The Well posedness and Regularity of a Batch Arrival Poisson Queue with Generalized Vacation

2019 International Conference on Artificial Intelligence, Control and Automation Engineering (AICAE 2019) ISBN: 978-1-60595-643-5

The Well-posedness and Regularity of a Batch Arrival Poisson

Queue with Generalized Vacation

Xiao-hua WANG

College of Humanities, Shaoxing University, Zhejiang Shaoxing, 312000, P. R. China

*Corresponding author

Keywords: Well-posedness, Regularity, Poisson queue, 𝐶₀-semigroup, Dissipative operator, Dispersive operator.

Abstract. In this paper the solution of a batch arrival Poisson queue with generalized vacation is investigated. By using the method of functional analysis, especially, the linear operator theory and the 𝐶0 semigroup theory on Banach space, we prove the well-posedness of the system, and show the

existence of positive solution.

The Well-posedness of Solution

The system of differential equations associated with this model is the following([1]):

{

𝑑𝑄(𝑡)

𝑑𝑡 = −𝜆𝑄(𝑡) + 𝜇𝑅0(𝑡) + ∫ 𝑣 +∞

0 (𝑥)𝑃0(𝑥, 𝑡)𝑑𝑥, 𝑑𝑅0(𝑡)

𝑑𝑡 = −(𝜆 + 𝜇)𝑅0(𝑡) + 𝜇𝑅1(𝑡) + ∫ 𝑣 +∞

0 (𝑥)𝑃1(𝑥, 𝑡)𝑑𝑥, 𝑑𝑅𝑛(𝑡)

𝑑𝑡 = −(𝜆 + 𝜇)𝑅𝑛(𝑡) + 𝜇𝑅𝑛+1(𝑡) + 𝜆 ∑ 𝑎𝑖 𝑛

𝑖=1 𝑅𝑛−𝑖(𝑡) + ∫₀+∞𝑣(𝑥)𝑃𝑛+1(𝑥, 𝑡)𝑑𝑥, 𝑛 ≥ 1, 𝜕𝑃0(𝑥,𝑡)

𝜕𝑡 +

𝜕𝑃0(𝑥,𝑡)

𝜕𝑥 = −[𝜆 + 𝑣(𝑥)]𝑃0(𝑥, 𝑡), 𝜕𝑃𝑛(𝑥,𝑡)

𝜕𝑡 +

𝜕𝑃𝑛(𝑥,𝑡)

𝜕𝑥 = −[𝜆 + 𝑣(𝑥)]𝑃𝑛(𝑥, 𝑡) + 𝜆 ∑ 𝑎𝑖 𝑛

𝑖=1 𝑃𝑛−𝑖(𝑥, 𝑡), 𝑛 ≥ 1.

(1)

where the following notations are defined:

𝜆: group arrival rate; 𝜇: service rate; 𝑋: arrival size random variable; 𝑎_𝑘: 𝑃𝑟𝑜𝑏(𝑋 = 𝑘); 𝑉: vacation time random variable; 𝑉(𝑥): the distribution function of 𝑉.

The above set of equations is to be solved under the following boundary conditions:

𝑃0(0, 𝑡) = 𝜆𝑄(𝑡), 𝑃𝑛(0, 𝑡) = 0, 𝑛 ≥ 1. (2)

Equations (1)(2) should be solved together with the normalizing condition

𝑄(𝑡) + ∑ 𝑅𝑛 ∞

𝑛=0

(𝑡) + ∑ ∫ 𝑃𝑛 +∞

0 ∞

𝑛=0

(𝑥, 𝑡)𝑑𝑥 = 1,

and an initial conditions 𝑄(0) = 1, 𝑅_𝑛(0) = 0, 𝑃_𝑛(𝑥, 0) = 0, 𝑛 ≥ 0.

