ISSN 2319-8133 (Online)
(An International Research Journal), www.compmath-journal.org
A Single Server State Dependent Queue with Two Types of Customer
R. Sakthi
1*and V. Vidhya
21
Assistant Professor, Department of Science & Humanities, Saveetha School of Engineering, SIMATS, Chennai - 602105, INDIA.
1
Research Scholar,
Bharathiar University, Coimbatore, INDIA.
2
Assistant Professor, School of Advanced Sciences, VIT- Chennai, INDIA.
Correspondence author email: [email protected].
(Received on: May 9, 2019) ABSTRACT
In this article we consider a queuing system M/M/1 with the service rate varying according to types of customers like normal and tagged. We develop a state dependent queuing model in which the service rate depends on customer type with quasi-birth-death structure and using matrix geometric method we develop the system performance measures with numerical illustrations.
MSC 60K25, 90B22, 68M207.
Keywords: Tagged customer; state dependent; quasi-birth-death process; stationary probability.
INTRODUCTION
Customers waiting in queue is an accepted fact in modern society human environment,
communication network, production process etc. In communication system during transfer of
data the packets gets congested that results in a block up of resources which leads to loss of
time, excess cost, overload etc. In most of the queuing system the server's rate of service has
variations due to the different type of customers, especially customer with high importance are
given preference compared with normal customer. A common situation is observed in service
facility of 𝑛(≥ 1) class of customers in a single queue with some customer having significant
importance.
In this paper we develop a M/M/1 state dependent queuing model with the following novelty: Consider a queuing system with the customer arrival having two types normal and tagged customers. The normal customer enters to the system gets service at the rate 𝜇
1till it reaches a tagged customer. When the server attains a moment to serve a tagged customer denoted by a pair of numbers (𝑛, 𝑚), where 𝑛 stands for the position of the tagged customer in the queue, and where m denotes the number of tagged customers behind it, the server increases its speed and finish the service as early as possible because of high importance. After the service completion the server observes the heavy tail queue, in order to maintain the system stability the server continues with the same rate of service that has been rendered to tagged customer till the heavy tailed reduces its length.
Many researchers Abou (1991), Bekker (2004), Bekker (2011), Delasay (2016), George (2001), Hopp (2007), contributed their work related to control of queue with dependent service rate. Queuing model with customer dependent service rate was investigated initially by Jackson (1963) in which the joint probability distribution of the queue length is obtained in a network with Markovian routing and each station has Markovian service rate that depends on the length of the queue at that station. These research work assert that server speedup is optimal in certain situation particularly George (2001) obtained the optimal single server service rate with increase in queue length. Quasi-Birth-Death process was first studied by Evans (1967) and Wallace (1969) coined the term quasi-birth-death process. The most extensive study of quasi birth death using matrix geometric invariant vectors was done by Neuts (1981). The methodology used by Neuts1981 is based on the fact that the sub vectors of steady state probability vector are connected with one another in matrix geometric structure.
The rate matrix is of matrix geometric structure which is obtained as the minimal positive solution of a non-linear matrix equation. Computation of the steady state distribution using matrix geometric solution of state dependent queue can be referred in Yue2006, in which the various performance measures of 𝑀/𝐸
𝑘/1 are investigated. Recently in Delasay (2016) the authors developed a state dependent queuing system in which the service rate depends on the system such as load and overwork, where overwork refers to system that has been with heavy load for an extended period of time in quasi-birth-death structure. In this work we develop the single server queuing model with state dependent service rate described as in the next section to obtain the performance measure and the state with maximum efficiency of the server.
