Two-class priority queueing system with restricted ...

Int. J. Electron. Commun. (AEÜ) 61 (2007) 534 – 539 www.elsevier.de/aeue

Two-class priority queueing system with restricted number of priority customers A.M.K. Tarabia Mathematics Department, Damietta Faculty of Science, New Damietta, Egypt Received 14 June 2006

Abstract In this paper, we analyze a two-class single-server preemptive priority queueing system with arrivals to each class are assumed to follow a Poisson process with exponentially distributed service times. Customers are served on a first-come, firstserved basis within their own queue. Explicit expressions for the mean queue length and the steady-state joint distribution of the number of high and low priority customers in the system are derived. The analysis is based on the generating function technique. The obtained expressions are free from Bessel functions or any integral functions. Moreover, numerical values testing the quality of our analytical results are also presented. 䉷 2006 Elsevier GmbH. All rights reserved. Keywords: Queueing; Queue length; Preemptive resume; Generating function

1. Introduction Priority queues occur in many facets of everyday life, particularly in situations where preferential treatment is granted to certain kinds of individuals, e.g. telecommunications field. Priority mechanisms are an invaluable scheduling method that allows messages of different classes to receive different quality of service. For this reason, the priority queue has received considerable attention in the literature. Cobham [1] was the first to consider the non-preemptive priority for queues with Poisson input and exponential holding time and discussed them for single and multi-channel cases. Holley [2] simplified Cobham’s work. Barry [3] and Stephan [4] studied the problem of preemptive priority system with two priorities and Poisson input and exponential holding time. White and Christie [5] and Heathcote [6] studied the same problem for two channels and have used the method of the generating functions to describe the steady-state and

E-mail address: [email protected]. 1434-8411/$ - see front matter 䉷 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2006.09.006

transient queue length distributions, respectively. The main difficulty extending those results to multiple classes with general service time distributions is the manipulation of the Laplace transforms that characterize the dynamics of the system. An extensive study on two priorities scheme can be cited in Miller [7] where he obtained several results on this model using generating functions and Laplace transformations. A different approach concerning the Laguerre transform has also been applied to this model, see for example Keilson and Nunn [8] and Keilson and Sumita [9]. Miller [10] considered exponential single server priority queues with two classes of customers. He obtained recursive computation formulae for steady distributions using Neuts [11] theory of matrix-geometric invariant probability vectors. Schaak and Larson [12] dealt with multi-server, non-preemptive, multi-stationary queueing system with Poisson arrivals and exponential servers. They assumed a cut-off priority queue discipline and derived several performance measures. Franti [13,14] developed algorithms for a dynamic priority queue for computing performance measures such as queue length distribution and moments. Furthermore, he studied

A.M.K. Tarabia / Int. J. Electron. Commun. (AEÜ) 61 (2007) 534 – 539

algorithmic analysis for a dynamic, non-preemptive priority queueing system with two independent Poisson streams of customers with general service time distributions. Nishida [15] gave an approximation analysis for a heterogeneous multiprocessor system with preemptive and non-preemptive priority discipline and also discussed the numerical comparison between simulation and approximation analysis. Recently, Bitran and Caldentey [16] analyzed a two-class single queueing system that features state dependent arrivals and preemptive priority service discipline. For this system, they computed the mean equilibrium queue length and studied the effective service time and other first passage time quantities for both classes. More recently, Drekic and Woolford [17] analyzed a two class, single-sever preemptive priority queueing model with low priority balking customers. They obtained the steadystate joint distribution of the number of high and low priority customers in the system. Our motivation is to consider the model considered by Franti [13] with assuming that the number of high priority is restricted to L customers. The paper is organized as follows. In Section 2, we describe the model along with the assumptions and several notation used throughout the paper. Moreover, we obtain the generating function for queue size of each priority classes. Using the obtained generating function and partial fractions technique, we derive explicit expressions for the steady-state probabilities which are free from Bessel functions or any integral forms. Finding explicit expressions for Pi,. and P.,j the marginal distributions of the high and low priority classes, respectively is presented in Section 3. In Sections 4 and 5, include a simple way to obtain and compute the mean queue length of the high and low priority classes, respectively. In Section 6, several special cases obtained to support our work and conclusion. Also, some numerical results conclude the paper in Section 7.

