸[𝐿₁] −^𝜇_𝛾^𝑏 𝑝_0,2 (15)

So that 𝐸[𝐿

₀

] can be obtained.Adding (2), (4), (6) and rearranging the terms, we get

𝜆𝑃𝑛,0+ 𝜆𝑃𝑛,1+ 𝜆𝑃𝑛,2− [(𝜇𝑣+ 𝑛𝛼)𝑃𝑛+1,0+ 𝜇𝑏𝑃𝑛+1,2] =

−[(𝜇_𝑣 + (𝑛 − 1)𝛼)𝑃_𝑛,0+ 𝜇_𝑏𝑃_𝑛,2], 𝑛 ≥ 1

𝜆𝑃_𝑛−1,0+ 𝜆𝑃_𝑛−1,1+ 𝜆𝑃_𝑛−1,2

(16) Using (16) in (2),(4),(6), we get

𝜆𝑝

_𝑛,0

+ 𝜆𝑝

_𝑛,1

+ 𝜆𝑝

_𝑛,2

= (𝜇

_𝑣

+ 𝑛𝛼)𝑝

_𝑛+1,0

+ 𝜇

_𝑏

𝑝

_𝑛+1,2

, 𝑛 ≥ 0 (17) Adding over all possible values n in (17), we get

𝜆𝑄₀(1) + 𝜆𝑄₁(1) + 𝜆𝑄₂(1) = 𝜇𝑏(𝑄2(1) − 𝑝0,2) + 𝜇𝑣(𝑄0(1) − 𝑝0,0) + 𝛼

∑^∞_𝑛=0(𝑛 + 1)𝑃_𝑛+1,0− 𝛼(𝑄₀(1) − 𝑝_0,0), 𝑛 ≥ 0

𝐸[𝐿0] = _,

0

( 1)

_n 1 0 ⁿ n

n p z and

𝑄2(1) = 1 − 𝑄0(1) − 𝑄1(1)

, 𝐸[𝐿

₀

] in (15) and we get

𝜆𝑄0(1) + 𝜆𝑄1(1) + 𝜆 − 𝜆𝑄0(1) − 𝜆𝑄1(1) = 𝜇𝑏− 𝜇𝑏𝑄0(1)

−𝜇_𝑏𝑄₁(1) + 𝜇_𝑣𝑄₀(1) − 𝜇_𝑣𝑃_{0 ,0}− 𝜇_𝑏𝑝_0,2− 𝛼𝑄₀(1) + 𝛼𝑃_0,0+ 𝛼𝐸[𝐿₀]

𝜆 = 𝜇𝑏− 𝜇𝑏𝑄0(1) − 𝜇𝑏𝑄1(1) − 𝜇𝑏𝑝0,2+ 𝜇𝑣𝑄0(1) − 𝜇𝑣𝑝0,0− 𝛼𝑄0(1) + 𝛼𝑝0,0

+^{𝛼(𝜇𝑏}_𝛾^−𝜆)(1 − 𝑄0(1) − 𝑄1(1)) −^𝛼𝛽_𝛾 𝐸[𝐿1] −^𝛼𝜇_𝛾^𝑏𝑝0,2

𝜆𝛾 − 𝛼(𝜇

_𝑏

− 𝜆) = ((𝜇

_𝑣

− 𝜇

_𝑏

)𝛾 − 𝛼(𝛾 + 𝜇

_𝑏

− 𝜆))𝑄

₀

(1) − (𝛼(𝜇

_𝑏

− 𝜆) + 𝜇

_𝑏

𝛾) 𝑄

₁

(1) − 𝛾(𝜇

_𝑣

− 𝛼)𝑝

_0,0

− 𝜇

_𝑏

(𝛾 + 𝛼)𝑝

_0,2

− 𝛼𝛽𝐸[𝐿

₁

] (18) For z=1, from equation (9), we get

𝑝

_0,0

=

_(𝜇^𝛾

𝑣−𝛼) 𝑘₂(1)

𝑘₁(1)

𝑄

₀

(1) (19)

