Analysis of an M/G/1 Stochastic Clearing Queue in a 3-Phase Environment

Xiaoyan Zhang; Liwei Liu; Tao Jiang

doi:10.1515/JSSI-2015-0374

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Analysis of an M/G/1 Stochastic Clearing Queue in a 3-Phase Environment

Xiaoyan Zhang , Liwei Liu and Tao Jiang

Published/Copyright: August 25, 2015

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From the journal Journal of Systems Science and Information Volume 3 Issue 4

Abstract

This paper studies a single server M/G/1 stochastic clearing queue operating in a 3-phase environment, where the time length of the first and third phase are assumed to follow exponential distributions, and the time length of the second phase is a constant value. At the completion of phase 1, the system moves to phase 2, and after a fixed time length, the system turns to phase 3. At the end of phase 3, all present customers in the system are forced to leave the system, then the system moves to phase 1 and restarts a new service cycle. Using the supplementary variable technique, we obtain the distribution for the stationary queue at an arbitrary epoch. We also derive the sojourn time distribution and the length of the server’s working time in a cycle.

Keywords: M/G/1 queue; fixed time length; supplementary variable technique; sojourn time distribution

1 Introduction

In most of our queueing models, the capacity of the facility is assumed to be largely enough. But in the life situation, things are not the same, the capacity of the queue is finite in general, so the probability that some waiting customers cannot be accommodated is negligibly small. Besides, the waiting customers that can’t be served are not always willing to retry for its next service, instead, they usually abandon the system. So, it is realistic to consider that all present customers are removed at the visit points of the facility. Such systems are referred to as stochastic clearing systems.

Stochastic clearing systems have important applications in health center and banks. Since the introduction of this concept, there are amount of papers were studied. Some classical papers can be referred to [1-4]. Furthermore, Kyriakidis and Dimitrakos^[5] considered a Markov decision model and derived the optimal control of a compound immigration process. In [6], Boudali and Economou considered a Markovan queue from an economic viewpoint, where the arrivals stop when the system is under repair. Later, they^[7] extended the model to that the new arrivals continue to arrive when the system is in the repair period. Yechiali^[8] considered an M/M/c queue, and obtained some performance measures.

However, queues with deterministic time length is also an interesting aspect. Such system has wide application in analyzing the performance of various real life traffic situations as well as industrial queues. Jeet Singh^[9] studied an M/G/1 queueing model with a vacation of fixed duration, using the probability argument, they gained the probability generating function of the units present in the system and some other performance measures. Madan[¹⁰] discussed the steady state behavior of an arbitrary service time queue with deterministic server vacation. In his paper, what he has considered is that the customers arrive at the system with uniform arrival rates. Later, he^[11] considered an M/G/1 queuing model with time homogeneous and deterministic repair times.

The model in consideration can be used in real life situations, such as the application in bank account. In the bank, after a period of service time before noon, the bank will close for a deterministic time, then continue to serve in the afternoon. This is the same as the model we studied in this paper. And to the best of our knowledge, this model hasn’t been studied before. In this paper, we give the stationary queue size distribution at an arbitrary epoch. Following the idea presented by Jiang et al.^[12], which provided an approach to analyze the sojourn time distribution and the length of the server’s working time in a cycle, we also derive these performance measures in our paper.

The rest of this paper is organized as follows: Section 2 is the model description. Using supplementary variable technique, Section 3 and Section 4 gives the stationary queue size distribution at an arbitrary epoch. Some performance measures are presented in Section 5. At last, Section 6 is the conclusion.

2 Model Description

Consider a single server M/G/1 queue operating in a 3-phase environment. Under environment i, i = 1, 2, 3, the Poisson arrival rate is λ_i, and the service times in phase 1 and 3 follow a general (arbitrary) distribution with mean 1μ1,1μ3, the distribution function B₁(ν), B₃(ν), and Laplace Stieltjes transform B1∗(s),B3∗(s). We also assume that the system resides in phase 1 and 3 follow an exponential distribution with parameter θ₁and θ₃, respectively. At the completion epoch of phase 1, the system moves to phase 2, and if there is arrival occurs when the system resides in phase 2, the new arriving customer enters the system with probability p or leave the system with q = 1 – p. and all the customers in phase 2 undergoes a fixed duration d without receiving service. After phase 2, the system enters into phase 3. As soon as the completion of phase 3, the present customers are forced to leave the system without accepting service, then the system moves to phase 1 and restarts a new service cycle.