𝒜1

( 𝑄 𝑅₀ 𝑅𝑛

𝑃0(𝑥)

𝑃_𝑛(𝑥)) =

(

−𝜆𝑄 + 𝜇𝑅0+ ∫ 𝑣 +∞

0

(𝑥)𝑃0(𝑥)𝑑𝑥

−(𝜆 + 𝜇)𝑅₀+ ∫ 𝑣+∞

0

(𝑥)𝑃₁(𝑥)𝑑𝑥

−(𝜆 + 𝜇)𝑅_𝑛+ ∫+∞𝑣

0

(𝑥)𝑃_𝑛+1(𝑥)𝑑𝑥

−𝑃0′(𝑥) − [𝜆 + 𝑣(𝑥)]𝑃0(𝑥)

−𝑃_𝑛′(𝑥) − [𝜆 + 𝑣(𝑥)]𝑃_𝑛(𝑥), 𝑛 ≥ 1 )

ℬ

( 𝑄 𝑅₀ 𝑅𝑛

𝑃0(𝑥)

𝑃_𝑛(𝑥)) =

( 0 𝜇𝑅₁

𝜇𝑅𝑛+1 + 𝜆 ∑ 𝑎𝑖 𝑛

𝑖=1

𝑅𝑛−𝑖, 𝑛 ≥ 1

0

𝜆 ∑ 𝑎𝑖 𝑛

𝑖=1

𝑃𝑛−𝑖(𝑥), 𝑛 ≥ 1

) ,

with the domain 𝐷(ℬ) = 𝑋, 𝐷(𝒜) = 𝐷(𝒜1), and

𝐷(𝒜) = {(𝑄, 𝑅_𝑛, 𝑃_𝑛(𝑥)) ∈ 𝑋: 𝑃_𝑛′(𝑥) ∈ 𝐿1_(ℝ+_{), 𝑛 ≥ 0; 𝑃}

0(0) = 𝜆𝑄, 𝑃𝑛(0) = 0, 𝑛 ≥ 1. }

Then the equation system (1.1)(1.2) can be rewritten as an abstract Cauchy problem on 𝑋:

{𝑑𝑃(𝑡)_𝑑𝑡 = 𝒜𝑃(𝑡), 𝑡 > 0, 𝑃(0) = 𝑃̃0.𝑃(𝑡) = (𝑄, 𝑅𝑛, 𝑃𝑛(𝑥)), 𝑃̃0 = (1,0,0, ⋯ ). (3)

Theorem 1.1𝒜₁ is a linear closed and densely defined operator on 𝑋. Proof of Theorem 1.1 is a direct verification, so we omit the details.

Let 𝑋∗ be the dual space of 𝑋, and 𝒜₁∗ be the dual operator of 𝒜₁, then 𝑋∗ = ℝ × 𝑙∞(ℝ) × 𝐿∞_(ℝ+_{× ℕ). From (𝒜}

1𝑃, 𝑊) = (𝑃, 𝒜1∗𝑊), we can obtain

𝒜₁∗

( 𝑞 𝑟0

𝑟𝑛

𝑝₀(𝑥)

𝑝_𝑛(𝑥)) =

(

𝜆[𝑝₀(0) − 𝑞] 𝜇𝑞 − (𝜆 + 𝜇)𝑟₀ −(𝜆 + 𝜇)𝑟𝑛, 𝑛 ≥ 1

𝑝₀′(𝑥) − [𝜆 + 𝑣(𝑥)]𝑝₀(𝑥) +𝑞𝑣(𝑥)

𝑝_𝑛′(𝑥) − [𝜆 + 𝑣(𝑥)]𝑝_𝑛(𝑥) +𝑟𝑛−1𝑣(𝑥), 𝑛 ≥ 1 )

,

with the domain 𝐷(𝒜₁∗) = {(𝑞, 𝑟_𝑛, 𝑝_𝑛(𝑥)) ∈ 𝑋∗: 𝑝_𝑛′(𝑥), 𝑣(𝑥)𝑝_𝑛(𝑥) ∈ 𝐿∞(𝑅+), 𝑛 ≥ 0. }

Theorem 1.2 1 is not a eigenvalue of 𝒜₁∗.