2. Model description and analysis We consider a single server queueing system serving two types of customers; class-1 and class-2 each having its own respective line and the arrival process for both types is state independent. A higher priority is assigned to class-1. Suppose that the service rule within each class is FIFO and the priority system is preemptive resumed, i.e. during the service of a low priority customer, if high priority customers joins the system, then the low priority customer’s service is interrupted and will be resumed again when there is no high priority customers in the system. We denote by the number of the customers of class i (i = 1, 2). Let the number of customers in the first class is restricted to a finite number L including the one being served, if any, and the number of the second class is infinite. Let also 1 , 2 denote the arrival rates for the two classes and let 1 , 2

535

denote the service rates for the two classes. Denote the traffic intensities by 1 = 1 /1 , 2 = 2 /2 and the steady state probability that the system is in state (i, j ) by, where i is the number of the high priority customers and j is the number of low priority customers in the system. Clearly, the governing difference equations of the system under consideration are given by (1 + 2 )p0,0 = 1 p1,0 + 2 p0,1 ,

(1)

(1 + 2 + 2 )p0,j = 2 p0,j −1 + 1 p1,j + 2 p0,j +1 , j 1,

(2)

(1 + 2 + 1 )pi,0 = 1 pi−1,0 + 1 pi+1,0 , 1 i L − 1,

(3)

(1 + 2 + 1 )pi,j = 1 pi−1,j + 1 pi+1,j + 2 pi,j −1 , 1 i L − 1, j 1, (4) (2 + 1 )pL,0 = 1 pL−1,0

(5)

(2 + 1 )pL,j = 1 pL−1,j + 2 pL,j −1 ,

j 1.

(6)

Then the above system (1)–(6) can be written as [1 s + (2 − 2 s)(s − 1)]G0 (s) − 1 sG1 (s) = 2 (s − 1)p0,0 ,

(7)

− 1 Gi−1 (s) + [(1 − 2 (s − 1) + 1 ]Gi (s) − 1 Gi+1 (s) = 0, 1 i L − 1,

(8)

−1 GL−1 (s) + [1 − 2 (s − 1)]GL (s) = 0,

(9)

where Gi (s) is the generating function for the steady state probabilities pi,j . Now, these equations can be rewritten as A(s)G(s) = B(s),

(10)

where G(s) = [G0 (s), G1 (s), . . . , GL (s)]t is the generating function vector for pi,j , B(s) = [2 (s − 1)p0,0 , 0, . . . , 0]t and A(s) = [ai,j ] is a (L + 1) × (L + 1) order tridiagonal matrix and its coefficients are given by ⎧ 1 s + (2 − s)(s − 1), i = j = 0, ⎪ ⎪ ⎪ ⎪ −1 s, i = 0, j = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 (1 − i,L ) − 2 (s − 1) + 1 , ai,j = j = i, i = 1, 2, . . . , L, ⎪ ⎪ ⎪ −1 , j = i − 1, i = 1, 2, . . . , L, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 , j = i − 1, i = 1, 2, . . . , L, ⎪ ⎩ 0, Otherwise. In the following theorem we find explicit expression for Gi (s). Theorem 1. For any non-negative integer i, 0 i L, we have Gi (s) = 2 i1 p0,0 L

k=0 (1

CL−i (s)

, + 2 + 1 − L,k 1 1 − 2 s) (11)

536


where

So

Cm (s) = Cm (s) − 1 1 Cm−2 (s),

2 m L

(12)

with

Gi (s) =

2 i1 p0,0 CL−i (s) , DL+1 (s)

i = 0, 1, 2, . . . , L.

(18)

Now, to compute p0,0 explicitly, we multiply all the Eqs. (7)–(9) except the first one by s and adding all, we have

C1 (s) = 1 − 2 (s − 1),

2 G0 (s) − 2

C2 (s) = [1 − 2 (s − 1)] − 1 1 ,

L

Gi (s) = 2 p0,0 ,

i=0

= 1 + 2 + 1 − 2 s, p0,0 =

(1 − 1 ) (1 − L+1 ) 1

Putting s = 1, using the fact L

− 2 ,

1 = 1 /1 , 2 = 2 /2 and L,k , k = 1, 2, 3, . . . , L are the roots of Chebychev’s polynomials of second kind. Proof. The proof closely parallels the elementary construction of continuous-time Markovian queue, see for instance Tarabia [18]. We start by solving the above system (10) using Crammer’s rule, we get Gi (s) =

|A(s)|i , |A(s)|

i = 0, 1, 2, . . . , L,

Cm (s) = Cm−1 (s) − 1 1 Cm−2 (s), 2 m L,

(15)

where C1 (s) = 1 − 2 (s − 1), C2 (s) = [1 − 2 (s − 1)] − 1 1 , and = 1 + 2 + 1 − 2 s. On the other hand, multiply all rows of the determinant |A(s)| except the first row by s and add all rows to the first row, we can write |A(s)| = (s − 1)DL+1 (s),

D2 (s) = (2 − 2 s) − 1 2 .