3 Stationary Queue Size Distribution

In this section, we use the method of supplementary variable technique to derive the stationary queue size distribution at an arbitrary epoch. It is worth notice that as long as θ₃ > 0, the system in consideration is stable. So the system can be analyzed in steady state.

Let µ_i(x)dx be the conditional probability (hazard rate) of completion of a service during the interval (x, x + dx) with elapsed time x in phase 1 and 3, respectively. So we have

μi(x)dx=dBi(x)1−Bi(x),Bi(v)=1−exp⁡{−∫0vμi(x)dx},

where i = 1,3.

At an arbitrary time t, the system can be described by {L(t), J(t), B^–(t)}, where L(t) denotes the number of customers in the system at time t, J(t) denotes the phase in which the system operates at time t, and if L(t) ≥ 1 and J(t) = i, B−(t)=Bi−(t) represents the elapsed service time of the customer currently being served in operative phase i, i = 1,3. Then {L(t), J(t), B^–(t)} is a Markov chain with state space

{((m,2),m≥0)}∪{(0,i),(k,i,x),k≥1,x≥0,i=1,3}.

Next, we introduce the following definitions.

Let P_i,n(x, t)dx, i = 1, 3, n ≥ 1 be the probability that there are n customers in the system (including the one being served) at time t when the server is busy in serving the customer with elapsed service time lying between x and x + dx in phase i. Let Q_i_,0(t), i = 1,3 be the probability that there is no customer in phase i and the server is idle at time t, V_n(t), n ≥ 0 be the probability that there are n customers in the system and the system in phase 2. Last, let

Nr=e−λ2pd(λ2pd)rr!

be the probability of r, r ≥ 0 arrivals during phase 2.

In steady state, we define the following limiting probabilities

Pi,n(x)dx=limt→∞⁡Pi,n(x,t)dx,Qi,0=limt→∞⁡Qi,0(t),Vn=limt→∞⁡Vn(t),i=1,3,n≥0.

Using the probabilities above and by the approach of supplementary variable technique, we can easily get the steady state equations for the system:

ddxPi,n(x)+[λi+θi+μi(x)]Pi,n(x)=λi(1−δn,1)Pi,n−1(x),i=1,3,n≥1(1)

(λ1+θ1)Q1,0=θ3∑n=1∞∫0∞P3,n(x)dx+θ3Q3,0+∫0∞P1,1(x)μ1(x)dx(2)

(λ3+θ3)Q3,0=V0N0+∫0∞P3,1(x)μ3(x)dx(3)

Vn=θ1∫0∞P1,n(x)dx,n≥1,V0=θ1Q1,0(4)

P1,n(0)=λ1δn,1Q1,0+∫0∞P1,n+1(x)μ1(x)dx,n≥1(5)

P3,n(0)=λ3δn,1Q3,0+∫0∞P3,n+1(x)μ3(x)dx+V0Nn+V1Nn−1+⋯+VnN0,n≥1(6)

4 The Steady State Solution

First, we define the following equations:

Pi(x,z)=∑n=1∞Pi,n(x)zn,V(z)=∑n=0∞Vnzn,Pi(z)=∑n=1∞Pi,nzn,Pi,n=∫0∞Pi,n(x)dx,i=1,3.

Solving Equation (1), we have

Pi(x,z)=Pi(0,z)exp⁡{−[θi+λi(1−z)]x}[1−Bi(x)],i=1,3(7)

According to Equation (4), we derive

V(z)=θ1∑n=1∞P1,nzn+V0=θ1P1(z)+θ1Q1,0(8)

Multiplying Equation (5) by zⁿ and summation of n ≥ 1, we have

zP1(0,z)=z2λ1Q1,0+∫0∞P1(x,z)μ1(x)dx−z∫0∞P1,1(x)μ1(x)dx(9)

From (7) and (9), we obtain

[z−B1∗(θ1+λ1(1−z))]P1(0,z)=z2λ1Q1,0−z∫0∞P1,1(x)μ1(x)dx(10)

Lemma 1

The equation z=Bi∗(θi+λi(1−z)) has a unique root in |z| < 1.