Proof Let 𝑊 = (𝑞, 𝑟_𝑛, 𝑝_𝑛(𝑥)) ∈ 𝑋∗, 𝒜₁∗𝑊 = 𝑊, i.e.,

0 0 0

[p (0) q] q, q ( )r r, ( )r_n r n_n, 1

             (4)

0'( ) [ ( )] 0( ) ( ) 0( )

p x   v x p x qv x  p x (5)

1

'( ) [ ( )] ( ) ( ) ( ), 1

n n n n

𝑝0(𝑥) = 𝑒∫ [

𝑥

0 1+𝜆+𝑣(𝑠)]𝑑𝑠{𝑝₀(0) − ∫ 𝑞

𝑥

0

𝑣(𝑢)𝑒− ∫ [0𝑢1+𝜆+𝑣(𝑠)]𝑑𝑠𝑑𝑢},

𝑝_𝑛(𝑥) = 𝑒∫ [₀𝑥1+𝜆+𝑣(𝑠)]𝑑𝑠_{𝑝

𝑛(0) − ∫ 𝑟𝑛−1 𝑥

0

𝑣(𝑢)𝑒− ∫ [₀𝑢1+𝜆+𝑣(𝑠)]𝑑𝑠_{𝑑𝑢}, 𝑛 ≥ 1.}

Let 𝑥 → +∞ , observing 𝑝₀(𝑥), 𝑝_𝑛(𝑥) ∈ 𝐿∞, we get

𝑝0(0) = ∫ 𝑞 +∞

0

𝑣(𝑢)𝑒− ∫ [0𝑢1+𝜆+𝑣(𝑠)]𝑑𝑠𝑑𝑢, 𝑝_𝑛(0) = ∫ 𝑟_𝑛−1

+∞

0

𝑣(𝑢)𝑒− ∫ [0𝑢1+𝜆+𝑣(𝑠)]𝑑𝑠𝑑𝑢, 𝑛 ≥ 1.

Let ∫₀+∞𝑣(𝑢)𝑒− ∫ [

𝑢

0 1+𝜆+𝑣(𝑠)]𝑑𝑠𝑑𝑢 = 𝑎, _then 0 < 𝑎 < 1, _{and we obtain} 𝑝₀(0) = 𝑎𝑞, 𝑝_𝑛(0) =

𝑎𝑟𝑛−1, 𝑛 ≥ 1. It is combined with (4), we get 𝑞 = 0. Furthermore, we also get 𝑟𝑛 = 0, 𝑛 ≥ 0,

therefore, 𝑝𝑛(0) = 0, 𝑛 ≥ 0. Thus we get 𝑝𝑛(𝑥) ≡ 0, 𝑛 ≥ 0. Hence 𝑊 = 0 and 1 is not a

eigenvalue of 𝒜₁∗.

We can use the Lumer-Phillips Theorem(Ref.[2]) to show the well-posedness of the system (3).

Theorem 1.3. (1)𝒜 is a dissipative operator on 𝑋.

(2) The operator 𝒜1 generates a 𝐶0 semigroup of contraction.

Proof Firstly, 𝒜 is a dissipative operator on 𝑋. In fact, for any 𝑃 = (𝑄, 𝑅𝑛, 𝑃𝑛(𝑥)) ∈ 𝐷(𝒜), we

define 𝑊 = (𝑞, 𝑟_𝑛, 𝑝_𝑛(𝑥)) ∈ 𝑋∗ , where 𝑞 =∥ 𝑃 ∥ 𝑠𝑔𝑛(𝑄), 𝑟_𝑛 =∥ 𝑃 ∥ 𝑠𝑔𝑛(𝑅_𝑛), 𝑝_𝑛(𝑥) =∥ 𝑃 ∥ 𝑠𝑔𝑛(𝑃_𝑛(𝑥)), then (𝑃, 𝑊) =∥ 𝑃 ∥∥ 𝑊 ∥. In addition, we have

