International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-8 Issue-4, November 2019
Effects of Catastrophe on a Queueing System with Voice over Internet Protocol
M.Balasubramanian, Abel Thangaraja.G, Bharathidass.S
Abstract: Consider a retrial queue with VoIP calls and two third generation service providers. All these solutions kinds of heterogeneous services such as essential and optional provide a dynamic interconnection between the users on the services. The multiple vacation policy, retrial policy, customer’s internet. impatience and the concept of catastrophe are adopted to derive VoIP systems transmit audio streams over IP networks the required solutions. The steady state system size distribution and probability generating function under different level have with media delivery protocols. Some of the popular codecs been obtained. Based on some assumptions, special and are µ-law and a α-law versions of G.711, G.722, an open particular cases are discussed. source voice codec known as iLBC, and others. Recently, with the rapid usage of IP, a low cost transport mechanism Keywords: VoIP, Catastrophe, Impatience, Optional service, which can be used for both voice and data. Making use of Vacation and Retrial policies. the IP infrastructure and hardware the voice traffic on top of
the data network. I. INTRODUCTION The arriving calls are served by a one server in two non- The concepts of queueing theory have been applied in homogeneous modes such as the first one is essential but the various fields like phone systems, communication network second is optional. If the server is busy either with essential systems, computer systems, industrial sectors and so on. The or with optional service, the arriving call balks from the techniques adopted in the queueing models produce system or enters into an orbit to make trials to get desired remarkable solutions. In this paper, Voice and Internet service. After end of a service of a call, if there is no arrival Protocol (VoIP), Vacation policies, Catastrophe, retrial in the system, the server avails vacation. queue, optional service and impatience have been utilized in Vacation models of server‟s are applicable for the systems in proper place. Generally, in any queueing models, the which the server wants to utilize the idle time for various arriving units are considered as human beings, but, here, purposes. Vacation of the server is one of the concepts of VoIP calls are taken as arriving units. queueing theory and that leads to study new results. After the Voice over IP (VoIP) is a typical method for the completion of service, if the queue is empty, the server leaves delivering voice communications and multimedia sessions and engages other work but notice the new arrivals to the through Internet Protocol (IP) networks. VoIP is a toll-free system. The period of unavailable of the server in the counter is long distance voice, fax calls on IP data networks other than known as vacation period. On returning from vacation, if there public switched telephone network(PSTN). It saves the long were units in the system the server do the service otherwise he distance costs between two or more locations. Earlier people avails another vacation. These type of vacations are called depend on the PSTN for communicating voice. The single and multiple vacations. connection is dedicated to only two parties between two In communication networks, the system may failure due to locations when the call is made. Although there is an various factors, in particular, the occurrence of catastrophe is availability of many bandwidth, none of the information can one of the factors. The concept of catastrophe has played a vital pass over the call. Internet Protocol and internet and role in the areas of Science and Technology. It occurs at broadband telephone and broadband telephone services are random leading to extinction of all the units and activate the generally associated with VoIP. The principles of VoIP service facility until a new arrival is not unusual in most of the telephone calls are same as digital telephone calls. The real life situations. The catastrophe may exists from outside or digital information is gathered and transmitted as IP packets. within the system. In computer based systems, if a task is VoIP services begins with providing solutions for the infected, this infected task may transfer the virus to the other business and technical problems. In the second generation processers. These infected network of tasks may be imitated by era, like Skype with closed networks offered free calls. The the catastrophes which leads to construct queueing models with federated VoIP concept was adopted in the Google Talk, a catastrophes.
II. REVIEW OF LITERATURE Revised Manuscript Received on November 19, 2019 Falin and Templeton (1997) have summarized many M.Balasubramanian, Assistant Professor, Department of Statistics, Periyar E.V.R. College (Autonomous), Bharathidasan University( contributions relating to the queueing systems with retrial Affiliation), Tiruchy, Tamilnadu, India. queues. Artalejo and Choudhury (2004) have used Abel Thangaraja.G, Assistant Professor and Head of the Computer Science, KayPeeYes College of Arts and Science, Kotagiri, Tamilnadu, India. Bharathidass.S, Assistant Professor, Department of Statistics, Periyar E.V.R. College (Autonomous), Bharathidasan University( Affiliation), Tiruchy, Tamilnadu, India.
