Research Article Markovian Queueing System with...
Transcript of Research Article Markovian Queueing System with...
Research ArticleMarkovian Queueing System withDiscouraged Arrivals and Self-Regulatory Servers
K. V. Abdul Rasheed and M. Manoharan
Department of Statistics, University of Calicut, Kerala 673 635, India
Correspondence should be addressed to K. V. Abdul Rasheed; [email protected]
Received 28 September 2015; Revised 1 February 2016; Accepted 28 March 2016
Academic Editor: Ahmed Ghoniem
Copyright Β© 2016 K. V. Abdul Rasheed and M. Manoharan.This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.
We consider discouraged arrival of Markovian queueing systems whose service speed is regulated according to the number ofcustomers in the system. We will reduce the congestion in two ways. First we attempt to reduce the congestion by discouraging thearrivals 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 π
1, π2, and π (π
1β€ π2
< π), say, slow, medium,and fast rates, respectively. If the number of customers in the system exceeds a particular point πΎ
1or πΎ2, the server switches to the
medium or fast rate, respectively. For this adaptive queueing system the steady state probabilities are derived and some performancemeasures such as expected number in the system/queue and expected waiting time in the system/queue are obtained. Multipleserver discouraged arrival model having one service switch and single server discouraged arrival model having one and two serviceswitches are obtained as special cases. A Matlab program of the model is presented and numerical illustrations are given.
1. Introduction
Queueing theory plays an important role in modeling reallife problems involving congestion in wide areas of appliedsciences. A customer decides to join the queue only whena short wait is expected and if the wait has been sufficientlysmall he tends to remain in the queue; otherwise the customerleaves the system and then the customer is said to beimpatient. When this impatience increases and customersleave before being served, some remedial actions must betaken to reduce the congestion in the system.
Balking and reneging are the forms of impatience. If acustomer decides not to enter the queue upon arrival byseeing a long queue, the customer is said to have balked. Acustomer may enter the queue but after a while loses patienceand decides to leave and then the customer is said to bereneged.
In the study of queueing system the server is usuallyassumed to work at constant speed regardless of the amountof work existing. But in real life situation this assumptionmaynot always be appropriate as the system size may affect thesystem performance. That is, the servers adapt to the system
state by increasing the speed to clear the queue or decrease thespeed when fatigued, which means the service rate dependson the system size; see, for example, Jonckheere and Borst [1].Similarly we can see that the arrivals of customers into thesystem may also affect the level of congestion. For example,customers impatiencewill affect the arrivals of customers intothe system. A queueing system where the arrival rate and/orservice rate depends on the system size is called adaptivequeueing system.
Sometimes the arrival rate of customers into the systemdepends on the system size instead of a constant rate.Discouraged arrival is one formof state dependence. Here thearrivals get discouraged from joining the queue when moreandmore people are present in the system.We canmodel thiseffect by taking the birth and death coefficients, respectively,as ππ
= π/(π + 1), where π is the number of customers inthe system and π is a positive constant, and π
π= π, where
π is the constant service rate. Thus the arrival rate of thequeueing system is decreased by this discouragement; con-sequently the congestion of the system decreases. This typeof queueing systems is called discouraged arrival queueingsystem.
Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2016, Article ID 2456135, 11 pageshttp://dx.doi.org/10.1155/2016/2456135
2 Advances in Operations Research
Applications of queueing with impatience can be seenin traffic modeling, business and industries, computer com-munication, health sectors, medical sciences, and so forth.Customers with impatience and discouragement have theirown impact on the system performance of standard queueingsystems. It is important to note that customers impatiencehas a very negative impact on the queueing system underinvestigation.
We can see a large number of articles which discussingthe congestion control of queueing systems. For example, see,Haight [2], Ancker Jr. and Gafarian [3, 4], Boots and Tijms[5], Liu and Kulkarni [6], Kc and Terwiesch [7], Wang et al.[8], Kapodistria [9], and Kumar and Sharma [10].
In this paper we attempt to reduce the congestion intwo ways. First we attempt to reduce the congestion bydiscouraging the arrivals of customers from joining thequeue. In the second way we further reduce the congestionby introducing the concept of service switches which isdiscussed in Abdul Rasheed and Manoharan [11].
Discouraged arrival system was studied by manyresearchers. Raynolds [12] presented multiserver queueingmodel with discouragement and obtained equilibriumdistribution of queue length and derived other performancemeasures from it. A finite capacity M/G/1 queueing modelwhere the arrival and the service rates were arbitraryfunctions of the current number of customers in the systemwas studied by Courtois and Georges [13]. Natvig [14]studied the single server birth-death queueing process withstate dependent parameters π
π= π/(π + 1), π β₯ 0, and
ππ
= π, π β₯ 1. Van Doorn [15] obtained exact expressionsfor transient state probabilities of the birth-death processwith parameters π
π= π/(π + 1), π β₯ 0, and π
π= π, π β₯ 1.
Parthasarathy and Selvaraju [16] obtained the transientsolution to a state dependent birth-death queueing model inwhich potential customers are discouraged by queue length.
Narayanan [17] studied different linear and nonlinearstate dependent Markovian queueing models, in whicharrival rates and/or service rates are nonlinear and theirmodified forms obtain the transient/steady state probabilitydistribution 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 performancemeasures of state dependent queueing models. Ammar et al.[20] studied single server finite capacity Markovian queuewith discouraged arrivals and reneging usingmatrix method.