(𝜆𝛾 − 𝛼(𝜇𝑏− 𝜆)) = ((𝜇𝑣− 𝜇𝑏)𝛾 − 𝛼(𝛾 + 𝜇𝑏− 𝜆))𝑄0(1) − (𝛼(𝜇𝑏− 𝜆) +𝜇𝑏𝛾)𝛾

𝛽𝑝0,0−𝛼

𝛽𝑝0,0𝜆𝛾 − 𝛾(𝜇𝑣− 𝛼)𝑝0,0− 𝜇𝑏(𝛾 + 𝛼)𝛾 𝜂𝑄0(1) 𝑄₀(1) = 𝜆𝛾 − 𝛼(𝜇_𝑏− 𝜆)(((𝜇_𝑣− 𝜇_𝑏)𝛾 − 𝛼(𝛾 + 𝜇_𝑏− 𝜆) − 𝜇_𝑏(𝛾 + 𝛼)^𝛾_𝜂)

−_(𝜇^𝛾

𝑣−𝛼) 𝑘₂(1)

𝑘₁(1)(^𝜆𝛾𝛼_𝛽 + 𝛾(𝜇𝑣− 𝛼) + (𝛼(𝜇𝑏− 𝜆) + 𝜇𝑏𝛾)^𝛾_𝛽))⁻¹)

(20) From equation (11), we get

𝐸[𝐿1] = 𝑄1′(1) =^𝛾𝜆_𝛽2𝑝0,0

𝑄1′′(1) =^𝛾𝜆_𝛽3²𝑝0,0

(21) We find the values 𝑝

_0,1

, 𝑝

_0,2

and 𝑝

_1,0

in the equation (3), (5) and (1) we get

𝑝0,1=_𝜆+𝛽^𝛾 𝑝0,0

(𝜆 + 𝜂)𝑝_0,2= 𝜇_𝑏𝑝_1,2+_𝜆+𝛽^𝛽𝛾 𝑝_0,0((𝜆 + 𝜂)^(𝜇𝑣_𝜂^−𝛼)𝑘_𝑘¹⁽¹⁾

2(1)−_𝜆+𝛽^𝛽𝛾 𝑝_0,0= 𝜇_𝑏𝑝_1,2) 𝑝1,2=_𝜇¹

𝑏((𝜆 + 𝜂)^(𝜇𝑣_𝜂^−𝛼)𝑘_𝑘¹⁽¹⁾

2(1)−_𝜆+𝛽^𝛽𝛾) 𝑝1,0=_𝜇¹

𝑣((𝜆 + 𝛾) − (𝜇𝑣− 𝛼)^𝑘_𝑘¹⁽¹⁾

2(1)𝑝0,0)

(22)

Differentiating (12) with respect to z and using L' Hospital's rule as follows 𝐸(𝐿

₂

) = 𝑄

₂

′(1) =

^[

(𝜇_𝑏−𝜆)(2𝛾𝑄₀′(1)+2𝛽𝑄₁′(1)+𝛾𝑄₀′′(1)+𝛽𝑄₁′′(1)) +2𝜆(𝛾𝑄₀′(1)+𝛽𝑄₁′(1)+𝜇_𝑏𝑃₀₂) ]

2(𝜇_𝑏−𝜆)²

(23) Differentiating equation (8) twice and putting z=1, we get

𝑄

₀

′′(1) =

^{𝜆−𝛾−2𝛼}

2𝛼

𝑄

₀

′(1) +

^𝜆

𝛾

𝑄

₀

(1) (24) Substituting (15), (21) and (24) in (23) , we get 𝐸[L(2)] The expected number of customers in the system can be computed as 𝐸[𝐿] = 𝐸[𝐿

₀

] + 𝐸[𝐿

₁

] + 𝐸[𝐿

₂

] Let W denote the total sojourn time of a customer in the system.

𝐸[𝑊] = 1

𝜆 (𝐸[𝐿

₀

] + 𝐸[𝐿

₁

] + 𝐸[𝐿

₂

])

3. STOCHASTIC DECOMPOSITION RESULTS

Theorem: If 𝜌 < 1, the stationary queue length Q can be decomposed into the sum of two independent random variables: 𝑄 = 𝑄

₀

+ 𝑄

_𝑑

, where 𝑄

₀

is the stationary queue length of a 𝑀/𝑀/1 queue without vacation and the additional stationary queue length 𝑄

_𝑑

.

