Performance Analysis for Queueing systems with close down ... · Consider an M/M/1 queueing system with infinite waiting room capacity. The arrivals follow a Poisson process with

Performance Analysis for Queueing systems with close

down periods Subject to Catastrophe

B Thilaka1, B Poorani

2 *and S Udayabaskaran

3

1 Sri Venkateswara College of Engineering, Sriperumbudur

2 KCG College of Technology, Karapakkam

3Vel Tech Dr R.R &Dr S.R Technical University, Avadi

Email : 1: [email protected] , 2*: [email protected] and 3: [email protected]

Abstract—A single server queue with Poisson arrival and exponential service times subject to maintenance of the server, close down

period and catastrophes is considered, wherein catastrophes occur only when there are customers in the system and they wipe out the

entire system resulting in the system being rendered inactive for a random period of time. Explicit expressions for the transient

probabilities of the close down period, maintenance state and system size have been obtained. The corresponding steady state analysis

and performance measures are also obtained. The effects of various parameters on the system performance measures are studied using

numerical examples.

Keywords: Catastrophe, close down state, maintenance state, transient probabilities, steady state probabilities.

1.Introduction

The study of Queueing system has generated a lot of interest fundamentally due to the fact that Queueing systems have a wide range of

application in wireless networks, Telecommunication networks etc. A comprehensive over view of the fundamental techniques and classical

results in queueing theory are given in the monographs by Bertsekas and gallager (1987), Takagi (1991), Daigle (1992), Gelenbe and Pujolle

(1998), Chan (2000), Hayes and Ganesh Babu (2004), and Giambene (2005)Wireless communications is one of the fastest growing segments of

the communication industry. WiMAX evolved to satisfy the need of having a wireless internet access and other broadband services which work

well in those areas where it is difficult to establish wired infrastructure and economically not feasible.

The IEEE 802.16 standard is for the air interface between subscriber stations and a base station (BS). An amendment to the standard,

IEEE 802.16e, expands the IEEE 802.16 standard to allow for subscriber stations to move around. On account of the mobility of subscriber

stations, the idea of power saving is a very significant issue for the battery-powered mobile stations [MSs]. The IEEE 802.16 e standard defines

sleep mode and idle mode operations on MAC layer to save the energy of the MSs.Sleep Mode is the state in which the MS conducts pre-

negotiated periods of absence from the Serving Base Station air interface. Sleep Mode is intended to minimize MS power usage and minimize

the usage of the serving Base Station air interface resources. Idle mode benefits the MS by removing the requirement for handoff and other

normal operations and benefits the network and base station by eliminating air interface and network handoff traffic from essentially inactive

MSs while still providing a simple and tidy method for alternate the MS about pending DL traffic.

Several researchers have developed analytical models and obtained the performance of the sleep mode operations in the IEEE 802.16e system

(see Seo, Lee, Park, Lee, and Cho (2004); Xiao (2005); Zhang and Fujise( 2006); Dong, Zheng, Zhang, and Dai (2007);Zhang (2007);Lei and Nilsson

(2007); Turek, De Vuyst,Fiems, and Wittevronged (2008); Hwang, Kim, Son, and Choi (2009,2010); Baek, Son, and Choi (2011); Huo, Jin, and Wang

(2011)). Han and Choi (2006) modeled the BS as a continuous time finite-capacity queue with a Poisson arrival process and deterministic service times

and obtained expressions for the average packet delay and average energy consumption by the MS. Kim, Choi, and Kang (2008) have analyzed the

power consumption and average delay for both uplink and downlink traffic by focusing on sleep mode duration in the IEEE 802.16e. Han, Min, and

Jeang (2007) have used a simple train model for incoming traffic flow for the PSC I and PSC II in the IEEE 802.16e standard. Baek and Choi (2011)

have determined the average message delay and the average power consumption of an MS by using non-Markovianqueueing system. Krishna Kumar,

Anbarasu and Anantha Lakshmi (2013) have analyzed the transient analysis of a single server queueing system with the server under maintenance and

close down period according to the operating mechanism of the sleep mode of MS and the buffer content at the BS in the IEEE 802.16e standard.