Published By: Retrieval Number: D5289118419/2019©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.D5289.118419 7301 & Sciences Publication Effects of Catastrophe on a Queueing System with Voice over Internet Protocol embedded Markov chain of retrial queues consisting of bulk service Markovian queue with system disaster. The services with two phases. Also they derived steady state expressions for the average number of times the system solution, average number of customers and other related reaches to the capacity, the average number of customers in measures. the system and their respective factorial moments have been Doshi (1985) has discussed the system with infinite derived under different conditions. number of vacation types. After completing all the services to customers, the server avails a type-1 vacation. Coming III. DESCRIPTION AND ASSUMPTION back from type-1 vacation, if there are no customers, the OF MODEL server starts a type-2 vacation and so on. Choudhury and In this study a queue of single server with the VoIP calls is Madhuchanda (2004) have dealt with a queueing system of considered. Here the arrival times of the calls were assumed to bulk arrival with heterogeneous services which are governed follow Poisson distribution with mean arrival rate of „λ‟. The by a single server. They derived the solution for the calls which were arrived may balk due to impatience or make stationary queue size distribution, waiting time distribution retrial to receive service again. The server has two services of and related performance measures. Armuganathan et al., heterogeneous types in which the first one is essential for all (2008) have analysed steady state solutions of a bulk size calls and second one is optional For both the services the times system of non-Markovian queue with N-policy and various of service were assumed to have general types of vacation. distributions with probability density functions and Senthilkumar & Arumuganathan (2009) have analysed s1 ( x) the retrial queue with impatient subscribers under steady s ( x) respectively. 2 state conditions and heterogeneous services of two phases If a call, on arrival, sees the server busy, it balks the system and distinct vacation policies. Voice over IP calls are taken with probability (1 ) or it enters into an orbit with as arrival units. The average number of customers, averge waiting duration in retrial queue and some particular cases probability α in order to enter the system again. The time have been analysed. Jeyakumar and Arumuganathan gap between successive try of every call follows exponential (2011) have analysed a single server non- Markovian queue distribution with mean retrial rate „γ‟. with more than one vacations and re-service as optional of At the time of completion of a service of a call, if there is the steady state conditions. They mathematically derived the no call in the system, the server goes for th mean queue size, busy period, idle period and cost function j ( j 1, 2,..., M) vacation with probability or retains j and obtained numerical results. Ponnammal et al., (2013) M have considered a customer‟s arrival follow Hyper Poisson in the system with probability and distribution and two types of services. The principles of 0 j 1. j 0 balking, reneging and impatience are applied to estimate When the system is functioning well, unfortunately, the mean queue length for different positions of the server. catastrophe occurs either during the essential or optional Swaminathan et al., (2015) have studied two independent service and the system becomes down at