𝑄

_𝑑

(𝑧) =

¹

1−𝜌

[ 𝑄

₀

(𝑧) (1 − 𝜌𝑧 −

_𝜇 ^𝛾𝑧

𝑏(1−𝑧)

) + 𝑄

₁

(𝑧) (1 − 𝜌𝑧 −

_𝜇 ^𝛽𝑧

𝑏(1−𝑧)

) +

^{𝑧(𝛾𝑄0(1)+𝛽𝑄1(1))}

𝜇_𝑏(1−𝑧)

+

^{𝜇𝑣−𝛼}

𝜂 𝑘₁(1)

𝑘₂(1)

𝑝

₀₀

] (25)

Proof: consider

=

^1−𝜌

1−𝜌𝑧

[

_𝜇^𝜇𝑏

𝑏−𝜆

( 𝑄

₀

(𝑧) (1 − 𝜌𝑧 −

^𝛾𝑧

𝜇_𝑏(1−𝑧)

) + 𝑄

₁

(𝑧) (1 − 𝜌𝑧 −

^𝛽𝑧

𝜇_𝑏(1−𝑧)

) +

^{𝑧(𝛾𝑄0}_𝜇^{(1)+𝛽𝑄1(1))}

𝑏(1−𝑧)

+

^𝜇𝑣_𝜂^−𝛼𝑘_𝑘¹⁽¹⁾

2(1)

𝑝

_0,0

)]

=

^1−𝜌

1−𝜌𝑧

𝑄

_𝑑

(𝑧)

(26)

E(Q)=

_1−𝜌^𝜌

+𝐸(𝑄

_𝑑

)

Theorem: If 𝜌 < 1, the stationary waiting time can be decomposed into the sum pf two independent random variables: 𝑊 = 𝑊

₀

+ 𝑊

_𝑑

, where 𝑊

₀

is the stationary waiting time of a 𝑀/𝑀/1 queue without vacations which has an exponential distribution with the parameter 𝜇

_𝑏

(1 − 𝑃) and the additional stationary waiting time 𝑊

_𝑑

due to the effect of working vacation and set up period with its Laplace Stieltjes transform(LST):

𝑊𝑑∗ (𝑠) =_(𝜇 ¹

𝑏−𝜆)𝑠[[(𝜇𝑏− 𝜆 + 𝑠)𝑠 − 𝛾(𝜆 − 𝑠)]𝑄0(1 −_𝜆^𝑠) + [(𝜇𝑏− 𝜆 + 𝑠)𝑠 − 𝛽(𝜆 − 𝑠)]

𝑄1(1 −^𝑠_𝜆) + (𝛾𝑄0(1) + 𝛽𝑄1(1))(𝜆 − 𝑠) +^𝜇𝑣_𝜂^−𝛼𝑘_𝑘¹⁽¹⁾

2(1)𝜇𝑏𝑠𝑝0,0]

(27) Proof: The relationship between the Probability generating function L and LST of waiting time is given by

𝑄(𝑧) = 𝑊 ∗ (𝜆(1 − 𝑧))

Assume that 𝑠 = 𝜆(1 − 𝑧); so 𝑧 = 1 −

^𝑠

𝜆

and 1 − 𝑧 =

^𝑠

𝜆

. Applying the relations in (25), we obtain the desired result.

𝑊_𝑑∗ (𝑠) = 1

1 − 𝜌[𝑄₀(1 −𝑠

𝜆) (1 − 𝜌(1 −𝑠

𝜆) −𝛾(1 −_𝜆^𝑠) 𝜇𝑏𝑠

𝜆

) + 𝑄₁(1 −𝑠

𝜆) (1 − 𝜌(1 −𝑠

𝜆) −𝛽(1 −_𝜆^𝑠) 𝜇𝑏𝑠

𝜆

)

+(𝛾𝑄₀(1) + 𝛽𝑄₁(1)) 𝜇𝑏𝜇_𝑏

𝜆

+𝜇_𝑣− 𝛼 𝜂

𝑘₁(1) 𝑘2(1)𝑝0,0]

Differentiating above equation w.r.to s, we get

𝑊𝑑= 1

(𝜇𝑏− 𝜆)[𝑄0(1 −𝑠

𝜆)((𝜇𝑏− 𝜆 + 𝑠) + 𝑠 + 𝛾) −1

𝜆𝑄0′(1 −𝑠

𝜆)((𝜇𝑏− 𝜆 + 𝑠)𝑠 − 𝛾(𝜆 − 𝑠) + 𝑄1(1 −𝑠 𝜆) ((𝜇𝑏− 𝜆 + 𝑠) + 𝑠 + 𝛽) −1

𝜆𝑄₁^′(1 −𝑠

𝜆) ((𝜇𝑏− 𝜆 + 𝑠)𝑠 − 𝛽(𝜆 − 𝑠))– (𝛾𝑄0(1) + 𝛽𝑄1(1)) +𝜇𝑣− 𝛼

𝜂

𝑘1(1) 𝑘2(1)𝜇𝑏𝑝0,0]