An important aspect in the dynamics of communication networks is the study of dramatic but relatively infrequent events that result in abrupt

and often catastrophic changes in the network state. Such catastrophic events are commonly regarded to as phase transition. Dabrowski (2017) gives a

survey of research phase transition in communication networks and discuss characteristics of real-world communication networks that need to be

included in network models to improve their realism. When a catastrophe occurs, it invariably paralyses/ destroys the systems and the system needs to

start fresh. Examples of causal agents are computer viruses (Pastor-Satorras and Vespignani (2001); Moreno, Pastor-Satorras and Vespignani (2002);

Zou, Towsley, and Gong (2007)), events that cause site failure, which propagate through a network (Motter and Lai (2002); Moreno, Gomez, and

Pacheco (2002); Watts (2002); Zhao, park, and Lai (2004); Lee et al. (2005)), and increase in network-wide load and its congestive effects (Sole and

Valverde (2001); Arrowsmith et al., (2004); Ehentique, Gomes-Gardeness and Moreno (2005); Mukherjee and Manna (2005)).

International Journal of Pure and Applied MathematicsVolume 119 No. 7 2018, 39-57ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

39

This has motivated us to investigate and analyze the behavior of a single server Markovianqueueing system with the server under

maintenance and the close down period according to the operating mechanism of the sleep mode of MS and buffer context at the BS in the IEEE

802, subject to phase transition. This paper is organized as follows: The mathematical model is described and explicit expressions for the

transient probabilities of the system are focused in section 2 through the approach of integral equations; the Mean of the First-passage-time to

reach the Maintenance state is obtained in section 3. Steady state probabilities of the system are derived in section 4. Some key performance

measures under steady state conditions are listed out in section 5. Numerical examples are explained to illustrate the effect of system parameters

on the performance measures in section 6. Finally section 7, completes the article

2.Model description and analysis

Consider an M/M/1 queueing system with infinite waiting room capacity. The arrivals follow a Poisson process with arrival rate

while the service times are exponentially distributed with mean

1. Once the server completes the services of all customers in the system, the

system enters a close down state (sleep state) D. The close down period is assumed to be exponentially distributed with mean

1. If any new

customer arrives into the system during the close down period, the close down period is interrupted and the server resumes service. If no

customer arrives into the system for the entire duration of the close down period, then the system enters the preventive maintenance state M. The

server remains in the preventive maintenance state for a random duration which is exponentially distributed with mean

1. Any customer

arriving to the system during the preventive maintenance state will not be allowed to join the system and will be lost forever. Once the

maintenance of the server is completed, the server returns to its functioning state (active state), ready to serve new customers. Customers arrive

into the system as a Poisson process with rate , during the working state (on state), which consists of the idle period, busy period and close

down period. Catastrophes are assumed to arrive as a Poisson process with rate . Once a catastrophe strikes the system, all the customers are

wiped out from the system and the system enters the maintenance state.

Let )(tX denote the number of customers in the system at time t when the server is in active state. Let )(tJ denote the state of the

server at time t. Then ...2,1,0)( tX and

.int

,

,

)(

stateenancematheinisservertheifM

statesleeptheinisservertheifD

stateactiveinisservertheifA

tJ

The joint process 0),(),( ttJtX is Markov. The state space of the system is given by

,...,2,,1,,0,0,,0 AAAMD

For brevity, the states D,0 and M,0 are denoted by D and M respectively; and the states ,...2,1,0,, nAn are

simply denoted by 0,1,2,…

The state transition diagram is given below:

Let ...2,1,0,)(),( nntXPtnP denote the system size probabilities that there are n customers in the system at time t

when the server is in active state, MtXPtMP )(),( be the probability that theserver is in maintenance state (and that there is no

International Journal of Pure and Applied Mathematics Special Issue

40

customer arriving to the state) at time t, and DtXPtDP )(),( be the probability that the server is in close down period at time '' t .

Using Probability laws, we derive the following integral equations,

t

n

ut

t

ut dueunPdueuDPtMP0 1

)(

0

)( )1.2(),(),(),(

t

ut dueuPtDP0

))(( )2.2(),1(),(

)3.2(),(),0(0

)(

t

utt dueuMPetP

t

utut

t t

ut dueuPdueuPdueuDPtP0

))(())((

0 0

))(( )4.2(),2(),0(),(),1(

)5.2(2,),1(),1(),( ))((

0 0

))((

ndueunPdueunPtnP ut

t t

ut

Taking Laplace Transform of (2.1)… (2.5), we obtain

)6.2(),(),(),( *

1

** snPs

sDPs

sMPn

)7.2(),1(),( ** sPs

sDP

)8.2(),(1

),0( ** sMPss

sP

)9.2(),2(),0(),(),1( **** sPs

sPsDPs

sP

)10.2(...3,2,),1(),1(),( ***

nsnPs

snPs

snP

Let n

n

n ztPtCtVtzP )()()(),(0

be the p.g.f and )()( tXEtm be the mean number of customers in the system at

time t.