Abdul Rasheed andManoharan [11] studied a Markovianqueueing system in which the arrival rate is constant andthe service rate depends on the number of customers in thesystem.The server speed is regulated according to the systemsize by introducing the service switches to the model. Theauthors analyzed the system by calculating the performancemeasures such as expected number of customers in thesystem/queue and expected waiting time of customers in thesystem/queue. Some generalizations of the above models arealso presented therein.
A generalization of M/M/πΆ queueing system with serviceswitches is considered in this paper in which the arrival rateππand the service rate π
πare both functions of π, the number
of customers present in the system. In real life situations ππ
and ππchange whenever π 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 fasterthan when the queue is small. That is, the service rate π
π
depends on π, the number of customers present in the system.Similarly situationsmay occur where customers refuse to jointhe queue because of long waiting by seeing a large numberof customers in the queue.That is, the arrival rate π
πdepends
on π, the number of customers present in the system. Thesekinds of adaptive queueing systems where the arrival rate andthe service rate depend on the number of customers presentin the system are discussed in this paper.
2. Multiserver Multirate DiscouragedArrival Queueing System
A queue is an indication of congestion which we can beseen in a system or a network consisting of many systems.Congestion arises in many areas and our interest is to controlthe congestion in whatever situation it arises. Abdul Rasheedand Manoharan [11] used the concept of service switches asa tool to control congestion and use multiple servers andmultiple service switches if congestion is very high. Theydiscussed congestion control aspects when the arrival rate isconstant and the service rate depends on the number in thesystem. In this paper we discuss the congestion control usingservice switcheswhenboth the arrival rate and service rate arethe functions of the number of customers in the system. Weconsider a generalized model with πΆ servers and two serviceswitches at the point πΎ
1and πΎ
2(πΎ2
> πΎ1) and hence the
system works in three speeds, say, slow, medium, and fast.Here work is performed at the slow rates until there are πΎ
1
customers in the system, at which point there is a switch to themedium rate and work is performed at the medium rate untilthere are πΎ
2customers in the system at which point there is
a switch to the fast rate. Here the arrival rate ππis given as
ππ
=π
π + 1, π β₯ 0, (1)
which means the customers will be discouraged from joiningthe queue and the service rate π
πis given by
ππ
=
{{{{{{{
{{{{{{{
{
ππ1, 1 β€ π < πΆ,
πΆπ1, πΆ β€ π < πΎ
1,
πΆπ2, πΎ1
β€ π < πΎ2,
πΆπ, π β₯ πΎ2,
(2)
where π1
β€ π2
< π.
Advances in Operations Research 3
The steady state probabilities are given by
ππ
=
{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{
{
ππ
1
(π!)2
π0, 0 β€ π < πΆ,
ππ
1
π!πΆ!πΆπβπΆπ0, πΆ β€ π < πΎ
1,
ππΎ1β1
1ππβπΎ1+1
2
π!πΆ!πΆπβπΆπ0, πΎ
1β€ π < πΎ
2,
ππΎ1β1
1ππΎ2βπΎ1
2ππβπΎ2+1
π!πΆ!πΆπβπΆπ0, π β₯ πΎ
2,
(3)
where π1
= π/π1, π2
= π/π2, and π = π/π.
The idle probability π0can be obtained as
π0
= [
πΆβ1
β
π=0
ππ
1
(π!)2
+
πΎ1β1
β
π=πΆ
ππ
1
π!πΆ!πΆπβπΆ+
πΎ2β1
β
π=πΎ1
ππΎ1β1
1ππβπΎ1+1
2
π!πΆ!πΆπβπΆ
+
β
β
π=πΎ2
ππΎ1β1
1ππΎ2βπΎ1
2ππβπΎ2+1
π!πΆ!πΆπβπΆ]
β1
.
(4)
After a careful manipulation of the infinite series on the righthand side of the above expression and further simplification,we get
π0
= [(ππΎ2βπΎ1
2π1βπΎ2ππ/πΆ
πΆ!πΆβπΆ
+
πΆβ1
β
π=0
(ππβπΎ1+1
1
(π!)2
βππΎ2βπΎ1
2ππβπΎ2+1
πΆ!πΆπβπΆ)
+
πΎ1β1
β
π=πΆ
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
π!πΆ!πΆπβπΆ)
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
π!πΆ!πΆπβπΆ) ππΎ2βπΎ1
2) ππΎ1β1
1]
β1
.
(5)
The expected queue size (πΏπ) is given as
πΏπ
=
πΎ1β1
β
π=πΆ+1
(π β πΆ) ππ
+
πΎ2β1
β
π=πΎ1
(π β πΆ) ππ
+
β
β
π=πΎ2
(π β πΆ) ππ
= πΏπ1
+ πΏπ2
+ πΏπ3
,
(6)
where
πΏπ1
=
πΎ1β1
β
π=πΆ+1
ππ
1
(π β 1)!πΆ!πΆπβπΆβ
πΎ1β1
β
π=πΆ+1
ππ
1
π! (πΆ β 1)!πΆπβπΆ,
πΏπ2
=
πΎ2β1
β
π=πΎ1
ππΎ1β1
1ππβπΎ1+1
2
(π β 1)!πΆ!πΆπβπΆπ0
β
πΎ2β1
β
π=πΎ1
ππΎ1β1
1ππβπΎ1+1
2
π! (πΆ β 1)!πΆπβπΆπ0,
πΏπ3
=
β
β
π=πΎ2
πππ
β πΆ
β
β
π=πΎ2
ππ
= πΏπ31
β πΏπ32
,
πΏπ31
=ππΎ1β1
1ππΎ2βπΎ1
2π2βπΎ2
πΆ!πΆ1βπΆπ0
β
β
π=πΎ2
ππβ1
πΆπβ1 (π β 1)!,
πΏπ31
= [ππΎ1β1
1ππΎ2βπΎ1
2π2βπΎ2ππ/πΆ
π0
πΆ!πΆ1βπΆ
βππΎ1β1
1ππΎ2βπΎ1
2π0
πΆ!