once. The time at servers Markovian queue with blocking and catastrophes. It which catastrophe occurs is exponentially distributed with is assumed that there is no waiting time for customers. The steady state probabilities, expected queue length and mean catastrophe rate . After recovered from the system is down, the process of the system is going on. probability of lost customers have been derived. Jain and Bura (2010) have analysed the effect of For the above stated model, form a two dimensional catastrophe intensity of Markovian queue with restoration Markov chain as N (t ), C (t). for a single server with finite capacity. The intensity of catastrophes follows uniform distribution. The steady state Here, N (t ) n, (n 0) is the no. of calls at time „t‟ probabilities, the expected number of units in both queue and C (t) 0,1, 2,3 are the states of the server. and system were derived. Sudhesh (2010) has studied Let be the transient probes for single server Markovian queue by using Pi, n (t),(i=0,1,2,3; n=0,1,2,3,...) the concepts of balking, reneging and catastrophe. probability of the system with n( 0) calls at time t, when,
Dhanesh Garg (2013) has discussed Markovian queue of a the server is idle, on essential service, on optional service single server with finite capacity through catastrophe and and on vacation for restoration. He has derived the p.g.f. for the number of times i 0,1, 2,3. the system reaches its capacity and the number of units in the systems. The corresponding expected values have been IV. EQUATIONS RELATED TO STEADY STATE OF THE MODEL obtained. Ayyappan et al (2013) have analysed the effect of catastrophe in an M/M/I queue when the server is idle and On the basis of the above section, the required steady- busy. The probabilities under steady state conditons, average state equations are framed. length of a queue and the variance have been explicitly derived. The derived results are justified by using the numerical values. Arul and Vidhya (2014) have discussed a Markovian queue of multi-server with system disaster and impatient customers. They have derived transient probes of the model and justified some special cases. Balasubramanian et al., (2015) have studied a single server
Published By: Retrieval Number: D5289118419/2019©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.D5289.118419 7302 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-8 Issue-4, November 2019
(1) M 0 = ( +n ) P0, n P3, n (0) 0 P2,n (0) * * * D (u ) S1 ( u ) S 2 ( u ) 0 jV j ( u) j 1 d P (x ) ( ) P (x ) P s (x ) ( n 1) P s (x ) P (x) (2) Now, apply the expression for P ( z, 0) from equation dx 1, n 1, n 0, n 1 0, n 1 1 1, n1 1 (14) in the equation (11) which implies d P (x ) ( ) P (x ) P (x ) P (x ) s (x) (3) ( z ) P* (z , ) P (z ) P ' (z ) . S * ( z ) S* ( ) (19) dx 2, n 2, n 2, n 1 1, n 2 1 0 0 1 1
M Similarly, using the expressions (15) and (16) in the
d P3, n (x ) ( ) P3, n (x ) P3, n 1 (x ) P2,n (0) j vk (x) (4) equations (12) and (13) which provide respectively as dx j 1
Define Laplace Stieltjes Transform as 2 1 2 2 * ( z ) P* ( z , ) P ( z , 0) S * ( z ) S*( ) (20) and (5) LST P (x) P (), i=1,2,3 i , n * * * i ,n ( z ) P ( z , ) P ( z , 0) M V ( z ) M V () (21)
3 2 jj jj Apply the LST in the equations (2), (3) and (4) and using j 1 j 1 In this stage, define the probability generation function as the relation (5), we have, * * * * P1, n ( ) P1, n (0) ( ) P1, n ( ) P0, n ( ) s1 ( ) ( n 1) P0, n 1 ( ) s1 ( ) P1, n1 ( ) (6) * * n * * * * P ( z ,0) P (0) z , i = 1,2,3. P2, n ( ) P2, n (0) ( ) P2, n ( ) P2, n1 ( ) P1,n (0) s2 ( ) (7) i i ,n n0
