An M/G/1 Retrial Queue with Priority, Balking and Feedback Customers

Transcription

1 Journal of Convergence Information Technology Volume 5 Number April 1 An M/G/1 Retrial Queue with Priority Balking an Feeback Customers Peishu Chen * 1 Yiuan Zhu 1 * 1 Faculty of Science Jiangsu University Zheniang Jiangsu 113 China Mathematics epartment Chaohu University Chaohu Anhui 38 China cps8@sinacom yzhu@useucn oi: 14156/citvol5issue18 Abstract We consier an M/G/1 queue with general retrial times where the blocke customers may balk the system or oin the priority queue or enter the orbit As soon as a customer is serve he will ecie either to leave the system or oin the orbit for another service in accorance with an FCFS iscipline We assume that only the customer at the hea of the orbit is allowe to retry for service We analyze the ergoicity of the embee Markov chain an present some performance measures of the system in steay-state Numerical results are presente with a focus on the effect of balking an feeback on the system performance Keywors Retrial queue Priority Balking Feeback Steaystate istribution 1 Introuction Retrial queueing systems are characterize by the fact that an arriving customer who fins all the servers busy upon arrival must leave the service area an repeat his request for service after some ranom time Such queues play very important roles in the analysis of telephone switching systems computer an communication systems [1 ] Most of the investigations in retrial queueing systems consier an arriving customer who fins the server busy oins a retrial group (orbit)while ignore research balking an waiting customers If a new arriving customer fins the server busy he may either leave the service area an oin orbit or balk the system forever [3] We further consier the system has waiting space an some blocke customers may choose wait in the waiting space (calle priority queue) instea of oining the orbit or balking the system [ 4] The feeback phenomenon is another important issue that nees to be aresse for communication systems with repeate attempts [5] We assume that ust after completion of his service a customer either leaves the system or he may eman re-service in which case this customer has to oin the orbit in our system For example messages turne out as errors are sent again can be moele as retrial queues with feeback in telecommunication systems Customers may call again if their problems are not completely solve in a call center Communication protocol is basically following certain rules so that the system works properly [ 6] One of the more usual communication protocols in the local area networks (LAN) is the CSMA (carrier sense multiple access) protocol There exist several types of CSMA as for example persistent CSMA an nonpersistent CSMA It is obviously that when the channel is free a user with any type packet can be transmitte When the channel is busy a user with a packet: (1) in persistent CSMA the user sticks to wait an transmits the packet as soon as the channel becomes ile; () in non-persistent CSMA the user attempts the transmission again after some ranom perio of time; (3) the user may be lost or balk the system an turn to other channels In our moel we consier a LAN with two types of users (persistent an non-persistent) connecte by a single channel The communication between the two users is realize by the channel Once the channel is free the central system allows a persistent user with a packet (if any) to occupy the channel with the aim of sening its message The nonpersistent users with a packet attempt the transmission inepenently after a ranom amount of time The persistent users have higher priority than the nonpersistent users in transmissions Some users balk the system when the channel is busy The rest of this paper is organize as follows A etaile mathematical escription of the moel uner stuy is given in the next section In section 3 we get the embee Markov chain escribing the behavior of the queueing system at the service completion epochs In section 4 we analyze the steay-state probabilities of the system an obtain the explicit expressions of the generating functions of the stationary istribution of 155

