Further research of the eigenvalues of the M/GB/1 operator

2016-09-15EHMETAblet

贵州师范大学学报（自然科学版） 2016年4期

关键词：重数财经大学讲师

EHMET Ablet

(College of Application Mathematics, Xinjiang University of Finance and Economics,Urumqi,Xinjiang 830012,China)

Further research of the eigenvalues of theM/GB/1 operator

EHMET Ablet

(College of Application Mathematics, Xinjiang University of Finance and Economics,Urumqi,Xinjiang 830012,China)

M/GB/1queueing model; eigenvalue; geometric multiplicity

1　Introduction

The following M/GB/1 model is commonly used in queueing theory which can be expressed as (see[1]):

(1)

η(x))p0,1(x,t),

(2)

η(x))pn,1(x,t)+λpn-1,1(x,t), n≥1,

(3)

(4)

(5)

p0,0(0)=1, pn,1(x,0)=0, n≥0,

(6)

where p0,0(t) represents the probability that at time t the system is empty, pn,1(x,t)dx(n≥1) represents the probability that at time t there are n customers in the system with elapsed service time of the customer undergoing service lying between x and x+dx, B represents the maximum capacity of the service station, and η(x) is the conditional service rate and satisfies

We will use notations in [2] (see also [5]). For simplicity,we introduce

Choose the state space X as follows:

It is obvious that X is a Banach space. Let

pn,1(x)(n≥0) are absolutely continuous function,

For p∈D(A),we define

For p∈X,we define Up=Λp and

then the above system of equations (1)-(6) can be written as an abstract ordinary differential equation in the Banach space X:

(7)

p(0)=(1,0,0,0,…).

(8)

papers [2-6] obtained the following results:

Theorem 1A-U+E generates a C0-semigroup T(t).

Theorem 20 is an eigenvalue of A-U+E,

belongs to resolvent set of A-U+E when η(x)=η.Particularly all points on the imaginary axis except for zero belongs to the resolvent set of A-U+E.

2　Main Results

(9)

(10)

λpn-1,1(x), n≥1

(11)

(12)

(13)

Solving equation (10) and (11), we can obtain

p0,1(x)=a0e-(γ+λ+η)x,

(14)

pn-1,1(τ)dτ, n≥1.

(15)

From (15), we can obtain the following by Fubini theorem

(16)

Substituting (14) into (9) yields

(17)

Substituting (14), (15) and (16) into (12) and (13) yields

(18)

(19)

From (17), (18) and (19), we can derive

|a0|<∞,|a1|<∞,|a2|<∞,|a3|<∞.

Upon making use of (15) and (16) in (13), we can also derive

⟹

(20)

(21)

(21) can be rewritten as:

⟹

(22)

If we let

(23)

then

(24)

comparing (22) with (24), we can derive

⟹

(25)

⟹

⋮

⟹

(26)

From (26) and(see Remark)

we can obtain the following estimation

(27)

From (14), (15), (16), (27) and Fubini theorem by noting that

we can estimate ‖p‖, i.e.,

(28)

⟹

Acknowledgments

The author is grateful to professor Geni Gupur for his constructive criticisms, encouraging comments and helpful suggestions.

References：

[1] CHAUDHRY M L,TEMPLETON J G C.A First Course in Bulk Queues[M].New York:John Wiley Sons,1983.

[2] GUPUR G,LI X Z,ZHU G T.Existence and Uniqueness of Nonnegative Solution of M/GB/1 Queueing Model[J].Computers and Mathematics with Applications,2000(39):199-209.

[3] GENI G.Resolvent Set of the M/Mk,B/1 Operator[J].Acta Analysis Functionalis Applicata,2004(6):106-121.

[4] ABDUKERIM H J,AGNES R.Asymptotic Stability of the Solution of the M/MB/1 Queueing Model[J].Computers and Mathematics with Applications,2007(53):1411-1420.

[5] JIA H,SERIKBOL B.Another Eigenvalue of the M/M2/1 Operator[J].Journal of Xinjiang University(Natural Science Edition),2009(1):60-68.

[6] ZHANG L,GENI G.Another Eigenvalue of the M/M/1 Operator[J].Acta Analysis Functionalis Applicata,2008(10):81-91.

1004—5570(2016)04-0045-06

M/GB/1 算子的特征值的进一步研究

艾合买提·阿不来提

(新疆财经大学应用数学学院，新疆乌鲁木齐830012)