(19)

Solving the recurrence relations (15) with s =1, (see Tarabia [18]) we get Cm (1) = m 1,

1 m L.

(20)

Similarly, for the recurrence relations (17), we have

L k1 −k DL+1 = L 1 2 − 2 − 2 1 k=1 L

= L 1 2 1 − 2 − 2

k1 ,

(21)

k=1

Using Eq. (18) with i = 0, s = 1 and substituting from Eqs. (20)–(21), the marginal probability p0,. can be written as L 1 2 p0,0 k L 1 2 [1 − 2 − 2 L k=1 1 ] p0,0 = , k [1 − 2 L k=0 1 ]

(14)

where represents the determinant of the bottom right square matrix obtained by eliminating the first row and column of the matrix A(s) and it can be seen that satisfies the following recurrence relations:

D1 (s) = 2 − 2 s,

we obtain

G0 (1) = p0,.

|A(s)|i = 2 i p0,0 (s − 1)CL−i (s), i = 0, 1, 2, . . . , L,

where

i=1

i=0 Gi (s) = 1,

2 = 2 . 2

(13)

where |A(s)|i , i = 0, 1, 2, . . . , L is the determinant of the matrix A(s) with replacing its ith column by the elements of the vector B(s). Expanding the determinant |A(s)|i , we easily obtain

Dm (s) = Cm−1 (s) − 1 1 Cm−2 (s),

Gi (s) =

L

(16) 2 m L,

(17)

=

L

p0,k + p0,0 =

k=1

p0,0 L

[1 − 2

k k=0 1 ]

,

(22)

On the other hand, Eqs. (19) and (22) together with some little simplifications imply that 2 + p0,0 =

p0,0 L

[1 − 2

k k=0 1 ]

.

)) − 2 , 1 < 1. This gives p0,0 = ((1 − 1 )/(1 − L+1 1 It is useful to consider a particular case of the above equation. Assuming L → ∞ and 1 +2 < 1 then p0,0 =1−1 − 2 , which coincides with the value reported in Srivastava and Kashyap [19]. Using the definition of Chebychev’s polynomial of second kind (see Abramowitz and Stegun [20]) DL+1 can be written as DL+1 (s) =

L k=0

(k − 2 s),

(23)


where k = 1 + 2 + 1 − 1 1 L,k , k = 1, 2, . . . L. Hence i p0,0 CL−i (s) Gi (s) = 2 L1 , k=0 (k − 2 s)

i = 0, 1, 2, . . . , L.

L Ai,k

j

j +1 k=0 k

L

i = 0, 1, 2, . . . , L. Expanding in power of s and comparing the coefficient of in both sides leads to j pi,j = 2 i1 2 p0,0

L Ai,k

j +1

k=0

k

i1 (1 − 1 ) (1 − L+1 ) 1

,

i = 0, 1, 2, . . . , L.

,

P.,j = 22 i2 p0,0

L A0,k j +2

k=0

k

,

with sk =

k=i

p0,0 =

(1 − 1 ) (1 − L+1 ) 1

k 2

and

L

pi,0 + 1

L

Finding explicit expressions for Pi,. and P.,j the marginal distributions of the high and low priority classes, respectively is presented in the following propositions. Proposition 2. The marginal distribution of the high priority is given by (1 − L+1 ) 1

,

= 1

L−1

pi,0

i=1

pi,0 + 1

i=0

L

pi,0 + 2 p0,1

i=1

P.,0 = −1 2 p0,1 .

− 2 ,

3. The marginal distributions of the high and low priority classes

i1 (1 − 1 )

i = 0, 1, 2, . . . , L.

In general and using similar approach by adding Eqs. (2) and (4) for i, 0 i L, we get 2 P.,j + 1

L−1

pi,j + 1

i=0

= 1

L−1

L

pi,j + 2 p0,j

i=1

pi,j + 1

i=0

L

pi,j + 2 P.,j −1 + 2 p0,j +1 ,

i=1

which simply leads to 2 P.,j + 2 p0,j = 2 P.,j −1 + 2 p0,j +1 . By induction method, the marginal probabilities P.,j satisfy the following recursion relation: P.,j = −1 2 p0,j +1 ,

j 0.

The proposition follows directly from Proposition 1. Proof. As is evident its definition, Gi (1) represents the marginal distribution of the high priority, therefore one can put in Eq. (18), we may obtain Gi (1) =

2 i1 p0,0 CL−i (1) , DL+1 (1)

(24)

which implies

which are the steady state probabilities of the given system.