M Now, using the relation (22) in the equations (19), (20) and * * * * P () P (0) ( )P ()P () P (0) V () (8) 3,n 3, n 3, n 3, n 1 2,n j j j 1 (21) and they become respectively as
P ( z ) P ' ( z ) . S * ( z) 1 * 0 0 1
P (z, 0) (23) V. PROBABILITY GENERATING FUNCTIONS ( z ) 1 ' Now, define probability generating functions. * S * ( z ) 1 P (z ) P S ( z ) (z) * P (z, 0) 1 2 0 0 (24) 2 ( z ) and n M P ( z ) P z ' * * * (25) n 0 P0 (z ) P0 (z )S1 ( z )S 2 ( z ) jV j ( z) 1 0 0, n * P (z, 0) j 1 3 n P ( z , 0) P (0) z , i = 1,2,3., (9) ( z ) i i,n The p.g.f. P(z) of the number of calls at any interval is stated and
n 0 P* ( z, ) P* ( ) z n , i = 1,2,3. as
i i , n
n 0 * * * P(z ) P (z ) P (z ,0) P (z ,0) P (z,0) Multiply the equations (1), (6), (7) and (8) by the proper 0 1 2 3 powers of z and making use of the p.g.f given in (9) and we get, Substitute the equations (23) , (24) and (25) in the ' 1 ( ) P0 (z ) zP0 (z ) P3 (z , 0) 0 P2 (z, 0) (1 z) (10)condition (26) and get,
1
0 (K ( z) 1) (11) P (z) ( z )(z D (z ) ( z )(K (z ) 1)P (z ) (1 z ) (27) * * 1 * ( z ) P (z, ) P ( z , 0) P ( z ) S ( ) P (z)S () ( z )(z D (z) 1 1 0 1 0 1 where * * ( z ) P (z, ) P ( z, 0) P (z, 0)S (12) M () * * * 2 2 1 2 D(z)=S1 ( z ).S 2 ( z ). 0 jV j ( z) M (13) j 1 ( z ) P* (z, ) P ( z , 0) P ( z , 0) V*() and 3 3 2 j j j 1 M * * * K(z)=S1 ( z ).S 2 ( z ). jV j ( z) In order to estimate Pi ( z, 0) , i=1,2,3,…; Consider j 1 VI. SPECIAL CASES & RESULTS z and substitute it in the equations (11), 1. Erlangian Vacation time: (12) and (13) which yield P ( z , 0) P ( z ) S * ( z ) P ' ( z )S* ( z) (14) In this case, vacation times are distributed according to 1 0 1 0 1 * Erlang distribution with „k‟ phases. P (z ,0) P (z ,0).S ( z) (15) The pd.f. of k-Erlang is 2 1 2 kv x and k k 1 j M (kv j ) x e * v j (x) , j=1,2,3,...,M P3 ( z , 0) P2 ( z , 0). jV j ( z) (16) (k 1)! j 1 Its Laplace Stieltjes Transform is Substitute the expressions for P2 ( z, 0) and P3 ( z, 0) from k x the equations (15) and (16) in the expression (10) and get, kv j V ( z) (17) kv z j M * * * 1 [ S1 ( z ) S 2 ( z ){ 0 jV j ( z )} ( )] P0 ( z) (1 z) j ' j 1
P ( z)
o M * * * [ z S1 ( z ) S 2 ( z ){ 0 jV j ( z)} j 1 Using the method of solving linear differential equation in On considering the vacation times follow Erlang distribution, the p.g.f. of the number of calls P(z) shown in the equation (17) which gives (27), after applying the equation (29), becomes D (u ) ( ) (1 u )1 D (u) ( ) .exp P0 (z ) exp du . P0 (1) du du (18) D (u ) u u D (u ) D (u ) u Where
Published By: Retrieval Number: D5289118419/2019©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.D5289.118419 7303 & Sciences Publication Effects of Catastrophe on a Queueing System with Voice over Internet Protocol
( z )(z Dv (z)) (z )(Kv (z) 1)P0 (z) (1-z) (Kv (z) 1) 2. No optional service and no catastrophe in the system: 1
P (z) (30) ( z )(z Dv (z)) In this case, the arriving calls demand only essential where service but no one opted second service with the absence of M k * * v 1 2 0 D ( z ) S ( z ) S ( z ) j catastrophe. kv j
j 1 kV j z In this regard, it is assumed that * and k 0 and S ( z) 1. On assuming these M 2 * * kv j
K v ( z ) S1 ( z ) S 2 ( z ) j values, the probability generating function (27) becomes j 1 kv j z
1
P ( z ) ( z D2 ( z)) ( z D2 ( z )) K 2 ( z ) 1P0 ( z) (37) It is noted that the equation (18) is re-written for P0 ( z), where