Proof First, we define f(z)=z,g(z)=Bi∗(θi+λi(1−z)), for sufficiently small ε > 0, consider all z with |z| = 1 + ε, we have

|g(z)|=|Bi∗(θi+λi(1−z))|≤1<1+ε=|z|=|f(z)|.

Because both f(z) and g(z) are analytic for |z| = 1 + ε. Hence, by Rouche’s theorem, f(z) and f(z) – g(z) have the same number of roots in |z| < 1+ ε. Therefore, as ε tends to zero, it yields that f(z) – g(z) has only one root in |z| < 1, i.e., The equation z=Bi∗(θi+λi(1−z)) has a unique root in |z| < 1. By the result of Lemma 1, we have

γ2λ1Q1,0−γ∫0∞P1,1(x)μ1(x)dx=0,

where γ is the unique root of z=B1∗(θ1+λ1(1−z)). So the expression of P₁(0, z) can be written as

P1(0,z)=λ1Q1,0z(z−γ)z−B1∗(θ1+λ1(1−z))(11)

zP3(0,z)=∫0∞P3(x,z)μ3(x)dx−z∫0∞P3,1(x)μ3(x)dx+z2λ3Q3,0+∑n=1∞zn+1(V0Nn+V1Nn−1+⋯+VnN0)(12)

where

∑n=1∞zn+1(V0Nn+V1Nn−1+⋯+VnN0)=z[V(z)e−λ2pd(1−z)−V0N0].

The equation (12) can be simplified as

zP3(0,z)=∫0∞P3(x,z)μ3(x)dx−z∫0∞P3,1(x)μ3(x)dx+z2λ3Q3,0+z[V(z)e−λ2pd(1−z)−V0N0](13)

From (3), (7) and (13), we have

[z−B3∗(θ3+λ3(1−z))]P3(0,z)=z2λ3Q3,0+zV(z)e−λ2pd(1−z)−z(λ3+θ3)Q3,0(14)

By the result of Lemma 1, we also have

Q3,0=V(α)e−λ2pd(1−α)λ3(1−α)+θ3(15)

where α is the unique root of z=B3∗(θ3+λ3(1−z)). So the expression of P₃(0, z) can be written as

P3(0,z)=z[V(z)e−λ2pd(1−z)−(λ3(1−z)+θ3)Q3,0]z−B3∗(θ3+λ3(1−z))(16)

Then the expression of P₁(z) and P₃(z) can be obtained by

Pi(z)=∫0∞Pi(x,z)dx=Pi(0,z)1−Bi∗(θi+λi(1−z))θi+λi(1−z),i=1,3,

that is,

P1(z)=[λ1Q1,0z(z−γ)][1−B1∗(θ1+λ1(1−z))][z−B1∗(θ1+λ1(1−z))][θ1+λ1(1−z)](17)

P3(z)=z[V(z)e−λ2pd(1−z)−(λ3(1−z)+θ3)Q3,0][1−B3∗(θ3+λ3(1−z))][z−B3∗(θ3+λ3(1−z))][θ3+λ3(1−z)](18)

With the help of the normalizing condition

Q1,0+Q3,0+P1(1)+P3(1)+V(1)=1(19)

and

P1(1)=limz→1⁡P1(z)=limz→1⁡[λ1Q1,0z(z−γ)][1−B1∗(θ1+λ1(1−z))][z−B1∗(θ1+λ1(1−z))][θ1+λ1(1−z)]=λ1Q1,0(1−γ)θ1,P3(1)=limz→1⁡P3(z)=V(1)−θ3Q3,0θ3,V(1)=θ1P1(1)+θ1Q1,0=λ1Q1,0(1−γ)+θ1Q1,0.

We gain

Q1,0=θ1θ3θ1θ3+λ1(1−γ)(θ1+θ3)+θ12+θ1θ3λ1(1−γ)+θ12θ3.