Put 𝑠 = 0 in above equation, we get

E(W) =_(𝜇¹

𝑏−𝜆)[𝑄0(1)(𝜇𝑏− 𝜆 + 𝛾) + 𝑄1(1)(𝜇𝑏− 𝜆 + 𝛾) − (𝛾𝑄0(1) + 𝛽𝑄1(1)) + 𝑄0′(1)𝛾 + 𝑄1′(1)𝛽 +^𝜇𝑣_𝜂^−𝛼𝑘_𝑘¹⁽¹⁾

2(1)𝜇𝑏𝑝0,0]

Substituting the value of 𝑄

₀

′(1), 𝑄

₁

′(1), we get

0

lim

s 𝑊_𝑑∗ (𝑠) = 1

. 𝑊

_𝑑

is a random variable of the additional waiting time.

4. NUMERICAL RESULTS

In this section, we obtain the mean system length and the mean waiting time of an arbitrary customer. In this section, we illustrate the results obtained above numerically and discuss the effect of system parameters on system performance indices. We assume that the service rate 𝜇

_𝑏

in a regular busy period equals 4.0, arrival rate 𝜇

_𝑣

equals 2.0, at the same time, we assume that working vacation is an exponential distributed random variable with mean 𝛾

=0.8, 𝛽 =1.0, 𝜂 =0.4. We plot the values of mean queue length E(Q) and mean waiting time E(W) by fixed the arrival rate 𝜆 and impatient rate 𝛼 increases.

Figure.1 The changing curve of E(Q) Figure.2 .The changing curve of E(W)

REFERENCES

1. Al-Seedy, R. O., El-Sherbiny, A.A., El-Shehawy, S.A. and Ammar, S.I. Transient solution of the M/M/c queue with balking and reneging. Computers and Mathematics with Applications, 57:1280-1285, (2009).

2. Baba, Y. Analysis of a GI/M/1 queue with multiple working vacations, Operations Research Letters, 33:201-2009 (2005).

3. Benjaafar, S., Gayon, J. and Tepe. S. Optimal control of a production-inventory system with customer impatience. Operations Research Letters, 38:267-272, (2010).

4. Bonald, T. and Roberts, J. Performance modeling of elastic trac in overload. ACM Sigmetrics Performance Evaluation Review, 29:342-343 (2001).

5. Boxma, O.J., Schlegel, S. and Yechiali, U. A note on an M/G/1 Queue with a waiting server timer and vacations, Americal Mathematical Society Translations. series 2, 207.

25-35 (2002).

6. Doshi, B. Queueing systems with vacations A survey. Queueing Systems,1(1), 29–66 (1986).

7. Ke, J., Wu, C. and Zhang, Z.G. Recent developments in vacation queueing models: A short survey, International Journal of Operations Research,7(4), 3–8 (2014).

8. Levy, Y. and Yechiali, U. Utilization of idle time in an M/G/1 queueing system.

Management Science, 22(2), 202-211 (1975).

9. Obert, E.R. Reneging phenomenon of single channel queues. Mathematics of Operations Research, 4:162-178 (1979).

10. Padmavathy, R., Kalidass, K., Kamanath, K. Vacation queues with impatient customers and a waiting server, Int. J. Latest Trends. Softw. Eng. 1,10-19 (2011).

11. Palm, C. Methods of judging the annoyance caused by congestion. Tele, 4:189-208 (1953).

12. Takagi, H. Queueing analysis: A foundation of performance evaluation (Vol. 1).

Amsterdam: Elsevier, (1991).

13. Tian, N. and Zhang, Z. Vacation queueing models theory and applications (Vol. 1). New York: Springer,Verlag (2006).

14. Yue, D., Yue, W. and Xu, G. Analysis of customers impatience in an M/M/1 queue with

working vacations, J.Ind. Manag. Optim. 8, 895-908 (2012).