Pi,. =

j 0.

Proof. By adding Eqs. (1), (3) and (5) for i, 0 i L, we obtain

i=0

CL−i (sk ) , k=0 (k − i )

Clearly the analogous result for M/M/1/∞ queue can be immediately obtained as a special case when L → ∞, 1 < 1.

1 P.,0 + 1

i = 0, 1, 2, . . . , L, j 0

Ai,k =

i = 0, 1, 2, . . . , L,

Proposition 3. The marginal distribution of the low priority is given by

Ai,k , (k − 2 s)

k=0

Pi,. =

2 i1 p0,0 CL−i (1) , L k L 1 [2 − 2 k=0 1 ]

Which establishes the result.

,

Proof. The proof of this proposition is directly from the using of partial fractions, we obtain Gi (s) = 2 i1 p0,0

Pi,. = Gi (1)

Proposition 1. For any i = 0, 1, 2, . . . , L, j 0, we have pi,j = 2 i1 2 p0,0

now using Eqs. (20) and (23) we get

=

This establishes the theorem.

537

i = 0, 1, 2, . . . , L

4. Mean queue length of high priority class In this section, our main objective is to obtain the mean queue length of high priority class. To achieve it, we have

538


assumed that N1 be the queue length of the first class and define E(N1 ) =

∞ L

ip i,j =

i=0 j =0

L

ipi,. .

∞ 1 jp0,j +1 = −1 E(N2 ) = 2 Q0 − 1. 2 j =0

i=0

By substituting from Proposition 2 into the above equation, we have L i (1 − 1 ) E(N1 ) = i 1 (1 − L+1 ) 1 i=1

1 − L+1 1 (1 − 1 ) d 1 = 1 − 1 (1 − L+1 ) d1 1

=

Also, from Eq. (24) we can rewrite

L+1 ] 1 [1 − (L + 1)L 1 + L1

(1 − 1 )(1 − L+1 ) 1

Then Q0 = 2 (1 + E(N2 )). On account of Eq. (26) and the last equation, we easily obtain E(N2 ) = =

.

(25)

+

i=0 j =1

L

i i1 2 L+1−k i− + 1 1 (1 − L+1 ) 1 k=1

L

i1

i=0

L i 1 2 i 1 i 1 (1 − L+1 ) i=0 1

−

i

L+1−k 1

.

k=1

After simplification, it can be seen that

Now, let us define the mean queue length of low priority class as follows: jpi,j =

i1 Q0

= 2 (1 + E(N2 ))

E(N2 ) =

5. Mean queue of low priority class

L ∞

Qi

i=0

L i=0

This agrees with that mentioned in Tarabia [18] and Sharma and Gupta [21] when t → ∞. Moreover, it shows that high priority queue is functioning independently as an M/M/1/L queue.

E(N2 ) =

L

Qi ,

(1 − 1 ) [1 − 1 − 2 (1 − L+1 )] 1

2 (1 − L+1 ) 2 1 1 × + (1 − 1 ) 1 (1 − 1 )2 (1 − L+1 ) 1 ×(1 − 2L+1 − (2L + 1)(1 − 1 )L 1) . 1

i=0

where ∞ N2 be the queue length of the second class and Qi = j =1 j pi,j . Multiplying Eq. (2) by j and summing for all j, we obtain (1 + 2 + 1 )Q0 = 1 Q1 + 2 Q0 ∞ ∞ + 2 jp0,j +1 + 2 p0,j −1 j =1

j =1

In general, multiplying (4) by j and summing over all j, j 1 (1 + 2 + 1 )Qi = 1 Qi+1 + 2

(a) If we take L = 1, the system reduces to the well known queueing system M/M/1 and we obtain E(N2 ) =

using Proposition 2 and Eq. (24), we get

2 1 1 − L+1 1 Q1 = 1 Q 0 + . 1 1 − L+1 1

∞

6. Special cases

pi,j −1

2 (1 − 2 )

which agrees with the known result given by Tarabia [18]. (b) If we put 2 = 0, we get all the result of the well known M/M/1/L model such as pi =

i1 (1 − 1 ) (1 − L+! 1 )

,

i = 0, 1, 2, . . . L,

1 < 1.

j =1

+ 1 Qj −1 + 2 Qj using Proposition 2 and recursively, we obtain

i i 2 1 Qi = i1 Q0 + i− . L+1−k 1 1 (1 − L+1 ) 1 k=1

7. Numerical results

(26)

Some tedious numerical results were worked out, but for brevity we give only the following numerical results to show that how much our theoretical work is efficient to compute the steady state probabilities see Table 1.