Once P₁(z), P₃(z), V (z), Q_1,0 and Q_3,0 are obtained, the PGF of the stationary queue size distribution can be derived by

P(z)=P1(z)+P3(z)+V(z)+Q1,0+Q3,0.

5 Performance Measures

In this section, we will give some important performance measures of the system. Firstly, let L be the stationary random variable for the number of customers in the system. Then the expected number of customers in the system is

E[L]=limz→1⁡P′(z)=P1′(1)+P3′(1)+V′(1),

where

P1′(1)=λ1Q1,0θ1(2−γ)(1−B1∗(θ1))+λ1Q1,0(1−γ)[λ1(1−B1∗(θ1))−θ1]θ12(1−B1∗(θ1)),V′(1)=θ1P1′(1),P3′(1)=[V(1)−θ3Q3,0](θ3+λ3)+θ3[V′(1)+λ2pdV(1)+λ3Q3,0]θ32−[V(1)−θ3Q3,0](1−B3∗(θ3))θ3.

Secondly, we will give the working time of the server in a cycle. We consider two types of cases:

Case 1: No customer arrival occurs in phase 1.

Case 2: Customers’ arrival occur in phase 1.

We first define the following random variables. Let A_i, i = 1, 3 be the inter-arrival times of the customer in phase i, D_i, i = 1, 3 the duration times of phase i, R_k, k = 1, 2 denote the length of working times of the server in case k. We also let p_k, k = 1, 2 denote the probability of case i. Then we have

p1=P(A1>D1)=θ1λ1+θ1,p2=P(A1<D1)=λ1λ1+θ1.

First, we give a lemma as follows.

Lemma 2

(see Lemma 2 in [12]) Let X, Y, Z be the random variables, and Z = min(X,Y), where X follows a general (arbitrary) distribution with mean 1μ and distribution function X(ν) and Laplace Stieltjes transform X*(s), Y follows an exponential distribution with parameter η, then we have

E[e−sZ]=E[e−smin(X,Y)]=η+sX∗(s+η)s+η.

In case 1, we consider two subcases: No customer arrives in phase 2 and customers’ arrival occur in phase 2. Let F denote the length of working times of the server in phase 3 under the condition that no customer arrives in phase 2, F_r the length of the working times in phase 3 under the condition that customers’ arrival occur in phase 2, and B_i,k denote the length of busy period which start by k customers. We first consider the case no customer arrives in phase 2:

F={0,A3>D3,T,A3<D3,

where

T={T1,B3,1>D3,T2+F,B3,1<D3,

T₁ = (D₃|D₃ < B_3,1), T₂ = (B_3,1|D₃ > B_3,1). Then T1∗(s)andT2∗(s) are obtained as follows:

T1∗(s)=E[e−sD3|D3<B3,1]=1−B3,1∗(s+θ3)1−B3,1∗(θ3)θ3s+θ3,T2∗(s)=E[e−sB3,1|D3>B3,1]=B3,1∗(s+θ3)B3,1∗(θ3).

Then

T∗(s)=P(B3,1>D3)T1∗(s)+P(B3,1<D3)T2∗(s)F∗(s),

and

F∗(s)=P(A3>D3)+P(A3<D3)T∗(s),

that is,

F∗(s)=θ3λ3+θ3+λ3λ3+θ3{[1−B3,1∗(s+θ3)]θ3s+θ3+B3,1∗(s+θ3)F∗(s)}.

Simplifying the above equation, we have

F∗(s)=θ3(s+θ3)+λ3θ3[1−B3,1∗(s+θ3)][(λ3+θ3)−λ3B3,1∗(s+θ3)](s+θ3),

where Bi,1∗(s) satisfies Bi,1∗(s)=Bi∗(s+λi−λiBi,1∗(s)),i = 1,3.

Next, we consider the case customers’ arrival occur in phase 2. Let F_r denote the length of working times of the server in phase 3 under the condition that customers’ arrival occur in phase 2.