2 An M/G/1 retrial queue with priority balking an feeback customers Peishu ChenYiuan Zhu the chain Besies we present several performance measures of the system The effect of balking an feeback on the system performance is analyze numerically in section 5 Conclusions are presente in section 6 Moel escription We consier an M/G/1 retrial system with balking an feeback where the server applies a waiting space of infinite capacity The etaile escription of the moel is given as follows: (1) New customers arrive from outsie of the system accoring to a Poisson process with rate If an external customer fins the server ile he begins his service immeiately Otherwise the arriving customer balks the system with probability p or oins the waiting space (calle priority queue) with probability q or enters the retrial group (calle orbit) with probability r where p q r 1 A new arriving customer who obtains service immeiately is regare as a member of the priority queue We will assume that only the customer at the hea of the orbit is allowe access to the server If the server is busy upon retrial the customer oins the en of the orbit again Such a process is repeate until the customer fins the server ile an gets the requeste service at the time of a retrial Successive inter-retrial times of any customer are governe by an arbitrary istribution Ax with corresponing Laplace-Stieltes transform () s After completion of a service the customer at the hea of the priority queue (if any) is being serve immeiately Otherwise a possible new arrival an the one (if any) at the hea of the orbit queue compete for service Accoring to the above rule customers in the priority queue have non-preemptive priority over those in the orbit () There is a single server who provies service to all customers As soon as a customer is serve he will ecie either to leave the system with probability or oin the orbit for another service with complementary probability The service times are inepenently an ientically istribute with probability istribution function Bx Laplace-Stieltes transform () s an nth moments n (3) We suppose that inter-arrival times retrial times an service times are mutually inepenent of each other At an arbitrary time t the system can be escribe by means of the Markov process X(t)= {C(t) N (t) N ( t) (t) (t)} 1 1 where C(t) 1 enotes the server state ( or 1 epening if the server is free or busy) N (t) an N (t) are the number of customers in the 1 priority queue an in the orbit respectively If C(t) N1 (t) an N (t) then t represents the elapse retrial time If C(t) 1 then 1 (t) represents the elapse time of the customer currently being serve 3 The embee Markov chain N Let be the time of the th eparture N ) an N N represent the ( 1 1 number of customers in the priority queue an in the orbit respectively ust after the time For N1 an N we have the following funamental recursive equations: N1 1 1v1 i f N1 1 1 N1 (31) v1 if N1 1 N v i f N N N B v i f N N v i f N N (3) represents the numbers where the notations v 1 an v of customers arriving at the priority queue an the orbit respectively uring the th service time B represents the th serve customer procees from the priority queue an B 1otherwise is the serve customer fee back to the orbit immeiately after his service The sequence of ranom vectors X ( N N ) N forms a Markov chain with 1 N as the state space which is the embee Markov chain of our queueing system It is easy to see that { X N} is irreucible an aperioic Theorem 1The embee Markov chain { X N} is ergoic if an only if < r q Proof: To prove ergoicity we can use the following Foster s criterion [7] which states that an irreucible an aperioic Markov chain is ergoic if there exists a nonnegative function f N an such that the mean rift x E[ f ( x ) f ( x 1) x 1 ] is finite for all N an x for all N except perhaps for a finite number 156

3 Journal of Convergence Information Technology Volume 5 Number April 1 In our case { X N} is irreucible an aperioic Due to the recursive structure of equations (31) an (3) we consier the linear test function f ( m n) m n where is a positive parameter Then we have x E[ f ( x ) f ( x ) x ( m n)] m n 1 1 q r i f man n q r i f m an n 1 r (1 q) i f m1an n where is the loa of the system 1 If satisfies the following inequalities: q r< an r (1 q) < the Markov chain { X N} is ergoic These conitions r r mean that ( ) Such can be 1 q q foun iff this interval is not empty Then we have < is a sufficient conition for the ergoicity r q of X We will obtain from Remark 1 that this conition is also a necessary for ergoicity To erive the explicit expressions for the ile probability as well as some other performance measures the following lemma is necessary Lemma1 If < then equation r q z ( z ) ( q r qg( z ) rz ) 1 has a unique analytical solution z1 g( z) in the unit isk z 1an g(1) 1 r g ' (1) 1 q '' ( r q )[ (1 q) ( r q ) ] g (1) 3 (1 q) Proof: We now consier the function f ( z z ) z ( z ) ( q r qz rz ) For each fixe z with z 1 we regar f ( z1 z ) as a function of z If z we have f ( z z ) z ( z ) ( q r qz rz ) 1 z Accoring to Ruche s Theorem for each z with z 1 there exists a unique solution z g( z ) of the 1 equation f ( z z ) in the unit isk that is 1 f ( g( z ) z ) g( z ) ( z ) ( q r qg( z ) rz ) It is easy to get g(1) 1 an we can get g '' (1) by erivative of implicit functions g ' (1) an 4 Analysis of the steay-state probabilities In this section we stuy the steay state istribution of our queueing system Let us efine the following notations an probabilities: A( x) B( x) ax an bx 1 Ax 1 Bx P lim P[ C( t) N ( t) N ( t) ] 1 t P ( x) lim P[ C( t) 1 N ( t) N ( t) x ( t) () t 1 x x] 1 P ( x) lim P[ C( t) 1 N ( t) i N ( t) x ( t) (1) i t 1 1 for i x x x] By the metho of supplementary variable technique we reaily obtain the following equations: P P ( x) b( x) x (41) (1) P () () ( x ) ( a ( x )) P ( x ) 1 (4) x P (1) (1) (1) ( x i ) ( b ( x )) P i ( x ) pp i ( x ) x (1 ) qp ( x) (1 ) rp ( x) (1) (1) i i1 i 1 The bounary conitions are () (1) (1) 1 i (43) P () P ( x) b( x) x P ( x) b( x) x 1 P () P (1 ) P ( x) x (1) () i i i () (1) i 1 i1 P ( x) a( x) x P ( x) b( x) x (1) i1 1 (44) (1 ) P ( x) b( x) x i (45) An the normalization conition is 157

