Markovian Queueing System with Discouraged Arrivals and Self-Regulatory Servers

We consider discouraged arrival of Markovian queueing systems whose service speed is regulated according to the number of customers in the system. We will reduce the congestion in two ways. First we attempt to reduce the congestion by discouraging the arrivals of customers from joining the queue. Secondly we reduce the congestion by introducing the concept of service switches. First we consider a model in which multiple servers have three service rates μ 1 , μ 2 , and μ (μ 1 ≤ μ 2 < μ), say, slow, medium, and fast rates, respectively. If the number of customers in the system exceeds a particular point K 1 or K 2 , the server switches to the medium or fast rate, respectively. For this adaptive queueing system the steady state probabilities are derived and some performance measures such as expected number in the system/queue and expected waiting time in the system/queue are obtained. Multiple server discouraged arrival model having one service switch and single server discouraged arrival model having one and two service switches are obtained as special cases. A Matlab program of the model is presented and numerical illustrations are given.


Introduction
Queueing theory plays an important role in modeling real life problems involving congestion in wide areas of applied sciences.A customer decides to join the queue only when a short wait is expected and if the wait has been sufficiently small he tends to remain in the queue; otherwise the customer leaves the system and then the customer is said to be impatient.When this impatience increases and customers leave before being served, some remedial actions must be taken to reduce the congestion in the system.
Balking and reneging are the forms of impatience.If a customer decides not to enter the queue upon arrival by seeing a long queue, the customer is said to have balked.A customer may enter the queue but after a while loses patience and decides to leave and then the customer is said to be reneged.
In the study of queueing system the server is usually assumed to work at constant speed regardless of the amount of work existing.But in real life situation this assumption may not always be appropriate as the system size may affect the system performance.That is, the servers adapt to the system state by increasing the speed to clear the queue or decrease the speed when fatigued, which means the service rate depends on the system size; see, for example, Jonckheere and Borst [1].Similarly we can see that the arrivals of customers into the system may also affect the level of congestion.For example, customers impatience will affect the arrivals of customers into the system.A queueing system where the arrival rate and/or service rate depends on the system size is called adaptive queueing system.
Sometimes the arrival rate of customers into the system depends on the system size instead of a constant rate.Discouraged arrival is one form of state dependence.Here the arrivals get discouraged from joining the queue when more and more people are present in the system.We can model this effect by taking the birth and death coefficients, respectively, as   = /( + 1), where  is the number of customers in the system and  is a positive constant, and   = , where  is the constant service rate.Thus the arrival rate of the queueing system is decreased by this discouragement; consequently the congestion of the system decreases.This type of queueing systems is called discouraged arrival queueing system.

Advances in Operations Research
Applications of queueing with impatience can be seen in traffic modeling, business and industries, computer communication, health sectors, medical sciences, and so forth.Customers with impatience and discouragement have their own impact on the system performance of standard queueing systems.It is important to note that customers impatience has a very negative impact on the queueing system under investigation.
In this paper we attempt to reduce the congestion in two ways.First we attempt to reduce the congestion by discouraging the arrivals of customers from joining the queue.In the second way we further reduce the congestion by introducing the concept of service switches which is discussed in Abdul Rasheed and Manoharan [11].
Discouraged arrival system was studied by many researchers.Raynolds [12] presented multiserver queueing model with discouragement and obtained equilibrium distribution of queue length and derived other performance measures from it.A finite capacity M/G/1 queueing model where the arrival and the service rates were arbitrary functions of the current number of customers in the system was studied by Courtois and Georges [13].Natvig [14] studied the single server birth-death queueing process with state dependent parameters   = /( + 1),  ≥ 0, and   = ,  ≥ 1. Van Doorn [15] obtained exact expressions for transient state probabilities of the birth-death process with parameters   = /( + 1),  ≥ 0, and   = ,  ≥ 1. Parthasarathy and Selvaraju [16] obtained the transient solution to a state dependent birth-death queueing model in which potential customers are discouraged by queue length.
Narayanan [17] studied different linear and nonlinear state dependent Markovian queueing models, in which arrival rates and/or service rates are nonlinear and their modified forms obtain the transient/steady state probability distribution of queue length.Narayanan and Manoharan [18] considered nonlinear state dependent queueing models, in which arrival rates and/or service rates are nonlinear.Narayanan and Manoharan [19] derived the performance measures of state dependent queueing models.Ammar et al. [20] studied single server finite capacity Markovian queue with discouraged arrivals and reneging using matrix method.
Abdul Rasheed and Manoharan [11] studied a Markovian queueing system in which the arrival rate is constant and the service rate depends on the number of customers in the system.The server speed is regulated according to the system size by introducing the service switches to the model.The authors analyzed the system by calculating the performance measures such as expected number of customers in the system/queue and expected waiting time of customers in the system/queue.Some generalizations of the above models are also presented therein.
A generalization of M/M/ queueing system with service switches is considered in this paper in which the arrival rate   and the service rate   are both functions of , the number of customers present in the system.In real life situations   and   change whenever  changes, so that both arrival and departure have a bearing on the system state.
In many practical queueing systems, when there is a long queue, it is quite likely that a server will tend to work faster than when the queue is small.That is, the service rate   depends on , the number of customers present in the system.Similarly situations may occur where customers refuse to join the queue because of long waiting by seeing a large number of customers in the queue.That is, the arrival rate   depends on , the number of customers present in the system.These kinds of adaptive queueing systems where the arrival rate and the service rate depend on the number of customers present in the system are discussed in this paper.