Fr={Y1,D3<B3,r,Y2+F,D3>B3,r,

where Y₁ = (D₃|D₃ < B_3,r), Y₂ = (B_3,r|D₃ > B_3,r). Then,

Fr∗(s)=P(B3,r>D3)Y1∗(s)+P(B3,r<D3)Y2∗(s)F∗(s),

that is,

Fr∗(s)=[1−B3,r∗(s+θ3)]θ3s+θ3+B3,r∗(s+θ3)F∗(s),

where Bi,k∗(s)=[Bi,1∗(s)]k,i = 1, 3.

So in case 1,

R1∗(s)=K0F∗(s)+∑r=1∞KrFr∗(s).

In case 2, we also consider two subcases, one of which is the busy period ends before the completion of phase 1, and the other subcase is that phase 1 completed before the busy period ends. Let R_2,1 denote the working times under the condition that the busy period ends before the completion of phase 1, R_2,2 denote the working times under the condition that phase 1 completed before the busy period ends, and R the total working times of server in a cycle. Let K denote the number of existing customers at the end of the phase 1, with the result of Kim et al.^[14], K(z), the PGF of K, is given by

K(z)=θ1zθ1+λ1−λ1zz−B1,1∗(θ1)1−B1,1∗(θ1)B1∗(θ1+λ1−λ1z)−1B1∗(θ1+λ1−λ1z)−z.

Hence, we obtain

B1,K∗(s)=K(z)|z=B1,1∗(s)=θ1B1,1∗(s)θ1+λ1−λ1B1,1∗(s)B1,1∗(s)−B1,1∗(θ1)1−B1,1∗(θ1)B1∗(θ1+λ1−λ1z)−1B1∗(θ1+λ1−λ1z)−B1,1∗(s),R2∗(s)=B1,1∗(s+θ1)R∗(s)+[1−B1,1∗(s+θ1)]θ1s+θ1[K0FK∗(s)+∑r=1∞KrFK+r∗(s)],

where B3,K+r∗(s)=B3,K∗(s)[B3,1∗(s)]r. Then, the LST of total working times is obtained

R∗(s)=θ1λ1+θ1R1∗(s)+λ1λ1+θ1R2∗(s).

Finally, considering a tagged customer, we will derive the LST of the stationary sojourn time distribution of an arbitrary customer, where the sojourn time is the period from the time he enters the system to the time he leaves by either the service completion or the end of the thrid phase. Denote W and W^∗ (s) be the stationary sojourn time and its LST. In order to obtain W^∗ (s), we consider the following cases:

Case 1: The tagged customer arrives in state (0, 1).

Case 2: The tagged customer arrives in state (n, 1), n ≥ 1.

Case 3: The tagged customer enters the system in state (0, 2).

Case 4: The tagged customer enters the system in state (n, 2), n ≥ 1.

Case 5: The tagged customer arrives in state (0, 3).

Case 6: The tagged customer arrives in state (n, 3), n ≥ 1.

Let T_i be the random variable of the duration of time when the system resides in phase i (i = 1, 3), and let B_h_,i, h ≥ 1 be the service time of the customer at position h when the tagged customer arrives in phase i(i = 1,2,3). We also assume that when the customer doesn’t finish his service in phase 1, he will restart his service in phase 3, instead of continuing the service from the stopping point.

In case 1, on the tagged customer’s arrival, he receives the service immediately and leaves the system by either the service completion or the end of phase 3. Let W_0,1 be the sojourn time in this case. Define

W0,1∗(s)=E[e−sW0,1].

Using lemma 2, we have

W0,1∗(s)=E[e−sW0,1]=P(B1,1<T1)E[e−sB1,1|B1,1<T1]+P(B1,1>T1)E[e−sT1|B1,1>T1]e−sdE[e−smin(B1,1,T3)]=B1∗(S+θ1)+θ1e−sd(s+θ1)(s+θ3)(1−B1∗(s+θ1))(θ3+sB3∗(s+θ3))(20)

In case 2, let W_n,1 be the sojourn time, and let S_n_,i, i = 1, 2, 3 denote the unfinished work after the arrival epoch of the tagged customer, then

Sn,1=B1,1,n(+)+B2,1+⋯+Bn+1,1,

where B1,1,n(+) means the remaining service time of the customer being served at the arrival instant of a customer in phase 1, given that he finds n customers present in the system. Hence,

we have

Sn,1∗(s)=E[e−sSn,1]=[B1∗(s)]nB1,1,n(+)∗(s).