πΆ
β
π=1
ππβπΎ2+1
(π β 1)!πΆπβπΆ
βππΎ1β1
1ππΎ2βπΎ1
2π0
πΆ!
πΎ1β1
β
π=πΆ+1
ππβπΎ2+1
(π β 1)!πΆπβπΆ
βππΎ1β1
1ππΎ2βπΎ1
2π0
πΆ!
πΎ2β1
β
π=πΎ1
ππβπΎ2+1
(π β 1)!πΆπβπΆ] ,
πΏπ32
= πΆ [
β
β
π=0
ππ
β
πΆ
β
π=1
ππ
β π0
β
πΎ1β1
β
π=πΆ+1
ππ
β
πΎ2β1
β
π=πΎ1
ππ]
= [πΆ (1 β π0) β πΆ
πΆ
β
π=1
ππ
1
(π!)2
π0
β πΆ
πΎ1β1
β
π=πΆ+1
ππ
1
π!πΆ!πΆπβπΆπ0
β πΆ
πΎ2β1
β
π=πΎ1
ππΎ1β1
1ππβπΎ1+1
2
π!πΆ!πΆπβπΆπ0] .
(7)
Therefore πΏπ3becomes
πΏπ3
= [ππΎ1β1
1ππΎ2βπΎ1
2π2βπΎ2ππ/πΆ
π0
πΆ!πΆ1βπΆ
βππΎ1β1
1ππΎ2βπΎ1
2π0
πΆ!
πΆ
β
π=1
ππβπΎ2+1
(π β 1)!πΆπβπΆ
βππΎ1β1
1ππΎ2βπΎ1
2π0
πΆ!
πΎ1β1
β
π=πΆ+1
ππβπΎ2+1
(π β 1)!πΆπβπΆ
βππΎ1β1
1ππΎ2βπΎ1
2π0
πΆ!
πΎ2β1
β
π=πΎ1
ππβπΎ2+1
(π β 1)!πΆπβπΆβ πΆ (1 β π
0)
4 Advances in Operations Research
+ πΆ
πΆ
β
π=1
ππ
1
(π!)2
π0
+ πΆ
πΎ1β1
β
π=πΆ+1
ππ
1
π!πΆ!πΆπβπΆπ0
+ πΆ
πΎ2β1
β
π=πΎ1
ππΎ1β1
1ππβπΎ1+1
2
π!πΆ!πΆπβπΆπ0] .
(8)
After some steps we get expected number of customers in thequeue by using πΏ
π1, πΏπ2, and πΏ
π3as
πΏπ
= [ππΎ1β1
1ππΎ2βπΎ1
2π2βπΎ2ππ/πΆ
π0
πΆ!πΆ1βπΆ
+
πΆ
β
π=1
(πΆππβπΎ1+1
1
(π!)2
βππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ) ππΎ1β1
1π0
+
πΎ1β1
β
π=πΆ+1
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ) ππΎ1β1
1π0
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ) ππΎ1β1
1ππΎ2βπΎ1
2π0
β πΆ (1 β π0)] .
(9)
The expected system size (πΏ) is given as
πΏ =
πΆ
β
π=0
πππ
+
πΎ1β1
β
π=πΆ+1
πππ
+
πΎ2β1
β
π=πΎ1
πππ
+
β
β
π=πΎ2
πππ
= [
πΆ
β
π=0
πππ
+
πΎ1β1
β
π=πΆ+1
(π β πΆ) ππ
+
πΎ2β1
β
π=πΎ1
(π β πΆ) ππ
+
β
β
π=πΎ2
(π β πΆ) ππ
+ πΆ (
πΎ1β1
β
π=πΆ+1
ππ
+
πΎ2β1
β
π=πΎ1
ππ
+
β
β
π=πΎ2
ππ)] ;
(10)
that is,
πΏ = πΏπ
+ π, (11)
where
π =
πΆ
β
π=0
πππ
+ πΆ
β
β
π=πΆ+1
ππ,
π = [
πΆ
β
π=0
πππ
+ πΆ
πΎ1β1
β
π=πΆ+1
ππ
+ πΆ
πΎ2β1
β
π=πΎ1
ππ
+ πΆ
β
β
π=0
ππ
β πΆ
πΆ
β
π=0
ππ
β πΆ
πΎ1β1
β
π=πΆ+1
ππ
β πΆ
πΎ2β1
β
π=πΎ1
ππ] = πΆ +
πΆ
β
π=0
(π β πΆ)ππ
1
(π!)2
π0.