MODEL DESCRIPTION
Consider a single server queuing system with two types of customers normal type of
customer and tagged type. Assume that customers arrive according to a Poisson process with
rate 𝜆. The probability of arriving high importance tagged customer is given as 𝑝. Customers
arrive to the system in Poison arrival rate are served with service rate 𝜇
1, the service time are
exponentially distributed. When a tagged customer enters the queue the server immediately
turn to high rate setup, starts serving tagged customer with increased service rate 𝜇
2. Once the
tagged customer are served the server turns to the waiting customer and observes that the queue
is heavy tailed. Continuation of this scenario may lead the system to get instability as heavy
tail start to grow. To avoid the instability the server continues with the same rate of service 𝜇
2that has been rendered to tagged customer, as soon as the heavy tail decreases the server returns to the normal state with service rate 𝜇
1.
ASSUMPTION Let
𝑆 = {(𝐿𝑛(𝑡), 𝑆𝑛(𝑡)): 𝑛 ≥ 1}
represents the two dimensional Markov chain with state space
𝑆 = {(0,0) ∪ (𝑘, 𝑗): 𝑘 ≥ 1, 𝑗 = 0,1,2}The arrival rate 𝜆 is assumed to be Poisson process, the inter arrival time and service time are mutually independent. The server providing service has two service rates represented as 𝜇
1for normal customer and the service rate represented as𝜇
2(0 < 𝜇
1< 𝜇
2) the effective rate for tagged customer or at the moment of heavy tail.
Let 𝑃
𝑛,0: Probability that server is rendering service for normal customer with rate 𝜇
1𝑃
𝑛,1: Probability that server is rendering service for tagged customer with rate 𝜇
2𝑃
𝑛,2: Probability that server is rendering service for heavy tailed customer with rate 𝜇
2The steady state equations are
𝜆𝑃
0,0= 𝜇
1𝑃
1,0+ 𝑞𝜇
2𝑃
1,1+ 𝜇
2𝑃
1,2(𝜆 + 𝜇
1)𝑃
𝑛,0= 𝜇
1𝑃
𝑛+1,0+ 𝜆𝑃
𝑛−1,0, 𝑛 ≥ 1 (1)
(𝜆 + 𝜇
2)𝑃
1,1= 𝑞𝜇
2𝑃
2,1; 𝑛 ≥ 1 (2)
(𝜆 + 𝜇
2)𝑃
𝑛,2= 𝑞𝜇
2𝑃
𝑛+1,1+ 𝜆𝑃
𝑛−1,0; 𝑛 ≥ 1
(𝜆 + 𝜇
2)𝑃
𝑛,2= 𝜇
2𝑃
𝑛+1,2+ 𝑝𝜇
2𝑃
𝑛,1+ 𝜆𝑃
𝑛−1 ,2; 𝑛 ≥ 1 (3) The generator of the Quasi-Birth-Death process is given by