The Laplace transform of the generating function of ),( tzP

)11.2(),(),(),(),(0

**** n

n

usnPsMPsDPsuG

From equations (2.6)-(2.10), we have

)12.2(

)(

),1(),0(),(),0(),(),(),(

2

***2****

usu

suPsPsDPusPsMPsDPsuG


41

The Zeros of

0)(2 usu

are

2

4)()(

2

4)()( 2

2

2

1

ssand

ss

The zeros satisfy the condition that 21 10

Invoking the analyticity of ),(* suG, we have

0),1(),0(),( ***2 suPsPsDPu

)13.2(),1(

),0(),(1

***

sPsPsDP

On substituting (2.13) in (2.12),

)14.2(),1(),0(),(),(),( *

1

1

1

**** sPusPsMPsDPsuG n

n

n

Comparing (2.11) and (2.14), we get

)15.2(....3,2,),1(),( *

1

1

*

nsPsnP

n

Using (2.15) in (2.9),

)16.2(),0(),(),1( **1* sPsDPsP

Inversion of equation (2.16) gives,

)17.2(t)P(0,t)P(D,©)2(

),1( 1)(

t

tIetP t

(.)nI is the modified Bessel modified function of order n (see Watson (1962)),

where we have used the formula (see Abramowitz and Stegun)

n,convolutiodenotes©andwhere


42

t

tIssL n

nn

)2(

2

4)( 22

1

Using equation (2.7) & (2.8), equation (2.16) can be written as,

)18.2(11

),1(11*

ssP

1

1

11

))(())()(( n

n

sssssswhere

Equation (2.6), (2.7) & (2.8) can be written in terms of (2.18),

)19.2()(1

1

2

4)()(

),(*

2

* asH

s

ss

ssDP

)19.2()(1

1

),1(*

1

* bsH

ssP

)19.2()(1

11

),(*

1

1

11

* csH

ssssssMP

n

n

)19.2()(1

1

),(*

1

* dsH

ssnP

n

1

111*

))(())()(()(

n

n

sssssssHwhere

By taking inversion,

)20.2(e©)())((

©),( )u(-

0

(n)©1))((

0

1-)( aduuHut

utIeetDP

n

ut

t

ut


43

)20.2(e©)())((

),1( u-

0

(n)©1))((

0

1- bduuHut

utIetP

n

ut

t

)20.2(e©)())((

e©))((

e©e©),(

u-

0

(n)©))((

0

u)-t(-)(1))((

0

1-u)-t(-u)-)(t(-)(

cduuHut

utIen

eut

utIeetMP

n

nut

n

n

utut

t

ut

)20.2(e©)())((

),( u-

0

(n)©))((

0

n- dduuHut

utIetnP

n

nut

t

)20.2())((

e©

))((e©e©e

))((©e)(

)(

0

-

0

1)(

0

1---)u(-1))((

0

1-u)-)(t(-

eduut

utIene

duut

utIedu

ut

utIetH

nu

n

nuu

t

u

t

uuut

t

At ,0 the results of Krishna Kumar et al., (2013) are obtained as a special case.

3. Mean of First-Passage-time to reach the Maintenance state

The first-passage-time problems for queueing systems have been widely studied because of its application (Keilson 1979). Let

E(L) denote the first-passage-time to reach the maintenance state M,

The Laplace transform of (2.19c) is Consider the equation (2.19a),

),1(),1(),( *

1

1** sPs

sPss

sMP

n

n

)1.3(1

),(0

11

1

1*

n

n

n

n ssssssMP

)2.3(

41

1

41

1

41

)),(()(

22

2

2

222

0

*

sds

sMsPdLE

,0If


44

)4.3(2

4)()( 2

where


4. Steady state analysis

We study the behaviour of the steady state probabilities of the system size, the close down state probability and maintenance

state probability of the server for our queueing system.

Theorem:

,0,0, andFor the steady state probabilities of the system size ...2,1,0: nPn ,the close down probability

PD and the probability PM of the server under maintenance state of the queueing system subject to catastrophe are obtained as

)1.4(

1

1

111

1

DP

)2.4(

1

1

111

1

1

MP

)3.4(

1

1

111

1

1

0

P

)4.4(...3,2,1

1 nPP n

n

)5.4(11

11

1

1

P

Proof:

,00, andFor

From equation (2.16),

),0(),(),1( **1* sPsDPsP


45

Now using the Tauberion theorem (Widder 1946), we get

𝑃1 = lim𝑠→0

𝑠

𝜆

𝜇

2

4)()( 2 ss

𝑃∗ 𝐷, 𝑠 + 𝑃∗(0, 𝑠)

0

2

12

4)()(PPP D

01 PPP D

)7.4(1

11PPM

)8.4(1

110 PP

)9.4(,..3,2;1

1 nPP n

n

Nowusingthe normalization condition 12

10

n

nMD PPPPP , we find the unknown probability 1P as

1

1 11

11

P

Hence equations (4.6)-(4.9) determine the steady state probability PD of the close down state, the steady state probability PM of the

server under maintenance state and the steady state probabilities ...2,1,0: nPn of the system size under the influence of catastrophe.