(𝒜𝑃, 𝑊) =∥ 𝑃 ∥ {[−𝜆𝑄 + 𝜇𝑅0+ ∫ 𝑣 +∞

0

(𝑥)𝑃0(𝑥)𝑑𝑥]𝑠𝑔𝑛(𝑄) + [−(𝜆 + 𝜇)𝑅0

+ ∫ 𝑣

+∞

0

(𝑥)𝑃1(𝑥)𝑑𝑥 + 𝜇𝑅1]𝑠𝑔𝑛(𝑅0) − ∑[ ∞

𝑛=1

(𝜆 + 𝜇)𝑅𝑛− ∫ 𝑣 +∞

0

(𝑥)𝑃𝑛+1(𝑥)𝑑𝑥 − 𝜇𝑅𝑛+1

−𝜆 ∑ 𝑎𝑖 𝑛

𝑖=1

𝑅𝑛−𝑖]𝑠𝑔𝑛(𝑅𝑛) − ∫ [ +∞

0

𝑃0′(𝑥) + (𝜆 + 𝑣(𝑥))𝑃0(𝑥)]𝑠𝑔𝑛(𝑃0(𝑥))𝑑𝑥

− ∑ ∫ [+∞

0 ∞

𝑛=1

𝑃_𝑛′(𝑥) + (𝜆 + 𝑣(𝑥))𝑃_𝑛(𝑥) − 𝜆 ∑ 𝑎_𝑖

𝑛

𝑖=1

𝑃_𝑛−𝑖(𝑥)]𝑠𝑔𝑛(𝑃_𝑛(𝑥))𝑑𝑥} ≤∥ 𝑃

∥ {−𝜆|𝑄| + 𝜇|𝑅₀| + ∫ 𝑣

+∞

0

(𝑥)|𝑃₀(𝑥)|𝑑𝑥 − (𝜆 + 𝜇)|𝑅₀| + ∫ 𝑣

+∞

0

(𝑥)|𝑃₁(𝑥)|𝑑𝑥 + 𝜇|𝑅₁|

− ∑[

∞

𝑛=1

(𝜆 + 𝜇)|𝑅_𝑛| − ∫ 𝑣+∞

0

(𝑥)|𝑃_𝑛+1(𝑥)|𝑑𝑥 − 𝜇|𝑅_𝑛+1| − 𝜆 ∑ 𝑎_𝑖

𝑛

𝑖=1

|𝑅_𝑛−𝑖|] + 𝜆|𝑄|

− ∫ [

+∞

0

𝜆 + 𝑣(𝑥)]|𝑃0(𝑥)|𝑑𝑥 − ∑ ∫ [ +∞

0 ∞

𝑛=1

(𝜆 + 𝑣(𝑥))|𝑃𝑛(𝑥)| − 𝜆 ∑ 𝑎𝑖 𝑛

𝑖=1

|𝑃𝑛−𝑖(𝑥)|]𝑑𝑥} = 0.

Therefore, 𝒜 is dissipative. Observing (𝒜1𝑃, 𝑊) ≤ (𝒜𝑃, 𝑊), we known that 𝒜1 is dissipative

and hence 𝑅(𝐼 − 𝒜₁) is a closed subspace of 𝑋.

Furthermore, we have 𝑅(𝐼 − 𝒜₁) = 𝑋. If it is not true, then there exists a 𝑊 ∈ 𝑋∗, such that for any 𝐹 ∈ 𝑅(𝐼 − 𝒜₁), (𝐹, 𝑊) = 0. Hence, for any 𝑃 ∈ 𝐷(𝒜₁), ((𝐼 − 𝒜₁)𝑃, 𝑊) = 0, i.e., for any

Theorem 1.4. The operator 𝒜 generates a 𝐶0 semigroup on 𝑋. The system (3) is well-posed.

Proof Obviously, ℬ is a bounded linear operator on 𝑋, using the perturbation theory of semigroup ([2]), we know that the operator 𝒜 generates a 𝐶₀ semigroup on 𝑋. Therefore, the system (3) is well-possed.