4 An M/G/1 retrial queue with priority balking an feeback customers Peishu ChenYiuan Zhu () (1) i 1 i (46) P P ( x) x P ( x) x 1 To solve equations (41)-(46) we efine the probability generating functions () P ( x z ) P ( x) z (1) 1( 1 ) i i 1 i P x z z P x z z By using the above generating functions equations (4)-(43) become the following partial ifferential equations P ( x z) ( a( x)) P ( x z) x P1 ( x z1 z) ( q r b( x) x whose solutions are given by qz rz ) P( x z z ) P ( x z ) P ( z )[1 A( x)] e x (47) ( q r qz1 rz) ( ) ( )[1 ] P x z z P z z B x e (48) Using (41) an the previous generating functions equations (44)-(45) can be written as P ( z ) ( z ) P( x z ) b( x) x P 1 P( z z ) P P ( x z ) x 1 1 (49) 1 1 P ( x z ) a( x) x ( z ) z z 1 [ P 1( x z 1 z ) P 1( x z )] b ( x ) x (41) we get After substituting (47) an (48) into (49) an (41) P ( z ) ( z ) ( q r rz ) P( z ) 1 P (411) [ z ( z ) ( q r qz rz )] P( z z ) z [ z (1 z ) ] 1 P ( z ) P z 1 z ( z ) ( q r rz )] P( z ) (41) 1 Letting z1 g( z) in (41) we have ( z ) ( q r rz )] P( z ) 1 [ z (1 z ) ] g( z ) g( z ) P P ( z ) P z Substituting the above equation into (411) we get z (1 g( z )) P ( z ) (1 z ) g( z ) z (1 g( z )) Putting (413)into (411) we have ( z ) ( q r rz ) P( z ) 1 (1 z g) z( ) P (1 z g) z z (1 g z) Inserting (413)an (414) into(41)yiels (413) (414) (1 z) P P1 ( z1 z) z1 ( z) ( q r qz1 rz) ( z1 g( z)) (415) (1 z ) g( z ) z (1 g( z )) () () Now we efine P P ( x) x 1 as the probability that there are customers in the orbit when the server is ile Similarly we efine (1) (1) i i P P ( x) x i as the probability that there are i customers in the priority queue customers in the orbit when the server is busy For the limiting probability generating functions P( x z ) P1 ( x z1 z ) generating functions by we efine the corresponing 1 P z P z P x z x P z () ( ) 1 (416) 158

5 Journal of Convergence Information Technology Volume 5 Number April 1 (1) i 1 1 i i P ( z z ) P z z P ( x z z ) x 1 ( q r qz rz ) 1 P ( z z ) (417) 1 1 q r qz rz 1 Theorem If < the stationary istribution r q of the process { X N} has the following generating functions: ( r q ) P ( p) P z (1 ) z (1 g( z )) P (1 z ) g( z ) z (1 g( z )) 1 ( q r qz1 rz) P1 ( z1 z) q r qz rz 1 (418) (419) (1 z ) P z1 z q r qz1 rz z1 g( z) (1 z ) g( z ) z (1 g( z )) (4) where gz ( ) is the unique root of z1 of the equation z ( z ) ( q r qz rz ) 1 1 Proof: Inserting (411) an (415) into (416) an (417) we can obtain (419) an (4) We can write the normalization conition (46) as P P (1) P(11) 1 1 It follows that ( r q ) P ( p) Remark 1 Note that P is the ile probability of the system in steay-state From P we obtain that < is a necessary conition for the r q ergoicity of the Markov chain{ X N} The following corollary presents some performance measures for the system at the stationary regime Corollary 1 (1)The system is ile with probability ( r q ) P ( p) ()The system is occupie with probability P(1) P(11) 1 P (3)The server is ile with probability P P(1) 1 p (4)The server is busy with probability P(11) p (5)The mean number of customers in the priority queue q is EN ( 1) ( p)(1 q) (6)The mean number of customers in the orbit is r EN ( p)(1 q)[ ( r q ) ] where 1(1 )[1 ( p q) pq ] [ r( ) r( )] [(1 q) q ]( [ ] 1) { [ r (1 q q ) q ( r q ) ( r pq (1 p) q ) (( p q)(1 ) q ( r q ))]} (7)The mean number of customers in the system is E( N) E( N1) E( N) p (8)The arrival rate to the priority is P Pr( Server ile) q Pr( Server busy) priority ( r) p (9)The arrival rate to the orbit is P r Pr( Server busy) orbit r p (1) The arrival rate to the system is P Pr( Server ile) ( r q) Pr( Server busy) system p (11) A customer is lost with probability P pp r( Server busy) lost p p 159