5. System performance measures

Now we study some important performance measures, namely mean of the system size, availability of the server, system throughput,

mean waiting time and mean cycle time of the system under steady state condition.

Theorem:

,00, andFor then the steady state probability generating function )(zP of the number of customers in the

system is given by

)1.5(

1)1(11

1

11)1(11

1

1)(

z

zzzz

zzP

The mean )(XE of the system size is obtained as,

)6.4(1PPD


46

)2.5(

)1(

1

)1(

1

)1()1(

1

)(

XE

where is as given in (3.4)

IfQ denotes the number of customers in queue, then the steady state probability generating function )(z and mean )(QE are

determined as,

)3.5(

1

1

11

1

1

1

1

1

)()(

z

zEz Q

)4.5(

)1(

1

)1(

1

)1()1(

1

)(

2

QE

and

Proof:

Consider the generating function

n

n

nMD

n

n

n zPzPPPPzP

2

10

0

Substituting the equations (4.1)-(4.4), after some mathematical simplifications we get (5.1). The result (5.2) is derived directly from

(5.1) on differentiation with respect to z and substituting z=1.

Consider the generating function

1

1

0)(

n

n

nMD zPPPPz

Substituting the equations (4.1)-(4.4), after some mathematical simplifications we get (5.3). The result (5.4) is derived directly from

(5.3) on differentiation with respect to z and substituting z=1.

Corollary: The second moment )( 2XE and variance )(XVar of the system size, under steady state, are given as


47

)5.5(

)1(

1

1)1(

1

)1(

)1(1

)(3

2

XE

And

)6.5(

)1(

1

)1(

1

)1()1(

1

)1(

1

1)1(

1

)1(

)1(1

)(

2

3

XVar

Remark: The mean number )(XE of customers in the system includes the maintenance time when there are no customers in the

system.

The steady state probabilities are

)7.5(

)1(

1

1)1(

1

)1(

1

)(1

n

nPbusyisserverP

)8.5(

)1(

1

1)1(

1

)1(

1

1)1(

1

1)(

MPavailableisserverP

DPParrivaluponyimmediatelservediscustomerP 0)(


48

)9.5(

)1(

1

1)1(

1

)1(

1

1

MD PPPstateinorstatedowncloseorstateidleineitherisserverP 0) emaintenanc(

)10.5(

)1(

1

1)1(

1

)1(

1

111

n

nP

)( serviceforwaittohascustomerarrivinganPPw

)11.5(

)1(

1

1)1(

1

1)1(

1

)(1 0

DPP

The probability of at least k or more customers in the system is given as

)12.5(

)1(

1

1)1(

1

)1(

1

1)(

1

1

1

k

k

kn

n PPkXP

The conditional probabilities are:


49

)13.5(

)1(

1

1)1(

1

)1(

1

1)/( 1

M

n

n

P

P

availableisseverbusyisserverP

)14.5(

)1(

1

1)1(

1

)1(

1

1

1)/( 0

M

D

P

PPavailableisseveridleisserverP

The conditional probabilities in equations (5.13) and (5.14) depend only on the parameter and, are independent of

the other system parameters.

The conditional expectation of the mean number of customers in the system when the server is available is

MP

XEavailableisserverXE

1

)()/(

)16.5(

1

1

11

1

1

1

2

The conditional probability generation function of the number of customers in the queue is defined as

)17.5(

1

1

11

1

1

1

11

1

)(

zz

z

M

n

n

nDQ

P

zPPP

availableisserverzEzR

1)/()( 1

1

0


50

And its conditional expectation is

)18.5(

1

1

11

1

1

1

1

)()/(

2

MP

QEavailableisserverQE


System Throughput and other Characteristics:

We now introduce the following notations: Let U be the system throughput, the rate at which customers exit the queue; eff , the

effective arrival rate when the server is available; )( sWE , the mean queueing /sojourn time in the system when the system is available;

)( qWE , the mean waiting time in the queue when the system is available; )(TE , the mean cycle time of the system and )(NE , the

expected number of customers served during the busy period. The following theorem summarizes the results:

Theorem:

,00, andFor Under steady state,

)19.5(

1

1

11

1

1

1

U

Proof:

The system throughput, U is the rate at which customers exit the queue whenever there are one or more customers in the system, with

the exit rate

DM PPPU 01

Using equations (4.1)-(4.3),we get (5.19)

The effective arrival rate eff (the total arrival rate when the server is available) is defined as


51

1

1

11

1

11

1

1

0

n

nDeff PPP

Remark:


Numerical illustrations

We use the performance characteristic and probability descriptors obtained previously to study numerical results and to perform

sensitivity analysis on 0P ,the steady state probability that no customers in the system; ,U the system throughput; ,MP the probability that the

server is under maintenance state; ,DP the probability that the server is under close down period; ,wP the steady state probability that an

arriving customer has to wait for service; ),(XE the expected number of customers in the system when we vary the values of the system

parameters. All the parametric values are chosen so as to satisfy the stability condition .00,0, and

2.1 0 foroffunctionaasPFigure 4.2 0 foroffunctionaasPFigure

In figures 1 and 2, we represent the behavior of the probability 0P as a function of and , respectively, for different values of .

We have plotted curves for various values of . Clearly, figures 1 and 2 show that 0P increases smoothly for increasing values of and

whereas it decreases with increasing values of .

2.3 foroffunctionaasUFigure 4.4 foroffunctionaasUFigure


52

Figures 3 and 4 illustrate the impact of , and on the system throughputU . Here U is an increasing function of . Further, for a

fixed value of , the system throughputU increases values of increase as reflected in figure 3. Figure 4 reveals that U is decreasing slowly

for increasing values of but, for fixed value of , it is increasing for increasing values of as expected.

2.5 foroffunctionaasPFigure M 4.6 foroffunctionaasPFigure M

Figures 5 and 6 indicate the nature of variation of the probability MP . Specifically, from Figure 5, it is observed that MP is a

decreasing function of η whereas for ρ there can be seen to be two types of behaviour: the probability MP of the server under maintenance

state increases from ρ = 0.1 to ρ = 0.3 attaining the maximum at ρ = 0.3 for a fixed value of whereas the probability MP decreases as ρ

becomes high. A possible explanation for the phenomenon is as follows. The arrival rate being very small in the interval ρ = [0.1, 0.3], it results

in the close down period of the system ending with almost negligible arrival rate. Thus, the server goes immediately for the maintenance state

with high probability. In Figure 6, the curves show that the probability MP is an increasing function of whereas, for a fixed value of , it

decreases for increasing values of ρ except for the cases ρ = 0.1 to ρ = 0.3 as before.

2.7 foroffunctionaasPFigure D 4.8 foroffunctionaasPFigure D

In figures 7 and 8, we have plotted the close down probability DP as a function of and , respectively, for different offered loads

. Figures 7reveals that the probability DP increases smoothly for increasing values of whereas for fixed values of , it decreases

gradually for increasing values of except 1.0 . This is due to the fact that the close down period probability DP decreases and the

length of close down period has little influence for higher offered loads .However, for the case 1.0 the offered load is very small, so

that the close down probability becomes somewhat large. From Figure 8, it is clear that initially there is a steep decrease in the probability DP


53

for increasing values of and further it decreases slowly for increasing values of except 1.0 as noted before.

2.9 foroffunctionaasPFigure w 4.10 foroffunctionaasPFigure w

The influence of the parameters and on the probability wP is displayed in figure 9 and 10. In figure 9, it appears that as

increases, the probability wP decreases slowly whereas it increases always for increasing values of . Figure 10 shows the behavior of the

probability wP against . The probability wP increases always with increasing values of both and

2)(.11 foroffunctionaasXEFigure 4)(.12 foroffunctionaasXEFigure

The effects of and on the mean number )(XE of customers in the systems are represented in Figures 11 and 12, respectively, for

various values of . Figure 11 shows that )(XE is an increasing function of both and , which is a quite natural. But in figure 12,

)(XE decreases for increasing values of , for a fixed value of , it is increasing for increasing value of , which is counterintuitive.

7. Conclusion:

In this article, we have investigated a single server queuing system with close down period and server under maintenance subject to

catastrophes. Explicit expressions for the transient probabilities of the system size, the server under maintenance state and the close down period

are obtained. Further the mean first-passage-time to reach the maintenance state has been derived. The results of Krishna Kumar et. al.,(2013)

have been deduced as a special case in all the above. Under the steady state condition, the corresponding probabilities and some interesting

performance measures, namely, mean of the system size, availability of the server, system throughput and mean waiting time of an arbitrary

customer in the system have been obtained. Finally, graphical illustrations have been presented and the effects of various parameters in the

system performance measures are studied.


54

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