The Regularity of Solution

For a practical physical state, an important problem is the existence of positive solution.

Definition 2.1. ([3]) Let 𝑌 be a Banach lattice, 𝑌₊ be a positive cone of 𝑌 and 𝒯 be a linear operator on 𝑌. Denote 𝐺(𝑥) = {𝜑 ∈ 𝑌₊∗: ∥ 𝜑 ∥≤ 1, (𝑥, 𝜑) =∥ 𝑥₊ ∥2}. If, for any 𝑥 ∈ 𝐷(𝒯), there exists a 𝜑 ∈ 𝐺(𝑥), such that (𝒯𝑥, 𝜑) ≤ 0, then 𝒯 is called a dispersive operator.

Theorem 2.1. (1)𝒜 is a dispersive operator on 𝑋.

(2) The operator 𝒜 generates a positive 𝐶₀ contractive semigroup on 𝑋.

Proof It is well known that 𝑋 is a Banach lattice. According to Ref. [3], it is sufficient to prove that 𝒜 is a dispersive operator. For any 𝑃 = (𝑄, 𝑅_𝑛, 𝑃_𝑛(𝑥)) ∈ 𝐷(𝒜), we choose 𝑊 = (𝑞, 𝑟_𝑛, 𝑝_𝑛(𝑥)) ∈ 𝑋∗ _{, where 𝑞 =∥ 𝑃 ∥ 𝑠𝑔𝑛}

+(𝑄), 𝑟𝑛 =∥ 𝑃 ∥ 𝑠𝑔𝑛+(𝑅𝑛), 𝑝𝑛(𝑥) =∥ 𝑃 ∥

𝑠𝑔𝑛₊(𝑃_𝑛(𝑥)), and if 𝑎 > 0, then 𝑠𝑔𝑛₊𝑎 = 1; if 𝑎 ≤ 0, then 𝑠𝑔𝑛₊𝑎 = 0.

Similar to the proof of Theorem 1.3, a direct verification can show that (𝒜𝑃, 𝑊) ≤ 0. Observing 𝑊 ∈ 𝐺(𝑃), the desired result is obtained.

The following result studies the regularity of the system.

Theorem 2.2. Let 𝑇(𝑡) be a positive contractive semigroup with generator 𝒜, then 𝑇(𝑡) satisfies positive conserve property, i.e., for any 𝐻₀ ∈ 𝐷(𝒜) and 𝐻₀ > 0, ∥ 𝑇(𝑡)𝐻₀ ∥=∥ 𝐻₀ ∥, 𝑡 ≥ 0.

Proof Since 𝐻₀ ∈ 𝐷(𝒜) and 𝐻₀ > 0, then 𝑇(𝑡)𝐻₀ ∈ 𝐷(𝒜) is a classical solution of the system (3). Let 𝑃(𝑡) = (𝑄(𝑡), 𝑅_𝑛(𝑡), 𝑃_𝑛(𝑥, 𝑡)) = 𝑇(𝑡)𝐻₀ > 0, then 𝑃(𝑡) satisfies (1)(2). Note that