Table 1. Values of the idle probability in the first class for different densities values j 0 1 2 3 4 5 6 7 8 9 10

1 = 0.6, 2 = 0.35 0.06956 0.041064 0.031564 0.026819 0.036681 0.012121 0.019132 0.017320 0.015696 0.014233 0.012911

1 = 0.67, 2 = 0.25 0.0577062 0.01248461 0.00230997 0.00325133 0.00226431 0.00165306 0.00123117 0.00092547 0.000169878 0.00052877 0.0004010

8. Conclusions We remark that the explicit expressions obtained for the mean queue length and the steady state joint distribution of the number of high and low priority customers in the system are simple and easy to compute especially for small number (L < 30) of high priority customers. This because the computed roots L,k , k = 1, 2, . . . , L of Chebychev’s polynomials of second kind become identical for large L. Moreover, on the whole, the analysis of the first class behaves like M/M/1/L Markovian queue.

References [1] Cobham A. Priority assignment in waiting line problem. J Oper Res Soc Am 1954;2:70–6. [2] Holly JL. Waiting line subject to priorities. J Oper Res Soc Am 1954;2:341–3. [3] Barry JY. A priority queueing problem. Oper Res 1956;4: 385–6. [4] Stephan FF. Two queues under preemptive priority with Poisson arrival and service rates. Oper Res 1958;6:399–418. [5] White H, Christie L. Queueing with preemptive priorities or with breakdown. Oper Res 1958;6:79–96. [6] Heathcote C. The time-dependent problem for a queue with preemptive priorities. Oper Res 1959;7:670–80. [7] Miller RG. Priority queues. Ann Math Stat 1960;31:86–103. [8] Keilson J, Nunn W. Laguerre transformation as a tool for the numerical solution of integral equations of convolutions type. Appl Math Comput 1979;5:313–59. [9] Keilson J, Sumita U. Evaluation of the total time in system in a preempt/resume priority queue via modified Lindley process. Adv Appl Probab 1983;15:840–56.

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[10] Miller RD. Computation of steady-state probabilities for M/M/1 priority queues. Oper Res 1981;29:945–58. [11] Neuts MF. Matrix-geometric solutions in stochastic models. Baltimore: The Johns Hopkins University Press; 1981. [12] Schaak C, Larson RC. An N-server cut off priority queue. Oper Res 1986;34:257–66. [13] Franti SS. Algorithms for a dynamic priority queue with two types of customers. Drexel University, Philadelphia, 1985. [14] Franti SS. Analysis of a dynamic priority queue. Commun Stat Stoch Model 1990;6(3):415–44. [15] Nishida T. Approximate analysis for heterogeneous multiserver systems with priority jobs. Perform Eval 1992;15:77–88. [16] Bitran G, Caldentey R. Two-class priority system with statedependent arrivals. Queueing Syst 2000;40:4. [17] Drekic S, Woolford DG. A preemptive priority queue with balking. Eur J Oper Res 2005;164:387–401. [18] Tarabia AMK. Transient analysis of a non-empty M/M/1/N queue an alternative approach. J Oper Res Soc India 2001;38(4):431–9. [19] Srivastava HM, Kashyap BRK. Special functions in queueing theory and related stochastic processes. New York: Academic Press; 1982. [20] Abramowitz M, Stegun IA. Handbook of mathematical functions. New York: Dover; 1970. [21] Sharma OP, Gupta UC. Transient behaviour of an M/M/1/N queue. Stoch Process Appl 1982;13:327–31.

A.M.K. Tarabia is currently Associated Professor in the Department of Mathematics at Mansoura University, Egypt. He was born in Damietta, Egypt, in 1962. He received his B.ESc and B.Sc. and M.Sc. degrees in Mathematics from Mansoura University, in 1985, 1989 and 1993, respectively. Further he completed his Ph.D. from Indian Institute of Technology, Delhi, in 1998. Since August 2002, Dr. Tarabia has been traveling into various universities in Egypt and KSA and teaching various courses. Dr. Tarabia has contributed significantly in the area of modeling and analysis of continuous and discrete-time queueing systems. He has published several research articles in various journals such as Stochastic Processes and its Applications, Mathematical Scientist (Journal of Applied Probability), Applied Mathematics and Computation, An International Journal Computer & Mathematics with Applications, Sankhya, Journal of the Operational Research Society of India (Opsearch). He has authored many Statistics and Mathematics books in Arabic Language.