Multiserver Multirate Discouraged Arrival Queueing System
A queue is an indication of congestion which we can be seen in a system or a network consisting of many systems.Congestion arises in many areas and our interest is to control the congestion in whatever situation it arises.Abdul Rasheed and Manoharan [11] used the concept of service switches as a tool to control congestion and use multiple servers and multiple service switches if congestion is very high.They discussed congestion control aspects when the arrival rate is constant and the service rate depends on the number in the system.In this paper we discuss the congestion control using service switches when both the arrival rate and service rate are the functions of the number of customers in the system.We consider a generalized model with  servers and two service switches at the point  1 and  2 ( 2 >  1 ) and hence the system works in three speeds, say, slow, medium, and fast.
Here work is performed at the slow rates until there are  1 customers in the system, at which point there is a switch to the medium rate and work is performed at the medium rate until there are  2 customers in the system at which point there is a switch to the fast rate.Here the arrival rate   is given as which means the customers will be discouraged from joining the queue and the service rate   is given by where  1 ≤  2 < .

Advances in Operations Research
The steady state probabilities are given by where  1 = / 1 ,  2 = / 2 , and  = /.The idle probability  0 can be obtained as After a careful manipulation of the infinite series on the right hand side of the above expression and further simplification, we get ) The expected queue size (  ) is given as where Therefore  3 becomes After some steps we get expected number of customers in the queue by using  1 ,  2 , and  3 as The expected system size () is given as that is, where Hence the expected number of customers in the system () is obtained as The effective arrival rate ( * ) can be obtained by the following summation schemes: The expected waiting times in the system are  = ( Similarly the expected waiting times in the queue are For illustrating the analytical feasibility of the methods proposed we consider the following hypothetical example situation.
Example 1.A young hard worker started a beauty parlour.Customers are taken on a first come first serve basis.Inside the beauty parlour, sitting facility is available for waiting customers and in front of the beauty parlour there is a vast parking area, so no limitation on the number of customers who can wait for service.But the number of arrivals depends on the number of customers already present in the beauty parlour.If arriving customers see a large number in the system he may not join the queue.Since the congestion is very high the young man appointed one more worker and he decides to run the beauty parlour at three speeds, say, slow, medium, and fast.At the slow speed, it takes 40 minutes, on the average; at the medium speed, it takes 30 minutes; and at the fast speed, it takes 20 minutes to cut the hair with service switches at 5 and 7.That is, up to 4 customers in the system the beauty parlour runs at the slow speed.If the number of customers is more than 4 but less than 7, the beauty parlour runs at the medium speed.If the number of customers is more than 6, the beauty parlour runs at the fast speed.That is,  1 = 5 and  2 = 7.The interarrival time of customers is 35 minutes.Now we can calculate the measures of effectiveness.
Figure 1 gives the graph of steady state probability of number of customers in the system.
Figure 2 gives the graph of steady state probability of number of customers in the system.
From this example we can observe that waiting time of the customers decreases if the values of the switch point decrease and also by increasing the number of servers.Some special cases of the generalized  server model and two service switches are discussed in the following section.

Special Cases
3.1.Model with  Servers and One Service Switch.If  1 =  2 and hence  1 =  2 (which means one switch), the model with  server and two service switches reduces to the  server model with one service switch at the point  and hence the system works in two speeds, say, slow and fast.The following results are deduced from the  server model and two service switches.The arrival rate   and service rate   are given by where  1 <  and  > .
The steady state probabilities are given by The idle probability  0 is obtained as The expected number of customers in the queue is The expected number of customers in the system is The effective arrival rate can be calculated as follows: Advances in Operations Research 7 The expected waiting time in the system  is given by and expected waiting time in the queue   is The steady state probability of  customers in the system is !! −  0 ,  ≥ . (26) The idle probability can be obtained as The expected number of customers in the system () is