Q = (
L00 F0
B10 L1 𝐹0
B2
… L2
… F0
… … … …
B2 L1 F0
)
L00= −(𝜆), F00= (𝜆 0 0), F0= (𝜆 0 0 0 𝜆 0 0 0 𝜆
)
B10= (𝑞 𝜇1
𝜇2
𝜇2) , B2= (
𝜇1 0 0
0 𝑞𝜇2 0 0 0 𝜇2)
L1= (
−(𝜆 + 𝜇1) 0 0
0 −(𝜆 + 𝑞𝜇2) 0
0 0 −(𝜆 + 𝜇2)),
Theorem 1 If
𝜆𝜇2
< 1, then the matrix quadratic equation 𝑅
2𝐵
2+ 𝑅𝐿
1+ 𝐹
0= 0 has the
minimal non-negative solution given as
𝑅 = (
𝑟
11𝑟
12𝑟
130 𝑟
22𝑟
230 0 𝑟
33) (6)
where 𝑟
11= 𝜆 2𝜇
1, 𝑟
22= 𝜆 + 𝜇
2± √𝜆
2+ 𝜇
22+ 2𝜆𝜇
2− 4𝑞𝜇
22𝑞𝜆𝜇
22𝑟
33= 𝜆 𝜇
2, 𝑟
23= 𝑟
22𝑝𝜇
2𝜆 + 𝜇
2(1 + 𝑟
22+ 𝑟
33) , 𝑟
12= 0, 𝑟
13= 0 Theorem 2 The stationary probabilities of the system are determined as 𝜋
𝑘,0= 𝜋
0,0𝜆 (𝜆 + 𝜇
1) − 𝑟
11𝜇
1, 𝑘 ≥ 0 𝜋
𝑘,1= 𝜋
0,0𝜆𝑟
12𝑞 𝜇
2((𝜆 + 𝜇
1) − 𝑟
11𝜇
1)((𝜆 + 𝜇
2) − 𝑟
22𝑞𝜇
2) , 𝑘 ≥ 1 𝜋
𝑘,2=
𝜋0,0(𝜆+𝜇𝑙0𝑙1(𝑟23+𝑝)𝜇2 2)−𝑟33𝜇2(7)
where 𝑙
0= 𝜆 𝜆 + 𝜇
1− 𝑟
11𝜇
1𝑙
1=
𝑟12𝑞 𝜇2 𝜆+𝜇2−𝑟33𝜇2(8)
Proof The vector ∏ is partitioned as ∏ = (∏
0, ∏
1, … , )
𝑡, where each ∏
𝑘= (𝜋
𝑘,0, 𝜋
𝑘,1, 𝜋
𝑘,2), 𝑘 ≥ 0 Then (𝜋
𝑘,0, 𝜋
𝑘,1, 𝜋
𝑘,2) = (𝜋
1,0, 𝜋
1,1, 𝜋
1,2)𝑅
𝑘−1, 𝑘 ≥ 1 (9) Solving equation (9) 𝜋
𝑘,0= 𝜋
1,0𝑅
𝑘−1, 𝑘 ≥ 1 𝜋
𝑘,1= 𝜋
1,1𝑅
𝑘−1, 𝑘 ≥ 1 𝜋
𝑘,2= 𝜋
1,2𝑅
𝑘−1, 𝑘 ≥ 1 (10)
The probability (𝜋
0,0, 𝜋
1,0, 𝜋
1,1, 𝜋
1,2) satisfies the equation (𝜋
0,0, 𝜋
1,0, 𝜋
1,1, 𝜋
1,2)𝐵(𝑅) = 0 (11)
where 𝐵(𝑅) = ( 𝐿
00𝐹
0𝐵
10𝐿
1+ 𝑅𝐵
2) (12)
Using equation(11), the following equations are obtained as −𝜋
0,0𝜆 + 𝜋
1,0𝜇
1+ 𝜋
1,1𝑞𝜇
2+ 𝜋
1,2𝜇
2= 0 (13)
𝜆𝜋
0,0+ 𝜋
1,0(𝑟
11𝜇
1− (𝜆 + 𝜇
1)) = 0 (14)
𝜋
1,0𝑟
12𝑞 𝜇
2+ 𝜋
1,1(𝑟
22𝑞𝜇
2− (𝜆 + 𝜇
2)) = 0 (15)
𝜋
1,0𝑟
13𝜇
2+ 𝜋
1,1(𝑟
23𝜇
2+ 𝑝𝜇
2) + 𝜋
1,2(𝑟
33𝜇
2− (𝜆 + 𝜇
2)) = 0 (16)
Solving the equations (13)-(16), the vectors are obtained as 𝜋
1,0= 𝜋
0,0𝑙
1𝜋
1,1= 𝜋
0,0𝑙
1𝑙
2𝜋
1,2= 𝜋
0,0𝑙
2Substituting the values 𝜋
1,0, 𝜋
1,1and 𝜋
1,2in (10) the values of 𝜋
𝑘,0, 𝜋
𝑘,1and 𝜋
𝑘,2are obtained which is given in theorem (2). The normalizing condition is
𝜋
0,0+ (𝜋
1,0, 𝜋
1,1, 𝜋
1,2)(𝐼 − 𝑅)
−1𝑒 = 1 (17) 𝜋
0,0= 1 − (𝜋
1,0, 𝜋
1,1, 𝜋
1,2)(𝐼 − 𝑅)
−1𝑒 (18) In general, it's cumbersome to express the exact value of 𝑅 and the matrix 𝑅 can be approximated as follows:
(i) Initially 𝑅[0] = 0
(ii) 𝑅[𝑛 + 1] = −[F
0+ 𝑅
2(𝑛)𝐵
2](𝐿
1−1), 𝑛 ≥ 0.