(12)
Hence the expected number of customers in the system (πΏ) isobtained as
πΏ = [ππΎ1β1
1ππΎ2βπΎ1
2π2βπΎ2ππ/πΆ
π0
πΆ!πΆ1βπΆ
+
πΆ
β
π=1
(πΆππβπΎ1+1
1
(π!)2
βππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ) ππΎ1β1
1π0
+
πΎ1β1
β
π=πΆ+1
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ) ππΎ1β1
1π0
β πΆ (1 β π0)
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ) ππΎ1β1
1ππΎ2βπΎ1
2π0
+ πΆ
+
πΆ
β
π=0
(π β πΆ)ππ
1
(π!)2
π0] = [
ππΎ2βπΎ1
2π2βπΎ2ππ/πΆ
πΆ!πΆ1βπΆ
+
πΆ
β
π=1
(ππβπΎ1+1
1
(π β 1)!π!β
ππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ)
+
πΎ1β1
β
π=πΆ+1
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ)
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ) ππΎ2βπΎ1
2] ππΎ1β1
1π0.
(13)
The effective arrival rate (πβ
) can be obtained by the followingsummation schemes:
πβ
=
πΆβ1
β
π=0
ππππ
+
πΎ1β1
β
π=πΆ
ππππ
+
πΎ2β1
β
π=πΎ1
ππππ
+
β
β
π=πΎ2
ππππ
= [
πΆβ1
β
π=0
ππ
1
(π + 1)!π!+
πΎ1β1
β
π=πΆ
ππ
1
(π + 1)!πΆ!πΆπβπΆ
+
πΎ2β1
β
π=πΎ1
ππΎ1β1
1ππβπΎ1+1
2
(π + 1)!πΆ!πΆπβπΆ
+ππΎ1β1
1ππΎ2βπΎ1
2πβπΎ2
πΆ!πΆβπΆβ1(
β
β
π=0
ππ+1
(π + 1)!β
πΆβ1
β
π=0
ππ+1
(π + 1)!
β
πΎ1β1
β
π=πΆ
ππ+1
(π + 1)!)] ππ
0= [
[
ππΎ2βπΎ1
2πβπΎ2 (ππ/πΆ
β 1)
πΆ!πΆβπΆβ1
+
πΆβ1
β
π=0
(ππβπΎ1+1
1
(π + 1)!π!β
ππΎ2βπΎ1
2ππβπΎ2+1
(π + 1)!πΆ!πΆπβπΆ)
Advances in Operations Research 5
+
πΎ1β1
β
π=πΆ
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ1+1
(π + 1)!πΆ!πΆπβπΆ)
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π + 1)!πΆ!πΆπβπΆ) ππΎ2βπΎ1
2]
]
ππΎ1β1
1ππ0.
(14)
The expected waiting times in the system are
π = (ππΎ2βπΎ1
2π2βπΎ2ππ/πΆ
πΆ!πΆ1βπΆ
+
πΆ
β
π=1
(ππβπΎ1+1
1
(π β 1)!π!β
ππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ)
+
πΎ1β1
β
π=πΆ+1
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ)
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ) ππΎ2βπΎ1
2)
β ((ππΎ2βπΎ1
2πβπΎ2 (ππ/πΆ
β 1)
πΆ!πΆβπΆβ1
+
πΆβ1
β
π=0
(ππβπΎ1+1
1
(π + 1)!π!β
ππΎ2βπΎ1
2ππβπΎ2+1
(π + 1)!πΆ!πΆπβπΆ)
+
πΎ1β1
β
π=πΆ
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ1+1
(π + 1)!πΆ!πΆπβπΆ)
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π + 1)!πΆ!πΆπβπΆ) ππΎ2βπΎ1
2) π)
β1
.
(15)
Similarly the expected waiting times in the queue are
ππ
= ((ππΎ2βπΎ1
2π2βπΎ2ππ/πΆ
πΆ!πΆ1βπΆ
+
πΆ
β
π=1
(πΆππβπΎ1+1
1
(π!)2
βππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ)
+
πΎ1β1
β
π=πΆ+1
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ)
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π β 1)!πΆ!πΆπβπΆ) ππΎ2βπΎ1
2) ππΎ1β1
1π0
β πΆ (1 β π0)) ((
ππΎ2βπΎ1
2πβπΎ2 (ππ/πΆ
β 1)
πΆ!πΆβπΆβ1
+
πΆβ1
β
π=0
(ππβπΎ1+1
1
(π + 1)!π!β
ππΎ2βπΎ1
2ππβπΎ2+1
(π + 1)!πΆ!πΆπβπΆ)
+
πΎ1β1
β
π=πΆ
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ1+1
(π + 1)!πΆ!πΆπβπΆ)
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π + 1)!πΆ!πΆπβπΆ) ππΎ2βπΎ1
2) ππΎ1β1
1ππ0)
β1
.
(16)
For illustrating the analytical feasibility of the methodsproposed we consider the following hypothetical examplesituation.
Example 1. A young hard worker started a beauty parlour.Customers are taken on a first come first serve basis. Insidethe beauty parlour, sitting facility is available for waitingcustomers and in front of the beauty parlour there is a vastparking area, so no limitation on the number of customerswho can wait for service. But the number of arrivals dependson the number of customers already present in the beautyparlour. If arriving customers see a large number in thesystem he may not join the queue. Since the congestion isvery high the young man appointed one more worker and hedecides to run the beauty parlour at three speeds, say, slow,medium, and fast. At the slow speed, it takes 40 minutes, onthe average; at the medium speed, it takes 30 minutes; and atthe fast speed, it takes 20 minutes to cut the hair with serviceswitches at 5 and 7. That is, up to 4 customers in the systemthe beauty parlour runs at the slow speed. If the number ofcustomers is more than 4 but less than 7, the beauty parlourruns at the medium speed. If the number of customers ismore than 6, the beauty parlour runs at the fast speed. Thatis, πΎ1
= 5 and πΎ2
= 7. The interarrival time of customers is35 minutes.