* ( z ) * after replacing D(u) by D (u). D2 ( z ) S1 0 jV j ( z) M v j 1 * 2. Erlangian Essential Service time: K ( z ) S ( z ) M V * ( z) 2 1 jj j 1 Essential and optional service times follow Erlang with and 1 D2 (u) 1 parameters ( k, ) and exponential with parameter z D2 (u ) u 1 2 P0 ( z) P0 (1) exp du respectively. The probability density function of Erlang The derived result (37) is the result of Falin and Templeton
(1997). ( k, 1 ) for essential service is
k k 1 k 1x 3. No optional service, single vacation, no balking and no ( k 1 ) x e s1 ( x) (31) (k 1)! catastrophe in the system. Its Laplace Stieltjes Transform is The arriving calls demand only essential service and due to k * k 1 impatience, the calls do not balk but surely (certainty) enter
S1 ( z) , k>0, 1 0 (32)
k 1 z into an orbit for retrial. The server adopts only single vacation Similarly, the p.d.f. of Exponential 2 for optional service and the occurrence of catastrophe is not permitted. For this is, case, we consider
S2 ( z ) 1, M=1, =1 and 0 . x 2 s 2 (x ) 2e , 2 0 (33) * and its Laplace Stieltjes Transform is Based on these above values, the p.g.f. of the number of * 2 S 2 ( z) (34) 2 z calls P(z) is reduced from (27) we get, * z 0 S1 ( z ) 1 Po ( z) (38) Now apply the expressions (32) and (34) in the equation (27) P (z) z S1* ( z) 0 1V1* ( z) and get 1 where ( z )( z Ds ( z )) ( z )( K s ( z ) 1)P0 ( z ) (1-z) ( K s ( z) 1)
P (z) (35) ( z )(z Ds (z)) 1 D (u) 1 Where 3 k P (z ) P (1) exp du k M * 2 0 0 1 D (u ) u D (z) 0 V ( z) z s k j j 3 1 z 2 z j 1 and k and M k 1 2 * K (z) jV j ( z) s k 1 2 * * z z j 1 D (u ) S ( u ) V ( u) As in the case of (1), the equation (18) is re-written for 3 1 0 1 1
P0 (z) after replacing D(u) by Ds (u). The expression (38) is identical with the result given by II. Particular cases Artalejo and Choudhury (2004).
1. No Catastrophe in the system: 4. The System With No Vacation And No Catastrophe
Suppose the occurrence of catastrophe is not allowed ( When there are no calls in the system and the occurrence of catastrophe is not permitted with the server without 0), then the probability generating function (27) reduces to vacation and remains in the system. This implies that 1 (36) P (z ) (z D (z)) (z D1 (z )) K1 (z ) 1P0 (z) * 1 V j ( z ) 1, j =0 (j=1,2,3,...,M) and 0. where M * * * Then, the p.g.f. of the number of calls P(z) is obtained from D1 (z ) S1 ( z )S 2 ( z ) 0 jV j ( z) j 1 (27).
M * * * * * z) {z 0 S1 ( z )S 2 ( z )} 1 Po ( z)
K1 (z ) S1 ( z )S 2 ( z ) jV j ( P ( z) * * (39) { z S ( z )S ( z)} j 1 0 1 2 and where 1 D (u) 1 1 P0 (z ) P0 (1) exp D (u ) u du z 1 The result given in (36) is identical with the result of Senthilkumar and Arumuganathan (2009).
Published By: Retrieval Number: D5289118419/2019©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.D5289.118419 7304 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-8 Issue-4, November 2019
1 D (u) 1 P 4 0 (z ) P0 (1) exp D (u ) u du z 4 and
* * D4 (u ) S1 ( u )S 2 ( u).
VII. CONCLUSION AND SUGGESTION
A single server queue is studied with some concepts like VoIP, Catastrophe, essential and optional services, impatience, multiple vacations and retrial policy. The steady state system size distribution and probability generating functions under different level have been obtained. Some special and particular cases are discussed and the coincidence of the present work with some previous research works are justified. It is suggested that there are some open problems to be solved by introducing bulk size rules in arrival or services or both in this existing system.
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Published By: Retrieval Number: D5289118419/2019©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.D5289.118419 7305 & Sciences Publication