This iterative algorithm is convergent ,i.e 𝑅 = lim
𝑛→∞
𝑅[𝑛]
PERFORMANCE METRICS
The objective of this work is predict the various performance measures using the probability vector obtained in the previous section. The various performance measures are given as: The average number of customers when the service is in normal state is given by
𝐿
𝑠𝑛= ∑
∞𝑛=1𝑛 𝜋
0,𝑛(19)
The average number of customers in high priority service state is given by
𝐿
𝑠𝑡= ∑
∞𝑛=1𝑛 𝜋
1,𝑛(20)
The average number of customer in low priority service state is given by
𝐿
𝑠ℎ= ∑
∞𝑛=1𝑛 𝜋
2,𝑛(21)
The average number of customers in the system is given by
𝐸(𝑆) = 𝐿
𝑠𝑛+ 𝐿
𝑠𝑡+ 𝐿
𝑠ℎ(22)
The waiting time of normal customer, high prior customer, low prior customer in the system are calculated by using Little's theorem refer George and Harrison (2001)
𝑊
𝑠𝑛=
1𝜆
𝐿
𝑠𝑛, 𝑊
𝑠𝑡=
1𝜆
𝐿
𝑠𝑡, 𝑊
𝑠ℎ=
1𝜆
𝐿
𝑠ℎ(23)
ILLUSTRATION WITH NUMERICAL EXAMPLES
A numerical experimented is performed by coding the program in MATLAB to obtain
the maximum efficiency output and use it to test the sensitivity on parameters with different
performance measures. For computational purpose the parameters are arbitrarily selected as
𝜇
1= 3.1; 𝜇
2= 4 ; 𝜇
2= 4.6 ; 𝑝 = 0.5 ; 𝑞 = 0.5 and the arrival rate increased with
increment 0.1 from 𝜆 = 1.6 to 𝜆 = 2.5 as listed in Table 1. It is observed that the length at
each state increases as the arrival increases and simultaneously the waiting time to increases,
the same is represented in the diagram as shown in Figure 1 and Figure 2.
Table 1 – 𝝀 versus 𝑳𝒔𝒏, 𝑳𝒔𝒕, 𝑳𝒔𝒉, 𝑾𝒔𝒏, 𝑾𝒔𝒕, 𝑾𝒔𝒉
𝝀 𝑳𝒔𝒏 𝑳𝒔𝒕 𝑳𝒔𝒉 𝑾𝒔𝒏 𝑾𝒔𝒕 𝑾𝒔𝒉
1.6 0.1956 0.6476 0.6084 0.1222 0.4047 0.3802
1.7 0.2123 0.5698 0.6632 0.1248 0.3351 0.3901
1.8 0.2294 0.5072 0.72 0.1274 0.2817 0.4
1.9 0.2471 0.4560 0.7788 0.1300 0.2400 0.4098
2.0 0.2653 0.4137 0.8395 0.1326 0.2068 0.4197
2.1 0.2841 0.3782 0.9022 0.1352 0.1800 0.4296
2.2 0.3033 0.3482 0.9669 0.1378 0.1592 0.4395
2.3 0.3231 0.3226 1.0336 0.1404 0.1402 0.4493
2.4 0.3434 0.3006 1.1022 0.1430 0.1252 0.4592
2.5 0.3642 0.2814 1.1728 0.1456 0.1125 0.4691
Figure 1
Figure 2