Now we can calculate the measures of effectiveness.If πΎ1
= 5, πΎ2
= 7, π = 1/35, π1
= 1/40, π2
= 1/30,π = 1/20, and πΆ = 2. Then we get π
β= 0.0190, π
0= 0.3935,
πΏ = 0.795, πΏπ
= 0.032, π = 41.68 minutes, and ππ
= 1.7
minutes.Figure 1 gives the graph of steady state probability of
number of customers in the system.If πΎ1
= 3 and πΎ2
= 5 and using the same parameterswe get π
β= 0.0190, π
0= 0.3966, πΏ = 0.776, πΏ
π= 0.022,
π = 40.46 minutes, and ππ
= 1.18 minutes.If πΎ1
= 5, πΎ2
= 7, π = 1/35, π1
= 1/40, π2
= 1/30,π = 1/20, and πΆ = 1. Then we get π
β= 0.0170, π
0= 0.3195,
πΏ = 1.134, πΏπ
= 0.4544, π = 66.56 minutes, and ππ
= 26.65
minutes.Figure 2 gives the graph of steady state probability of
number of customers in the system.From this examplewe can observe that waiting time of the
customers decreases if the values of the switch point decreaseand also by increasing the number of servers.
Some special cases of the generalized πΆ server model andtwo service switches are discussed in the following section.
6 Advances in Operations Research
Steady state probability of number of customers in the system
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Stea
dy st
ate p
roba
bilit
y
5 10 15 200Number of customers in the system
Figure 1: [πΆ = 1, π = 1/35, π1
= 1/40, π2
= 1/30, π =
1/20, πΎ1
= 5, πΎ2
= 7].
Steady state probability of number of customers in the system
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Stea
dy st
ate p
roba
bilit
y
5 10 15 200Number of customers in the system
Figure 2: [πΆ = 2, π = 1/35, π1
= 1/40, π2
= 1/30, π =
1/20, πΎ1
= 5, πΎ2
= 7].
3. Special Cases
3.1. Model with πΆ Servers and One Service Switch. If π1
= π2
and hence π1
= π2(which means one switch), the model with
πΆ server and two service switches reduces to the πΆ servermodel with one service switch at the point πΎ and hence thesystem works in two speeds, say, slow and fast. The followingresults are deduced from the πΆ server model and two serviceswitches. The arrival rate π
πand service rate π
πare given by
ππ
=π
π + 1, π β₯ 0,
ππ
=
{{{{
{{{{
{
ππ1, 1 β€ π < πΆ,
πΆπ1, πΆ β€ π < πΎ,
πΆπ, π β₯ πΎ,
(17)
where π1
< π and πΎ > πΆ.
The steady state probabilities are given by
ππ
=
{{{{{{{{{{
{{{{{{{{{{
{
ππ
1
(π!)2
π0, 0 β€ π < πΆ,
ππ
1
π!πΆ!πΆπβπΆπ0, πΆ β€ π < πΎ,
ππΎβ1
1ππβπΎ+1
π!πΆ!πΆπβπΆπ0, π β₯ πΎ.
(18)
The idle probability π0is obtained as
π0
= [
πΆβ1
β
π=0
(ππ
1
(π!)2
βππΎβ1
1ππβπΎ+1
π!πΆ!πΆπβπΆ)
+
πΎβ1
β
π=πΆ
(ππ
1
π!πΆ!πΆπβπΆβ
ππΎβ1
1ππβπΎ+1
π!πΆ!πΆπβπΆ)
+ππΎβ1
1π1βπΎ
ππ/πΆ
πΆ!πΆβπΆ]
β1
.
(19)
The expected number of customers in the queue is
πΏπ
= [ππΎβ1
1π2βπΎ
ππ/πΆ
π0
πΆ!πΆ1βπΆ
+
πΆ
β
π=1
(πΆππβπΎ+1
1
(π!)2
βππβπΎ+1
(π β 1)!πΆ!πΆπβπΆ) ππΎβ1
1π0
+
πΎβ1
β
π=πΆ+1
(ππβπΎ+1
1β ππβπΎ+1
(π β 1)!πΆ!πΆπβπΆ) ππΎβ1
1π0
β πΆ (1 β π0)] .
(20)
The expected number of customers in the system is
πΏ = [ππΎβ1
1π2βπΎ
ππ/πΆ
πΆ!πΆ1βπΆ
+
πΆ
β
π=1
(ππβπΎ+1
1
(π β 1)!π!β
ππβπΎ+1
(π β 1)!πΆ!πΆπβπΆ) ππΎβ1
1
+
πΎβ1
β
π=πΆ+1
(ππβπΎ+1
1β ππβπΎ+1
(π β 1)!πΆ!πΆπβπΆ) ππΎβ1
1] π0.
(21)
The effective arrival rate can be calculated as follows:
πβ
=
πΆβ1
β
π=0
ππππ
+
πΎβ1
β
π=πΆ
ππππ
+
β
β
π=πΎ
ππππ
= [
πΆβ1
β
π=0
(ππβπΎ+1
1
(π + 1)!π!β
ππβπΎ+1
πΆ! (π + 1)!πΆπβπΆ)
+
πΎβ1
β
π=πΆ
(ππβπΎ+1
1β ππβπΎ+1
(π + 1)!πΆ!πΆπβπΆ) +
πβπΎ
(ππ/πΆ
β 1)
πΆ!πΆβπΆβ1]
β ππΎβ1
1ππ0.