According to Corollary 2.2.1 in the paper [13] of Kerner, which gives a method to compute the LST of the remaining service time at the arrival instant of a customer, given that he finds n customers present in the system. B1,i,n(+)∗(s) can be obtained by the following recursive equations:

B1,i,n(+)∗(s)=λis−λi(Bi∗(λi)1−B1,i,n−1(+)∗(s)1−B1,i,n−1(+)∗(λi)−Bi∗(s)),i=1,3,n≥2,

with the initial condition

B1,i,1(+)∗(s)=λiλi−sBi∗(s)−Bi∗(λi)1−Bi∗(λi).

Define

Wn,1∗(s)=E[e−sWn,1],

then Wn,1∗(s) can be gained by

Wn,1∗(s)=E[e−sWn,1]=P(Sn+1,1<T1)E[e−sSn+1,1|Sn+1,1<T1]+P(Sn+1,1>T1)E[e−sT1|Sn+1,1>T1]e−sd∑k=1n+1E[e−smin(Bk,3,T3)]=[B1∗(s+θ1)]nB1,1,n(+)∗(s+θ1)+θ1s+θ1(1−[B1∗(s+θ1)]nB1,1,n(+)∗(s+θ1))e−sd×∑k=1n+1θ3+s[B3∗(s+θ3)]k+1s+θ3=[B1∗(s+θ1)]nB1,1,n(+)∗(s+θ1)+θ1e−sd(s+θ1)(s+θ3)(1−[B1∗(s+θ1)]nB1,1,n(+)∗(s+θ1))×(θ3+s[B3∗(s+θ3)]2−[B3∗(s+θ3)]n+21−B3∗(s+θ3))(21)

In case 3, let W_0,2 be the sojourn time. Define

W0,2∗(s)=E[e−sW0,2],

then

W0,2∗(s)=E[e−sW0,2]=e−sdE[esmin(B1,2,T3)]=e−sdθ3+sB3∗(s+θ3)s+θ3(22)

In case 4, let W_n_,2 be the sojourn time. Define

Wn,2∗(s)=E[e−sWn,2],

then

Wn,2∗(s)=E[e−sWn,2]=e−sdE[esmin(Bn+1,2,T3)]=e−sdθ3+s[B3∗(s+θ3)]n+1s+θ3(23)

We can similarly compute the sojourn time in case 5, that is,

W0,3∗(s)=E[e−sW0,3]=E[e−smin(B1,3,T3)]=θ3+sB3∗(s+θ3)s+θ3(24)

In case 6, let W_n_,3 be the sojourn time. Then similarly to case 2, we have that

Sn,3∗(s)=E[e−sSn,3]=[B3∗(s)]nB1,3,n(+)∗(s).

Define

Wn,3∗(s)=E[e−sWn,3],

using Lemma 2, we have that

Wn,3∗(s)=E[e−sWn,3]=E[e−smin(Sn+1,3,T3)]=θ3+s[B3∗(s+θ3)]nB1,3,n(+)∗(s+θ3)s+θ3(25)

Then the LST of the stationary sojourn time distribution of an arbitrary customer can be written as

W∗(s)=∑i=13P0,iW0,i∗(s)+∑i=13∑n=1∞Pn,iWn,i∗(s).

The mean sojourn time of an arbitrary customer is gained by

E[W]=−dW∗(s)ds|s=0.

6 Conclusion

This paper studied a single server M/G/1 queue operating in a 3-phase environment, where the first and third phase is working phase, and the second phase is a determinstic-time phase without service. Using supplementary variable technique, we derived the probability generating function of the stationary queue size at an arbitrary epoch. The mean queue length and the server’s working time in a cycle are also obtained. Furthermore, we derived the LST of the stationary sojourn time distribution of an arbitrary customer.

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Received: 2015-5-31

Accepted: 2015-6-29

Published Online: 2015-8-25

Articles in the same Issue

https://doi.org/10.1515/JSSI-2015-0374

Keywords for this article

M/G/1 queue; fixed time length; supplementary variable technique; sojourn time distribution