6 An M/G/1 retrial queue with priority balking an feeback customers Peishu ChenYiuan Zhu (1) The mean waiting time a customer spens in the priority queue E( N1) q Wpriority P (1 q) ( r) priority (13) The mean waiting time a customer spens in the orbit EN ( ) Worbit P orbit (14) The mean time a customer spens in the system (incluing the service time) is W system E( N) ( p) E( N) P system 5 Numerical results In this section we present some numerical results to illustrate the effect of varying parameters on the main performance measures of our system We consier our moel with arrival rate 6 exponentially istribute service time with parameter 1 an exponentially istribute retrial time with parameter Then we have 1 1 All the values of the parameters have been chosen uner the stability conition In Figure 1 we stuy the relationship of the ile probability P with the parameters p an for q 4 As we expecte the ile probability increases rapily with increasing the balking probability p an leaving probability In Figure we stuy the relationship of the lost probability P lost with the parameters p an As intuition tells us the lost probability increases rapily with increasing the balking probability p an feeback probability ( 1 ) In Figure 3 we plot the mean priority queue size EN versus p for q 4 Though the 1 feeback customers return to the orbit instea of the priority queue the feeback probability still has impact to the priority queue size an the feeback probability influence the priority queue size lesser than the orbit size an the system queue size which we will see later In Figures 4-5 we plot the mean orbit queue size EN ( ) versus p an the mean system size EN versus p for q 4 4 As more arriving customers oin the orbit when the server is busy an the number of serve customers returning to the orbit increases the number of customers in the orbit an system increases apparently From Figures 3-5 we conclue that the leaving probability affect EN an EN ( ) particularly apparently than EN ( 1) In Figure 6 we plot the mean orbit size EN ( ) versus for p q 4 r 4 We observe that as approaches to the infinite the mean orbit size EN ( ) tens to stabilization Moreover we observe that as approaches the stability conition the mean orbit size tens to infinite (ue to the system becomes unstable) Figure 1 The relationship of P with p an Figure The relationship of P lost with p an 16

7 Journal of Convergence Information Technology Volume 5 Number April 1 Figure 3 EN ( 1) versus p for ifferent Figure 6 EN ( ) versus for ifferent 6 Conclusions Figure 4 EN ( ) versus p for ifferent In the foregoing analysis an M/G/1 queue with balking feeback an general retrial times is consiere to obtain expressions for various system performance measures of our interest Numerical examples have been carrie out to observe the tren of the ile probability an the mean system size for varying parametric values Base on the obtaine results the effect of balking an feeback are more apparent when the system is heavily loae For light loas balking has a minor effect on the ile probability an the mean system size of the priority queue The system performance measures are always sensitive to feeback 7 Acknowlegments The work was partly supporte by National Natural Science Founation of China (No75713) Yong Talents Funs of Anhui (1SQRL19) an Chaohu University Scientific Research Founation (XLZ-9 1) 8 References Figure 5 EN versus p for ifferent [1] BKrishna Kumar an DArivuainambi "The M/G/1 retrial queue with Bernoulli scheules an general retrial times" Computers an Mathematics with Application 48 pp 15-3 [] IAtencia PMoreno "A single-server retrial queue with general retrial times an Bernoulli scheule" Applie Mathematics an Computation 165 pp [3] A Aboul-Hassan SIRabia FATaboly "A iscrete time Geo/G/1 retrial queue with general retrial times an 161

8 An M/G/1 retrial queue with priority balking an feeback customers Peishu ChenYiuan Zhu balking customers" Journal of the Korean Statistical Society 348 pp [4] Zhu Yi-Juan Zhou Zong-Hao Feng Yan-Gang "M/G/1 retrial queue with priority an repair" Acta Automatica Sinica Vol348 pp x [5] Maan KC& Al-RawwashM "On the M / G / 1 queue with feeback an optional server vacations base on a single vacation policy" Applie Mathematics an computation 165 pp [6] Nahi Amani Peram Haipour an Farzaneh Seyemostafaei "An appropriate violation etection scenario for Service Level Agreements base on WS- Agreement Protocol" Journal of Convergence Information Technology Vol 5 No 1 1 pp 4-47 [7] Parke AG "Some conitions for ergoicity an recurrence of Markov chain" Operations Research pp