(22)
Advances in Operations Research 7
The expected waiting time in the system π is given by
π = [
[
(π2βπΎ
ππ/πΆ
πΆ!πΆ1βπΆ
+
πΆ
β
π=1
(ππβπΎ+1
1
(π β 1)!π!β
ππβπΎ+1
(π β 1)!πΆ!πΆπβπΆ)
+
πΎβ1
β
π=πΆ+1
(ππβπΎ+1
1β ππβπΎ+1
(π β 1)!πΆ!πΆπβπΆ))
β ((
πΆβ1
β
π=0
(ππβπΎ+1
1
(π + 1)!π!β
ππβπΎ+1
πΆ! (π + 1)!πΆπβπΆ)
+
πΎβ1
β
π=πΆ
(ππβπΎ+1
1β ππβπΎ+1
(π + 1)!πΆ!πΆπβπΆ) +
πβπΎ
(ππ/πΆ
β 1)
πΆ!πΆβπΆβ1)
β π)
β1
]
]
,
(23)
and expected waiting time in the queue ππis
ππ
= [
[
(ππΎβ1
1π2βπΎ
ππ/πΆ
π0
πΆ!πΆ1βπΆ
+
πΆ
β
π=1
(πΆππβπΎ+1
1
(π!)2
βππβπΎ+1
(π β 1)!πΆ!πΆπβπΆ) ππΎβ1
1π0
+
πΎβ1
β
π=πΆ+1
(ππβπΎ+1
1β ππβπΎ+1
(π β 1)!πΆ!πΆπβπΆ) ππΎβ1
1π0
β πΆ (1 β π0))
β ((
πΆβ1
β
π=0
(ππβπΎ+1
1
(π + 1)!π!β
ππβπΎ+1
πΆ! (π + 1)!πΆπβπΆ)
+
πΎβ1
β
π=πΆ
(ππβπΎ+1
1β ππβπΎ+1
(π + 1)!πΆ!πΆπβπΆ) +
πβπΎ
(ππ/πΆ
β 1)
πΆ!πΆβπΆβ1)
β ππΎβ1
1ππ0)
β1
]
]
.
(24)
3.2.Multiple Server AdaptiveQueueing System. Ifπ1
= π2
= π
and hence π1
= π2
= π (which means no switch), the modelwithπΆ server and two service switches reduces to theπΆ servermodel with no service switch. The following results can beobtained from the πΆ server model and two service switches.The service rate π
πis given by
ππ
={
{
{
ππ, 1 β€ π < πΆ,
πΆπ, π β₯ πΆ.
(25)
The steady state probability of π customers in the system is
ππ
=
{{{
{{{
{
ππ
(π!)2
π0, 0 β€ π < πΆ,
ππ
π!πΆ!πΆπβπΆπ0, π β₯ πΆ.
(26)
The idle probability can be obtained as
π0
= [
πΆβ1
β
π=0
ππ
(π!)2
+
β
β
π=πΆ
ππ
π!πΆ!πΆπβπΆ]
β1
. (27)
The expected number of customers in the system (πΏ) is
πΏ = [
πΆ
β
π=1
ππ
(π β 1)! (π!)+
β
β
π=πΆ+1
ππ
(π β 1)!πΆ!πΆπβπΆ] π0. (28)
The expected number of customers in the queue (πΏπ) is
πΏπ
= [
β
β
π=πΆ+1
ππ
(π β 1)!πΆ!πΆπβπΆβ
β
β
π=πΆ+1
ππ
π! (πΆ β 1)!πΆπβπΆ]
β π0.
(29)
The effective arrival rate πβ can be obtained by
πβ
= [(ππ/πΆ
β 1)
ππΆ!πΆβπΆβ1
β
πΆβ1
β
π=0
(ππ
πΆ! (π + 1)!πΆπβπΆβ
ππ
π! (π + 1)!)] ππ
0.
(30)
Waiting times of customers in the system (π) are given as
π = [(βπΆ
π=1(ππ/ (π β 1)! (π!)) + β
β
π=πΆ+1(ππ/ (π β 1)!πΆ!πΆ
πβπΆ))
((ππ/πΆ β 1) /ππΆ!πΆβπΆβ1 β βπΆβ1
π=0[ππ/πΆ! (π + 1)!πΆπβπΆ β ππ/π! (π + 1)!]) π
] . (31)
Waiting times of customers in the queue (ππ) are given as
ππ
= [(ββ
π=πΆ+1(ππ/ (π β 1)!πΆ!πΆ
πβπΆ) β ββ
π=πΆ+1(ππ/π! (πΆ β 1)!πΆ
πβπΆ))
((ππ/πΆ β 1) /ππΆ!πΆβπΆβ1 β βπΆβ1
π=0[ππ/πΆ! (π + 1)!πΆπβπΆ β ππ/π! (π + 1)!]) π
] . (32)
8 Advances in Operations Research
3.3. Model with Single Server and Two Service Switches. IfπΆ =
1, the model with πΆ server and two service switches reducesto the single server model with two service switches at thepoints πΎ
1and πΎ
2(πΎ1
< πΎ2). The following results can be
obtained from the multiserver model with two switches. Theservice rate π
πis given by
ππ
=
{{{{
{{{{
{
π1, 1 β€ π < πΎ
1,
π2, πΎ1
β€ π < πΎ2,
π, π β₯ πΎ2.