𝑑

𝑑𝑡∥ 𝑃(𝑡) ∥= 𝑑

𝑑𝑡∥ 𝑇(𝑡)𝐻0 ∥=

𝑑𝑄(𝑡) 𝑑𝑡 + ∑

𝑑𝑅_𝑛(𝑡) 𝑑𝑡

∞

𝑛=0

+ ∑ ∫ 𝜕𝑃𝑛(𝑥, 𝑡) 𝜕𝑡

+∞

0 ∞

𝑛=0

𝑑𝑥,

we get

𝑑

𝑑𝑡∥ 𝑃(𝑡) ∥ = −𝜆𝑄(𝑡) + 𝜇𝑅0(𝑡) + ∫ 𝑣

+∞

0

(𝑥)𝑃0(𝑥, 𝑡)𝑑𝑥 − (𝜆 + 𝜇)𝑅0(𝑡) + 𝜇𝑅1(𝑡) + ∫ 𝑣 +∞

0

(𝑥)𝑃1(𝑥, 𝑡)𝑑𝑥

− ∑[

∞

𝑛=1

(𝜆 + 𝜇)𝑅_𝑛(𝑡) − 𝜇𝑅_𝑛+1(𝑡) − 𝜆 ∑ 𝑎_𝑖

𝑛

𝑖=1

𝑅_𝑛−𝑖(𝑡) − ∫ 𝑣

+∞

0

(𝑥)𝑃_𝑛+1(𝑥, 𝑡)𝑑𝑥]

− ∫ {

+∞

0

𝜕𝑃₀(𝑥, 𝑡)

𝜕𝑥 + [𝜆 + 𝑣(𝑥)]𝑃0(𝑥, 𝑡)}𝑑𝑥 − ∑ ∫ {

+∞

0 ∞

𝑛=1

𝜕𝑃_𝑛(𝑥, 𝑡)

𝜕𝑥 + [𝜆 + 𝑣(𝑥)]𝑃𝑛(𝑥, 𝑡)

− 𝜆 ∑ 𝑎𝑖 𝑛

𝑖=1

𝑃𝑛−𝑖(𝑥, 𝑡)}𝑑𝑥

= −𝜆𝑄(𝑡) + 𝜇𝑅₀(𝑡) + ∫ 𝑣

+∞

0

(𝑥)𝑃₀(𝑥, 𝑡)𝑑𝑥 − (𝜆 + 𝜇)𝑅₀(𝑡) + 𝜇𝑅₁(𝑡) + ∫ 𝑣

+∞

0

− ∑[

∞

𝑛=1

(𝜆 + 𝜇)𝑅_𝑛(𝑡) − 𝜇𝑅_𝑛+1(𝑡) − 𝜆 ∑ 𝑎_𝑖

𝑛

𝑖=1

𝑅_𝑛−𝑖(𝑡) − ∫ 𝑣

+∞

0

(𝑥)𝑃_𝑛+1(𝑥, 𝑡)𝑑𝑥] + 𝜆𝑄(𝑡)

− ∫ [

+∞

0

𝜆 + 𝑣(𝑥)]𝑃0(𝑥, 𝑡)𝑑𝑥 − ∑ ∫ { +∞

0 ∞

𝑛=1

[𝜆 + 𝑣(𝑥)]𝑃𝑛(𝑥, 𝑡) − 𝜆 ∑ 𝑎𝑖 𝑛

𝑖=1

𝑃𝑛−𝑖(𝑥, 𝑡)}𝑑𝑥 = 0.

Hence ∥ 𝑃(𝑡) ∥=∥ 𝑃(0) ∥=∥ 𝐻₀ ∥.

Conclusion

By using the linear operator theory and the 𝐶₀ semigroup theory, we obtained the following results: (1) The operator 𝒜 generates a positive 𝐶₀ contractive semigroup on 𝑋. The system (3) is well-posed. (2) Let 𝑇(𝑡) be a positive contractive semigroup with generator 𝒜, then 𝑇(𝑡) satisfies positive conserve property, i.e., for any 𝐻0 ∈ 𝐷(𝒜) and 𝐻0 > 0, ∥ 𝑇(𝑡)𝐻0 ∥=∥ 𝐻0 ∥, 𝑡 ≥ 0.

After the mathematical modeling for the problem, our task is mainly to solve the following questions:

(1) the system under consideration has a unique nonnegative time-dependent solution; (2) approximate of solution; (3) the system has a steady state, and the dynamic solution of the system converges to the steady state.

Obviously, this paper only completed the above question (1).

References

[1] Borthakur A. and Choudhury G. On a batch arrival poisson queue with generalized vacation,

Indian J. Stat., 59(1997), B: 369-383.

[2] Pazy A. Semigroup of linear operators and applications to partial differential equations,

Springer-Verlag, New York, 1983.