(33)
The steady state probability of π customers in the system is
ππ
=
{{{{{{{{{{{
{{{{{{{{{{{
{
ππ
1
π!π0, 0 β€ π < πΎ
1,
ππΎ1β1
1ππβπΎ1+1
2
π!π0, πΎ
1β€ π < πΎ
2,
ππΎ1β1
1ππΎ2βπΎ1
2ππβπΎ2+1
π!π0, π β₯ πΎ
2.
(34)
We have the idle probability
π0
= [
[
πΎ1β1
β
π=0
(ππΎ1β1
1(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
)
π!)
+
πΎ2β1
β
π=πΎ1
(ππΎ1β1
1ππΎ2βπΎ1
2(ππβπΎ2+1
2β ππβπΎ2+1
)
π!)
+ ππΎ1β1
1ππΎ2βπΎ1
2π1βπΎ2ππ]
]
β1
.
(35)
The expected number of customers in the system (πΏ) is givenby
πΏ = [
[
πΎ1β1
β
π=1
(ππΎ1β1
1(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
)
(π β 1)!)
+
πΎ2β1
β
π=πΎ1
(ππΎ1β1
1ππΎ2βπΎ1
2(ππβπΎ2+1
2β ππβπΎ2+1
)
(π β 1)!)
+ ππΎ1β1
1ππΎ2βπΎ1
2π2βπΎ2ππ]
]
π0.
(36)
The expected number of customers in the queue (πΏπ) is given
by
πΏπ
= [
[
ππΎ1β1
1ππΎ2βπΎ1
2π2βπΎ2πππ0
+
πΎ1β1
β
π=1
((ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
)
(π β 1)!) ππΎ1β1
1π0
+
πΎ2β1
β
π=πΎ1
((ππβπΎ2+1
2β ππβπΎ2+1
)
(π β 1)!) ππΎ1β1
1ππΎ2βπΎ1
2π0
β (1 β π0)]
]
.
(37)
We can see that πΏπ
= πΏ β (1 β π0) from the above equation.
The effective arrival rate now can be obtained as
πβ
= [ππΎ1β1
1ππΎ2βπΎ1
2πβπΎ2 (ππ
β 1)
+
πΎ1β1
β
π=0
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ1+1
(π + 1)!) ππΎ1β1
1
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π + 1)!) ππΎ1β1
1ππΎ2βπΎ1
2] ππ0.
(38)
The expected waiting time in the system is
π = [
[
(ππΎ1β1
1ππΎ2βπΎ1
2π2βπΎ2ππ
+
πΎ1β1
β
π=1
[
[
ππΎ1β1
1(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
)
(π β 1)!
]
]
+
πΎ2β1
β
π=πΎ1
[
[
ππΎ1β1
1ππΎ2βπΎ1
2(ππβπΎ2+1
2β ππβπΎ2+1
)
(π β 1)!
]
]
)
β ((ππΎ1β1
1ππΎ2βπΎ1
2πβπΎ2 (ππ
β 1)
+
πΎ1β1
β
π=0
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ1+1
(π + 1)!) ππΎ1β1
1
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π + 1)!) ππΎ1β1
1ππΎ2βπΎ1
2) π)
β1
]
]
.
(39)
Advances in Operations Research 9
The expected waiting time in the queue is
ππ
= [
[
(ππΎ1β1
1ππΎ2βπΎ1
2π2βπΎ2πππ0
+
πΎ1β1
β
π=1
((ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ2+1
)
(π β 1)!) ππΎ1β1
1π0
+
πΎ2β1
β
π=πΎ1
((ππβπΎ2+1
2β ππβπΎ2+1
)
(π β 1)!) ππΎ1β1
1ππΎ2βπΎ1
2π0
β (1
β π0)) ((π
πΎ1β1
1ππΎ2βπΎ1
2πβπΎ2 (ππ
β 1)
+
πΎ1β1
β
π=0
(ππβπΎ1+1
1β ππΎ2βπΎ1
2ππβπΎ1+1
(π + 1)!) ππΎ1β1
1
+
πΎ2β1
β
π=πΎ1
(ππβπΎ2+1
2β ππβπΎ2+1
(π + 1)!) ππΎ1β1
1ππΎ2βπΎ1
2)
β ππ0)
β1
]
]
.
(40)
Also we can establish the relationship π = ππ
+ (1 β π0)/πβ,
so the expected service time is (1 β π0)/πβ
.
3.4. Model with Single Server and One Service Switch. If πΆ =
1, π1
= π, and hence π1
= π, the model with πΆ server andtwo service switches reduces to the single server model withone service switch at the point πΎ. We obtained the followingresults from the πΆ server two switch models. The service rateππis given as
ππ
={
{
{
π1, 1 β€ π < πΎ,
π, π β₯ πΎ.
(41)
The steady state probabilities are given by
ππ
=
{{{{{
{{{{{
{
ππ
1
π!π0, 0 β€ π < πΎ,
ππΎβ1
1ππβπΎ+1
π!π0, π β₯ πΎ.
(42)
The idle probability can be obtained from the resultββ
π=0ππ
=
1, as
π0
= [ππΎβ1
1(π1βπΎ
ππ
β
πΎβ1
β
π=0
(ππβπΎ+1
β ππβπΎ+1
1)
π!)]
β1
. (43)
The expected number of customers in the system (πΏ) is
πΏ = [ππΎβ1
1π2βπΎ
ππ
+
πΎβ1
β
π=1
ππΎβ1
1(
ππβπΎ+1
1β ππβπΎ+1
(π β 1)!)] π0. (44)
The expected number of customers in the queue (πΏπ) is
πΏπ
= [(πππ2βπΎ
β (π2βπΎ
β π2βπΎ
1)
β
πΎβ1
β
π=2
(ππβπΎ+1
β ππβπΎ+1
1)
(π β 1)!) ππΎβ1
1π0
β (1 β π0)] .
(45)
The effective arrival rate πβ can be obtained as
πβ
= [ππΎβ1
1πβπΎ
(ππ
β 1)
+
πΎβ1
β
π=0
(ππβπΎ+1
1β ππβπΎ+1
(π + 1)!) ππΎβ1
1] ππ0.
(46)
The expected waiting time in the system can be obtained as
π = [(ππΎβ1
1π2βπΎ
ππ
+ βπΎβ1
π=1ππΎβ1
1((ππβπΎ+1
1β ππβπΎ+1
) / (π β 1)!))
(ππΎβ1
1πβπΎ (ππ β 1) + β
πΎβ1
π=0((ππβπΎ+1
1β ππβπΎ+1) / (π + 1)!) π
πΎβ1
1) π
] . (47)
Similarly the expected waiting time of customers in the queuecan be obtained as
ππ
= [((πππ2βπΎ
β (π2βπΎ
β π2βπΎ
1) β βπΎβ1
π=2((ππβπΎ+1
β ππβπΎ+1
1) / (π β 1)!)) π
πΎβ1
1π0
β (1 β π0))
(ππΎβ1
1πβπΎ (ππ β 1) + β
πΎβ1
π=0((ππβπΎ+1
1β ππβπΎ+1) / (π + 1)!) π
πΎβ1
1) ππ0
] . (48)
10 Advances in Operations Research
3.5. Single Server Adaptive Queueing System. If πΆ = 1, π1
=
π2
= π, and hence π1
= π2
= π, the model with πΆ serverand two service switches reduce to the single server modelwith no service switch and hence the following results can beobtained from the πΆ server model and two service switches.The service rate π
πis given by π
π= π, π β₯ 1.
The steady state probability of π customers in the systemis
ππ
=ππ
π!π0, (49)
where
π0
= πβπ
. (50)
Hence steady state probability of π customers in the systembecomes
ππ
=πβπ
ππ
π!, (51)
which is a Poisson distribution with parameter π = π/π.The expected number of customers in the system (πΏ) is
πΏ =
β
β
π=1
πππ
= π, (52)
since ππfollows Poisson distribution.
Similarly the expected number of customers in the queue(πΏπ) is
πΏπ
=
β
β
π=1
(π β 1) ππ
= πβπ
+ π β 1. (53)
Expected waiting times of customers in the system (π) aregiven as
π =π
π (1 β πβπ). (54)
Expected waiting times of customers in the queue (ππ) are
given as
ππ
=πβπ
+ π β 1
π (1 β πβπ). (55)
Sensitivity Analysis. Now we investigate the nature of theexpected number of customers in the system and expectedwaiting time of customers in the system on the basis ofthe various values of the switch point πΎ by the followingsensitivity analysis.
FromTable 1 we can observe that as the value of the switchpoint πΎ increases, the results of the single server discouragedarrival model with one service switch tending to the results ofthe single server discouraged arrival model with no serviceswitch. That is, if πΎ β₯ 9, π
0= 0.3189 which is the π
0of
the single server discouraged arrival model with no serviceswitch. Similarly if πΎ β₯ 11, πΏ = 1.142857 which is the πΏ ofthe single server discouraged arrival model with no serviceswitch and if πΎ β₯ 12, π = 67.118963 which is the π ofthe single server discouraged arrival model with no serviceswitch.Hencewe conclude that there is no effect by the switchpoint if its value increases.
Table 1: [π = 1/35, π1
= 1/40, π = 1/30, πΆ = 1].
πΎ π0
Expected number inthe system (πΏ)
Expected waiting timein the system (π)
2 0.356 0.958 53.054 0.3217 1.114 65.049 0.3189 1.142850 67.11848611 0.3189 1.142857 67.11895712 0.3189 1.142857 67.11896325 0.3189 1.142857 67.11896350 0.3189 1.142857 67.118963100 0.3189 1.142857 67.118963
4. Summary and Concluding Remarks
In this paper we study a discouraged arrival Marko-vian queueing systems. To this system we introduce self-regulatory servers and analyzed the model by deriving steadystate characteristics. A generalized multiple server discour-aged arrival model with two service switches are discussed.By introducing service switches we could speed up the servicebased on the switch point if the number of arrivals increases.That is, the speed of the server can be slow, medium, and fast.Thus the congestion of customers can be reduced by the twofeatures mentioned above. The steady state probabilities andall the performance measures such as expected number ofcustomers in the queue/system and expected waiting time ofcustomers in the queue/system are derived.
From this general model we derivedπΆ server discouragedarrival model with one service switch, single server discour-aged arrival model with two service switches, single serverdiscouraged arrival model with one service switch, multipleserver adaptive queueing system, and single server adaptivequeueing system as a special case.
Numerical illustration of the model is given. A Matlabprogram was developed to help the numerical illustration. Asensitivity analysis is conducted to obtain the optimum valueof the switch point.
From the numerical illustration, we can observe that asthe value of the switch points decreases the waiting time ofcustomers also decreases.
For implementing the above models in real life situationswe need to include the cost of the various service rate of thesystem. An optimization problem can be a future work of themodel by finding the optimal choice of the service switchesand the number of servers by considering the costs involvedin the queueing system.
Competing Interests
The authors declare that they have no competing interests.
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