Conditions for stability and instability of retrial queueing systems with general retrial times
aa r X i v : . [ m a t h . P R ] F e b Conditions for stability and instability ofretrial queueing systems with general retrialtimes
Tewfik Kernane
Department of Probability and Statistics, Faculty of Mathematics, University ofSciences and Technology USTHB, Algiers, AlgeriaE-mail: [email protected]
Abstract
We study the stability of single server retrial queues under general distributionfor retrial times and stationary ergodic service times, for three main retrial policiesstudied in the literature: classical linear, constant and control policies. The approachused is the renovating events approach to obtain sufficient stability conditions bystrong coupling convergence of the process modeling the dynamics of the systemto a unique stationary ergodic regime. We also obtain instability conditions byconvergence in distribution to improper limiting sequences.
Key words:
Retrial queues; Stability; instability; Stochastic recursive sequence;Renovation events theory; Linear retrial policy; Constant retrial policy; Controlretrial policy; Strong coupling convergence
Introduction
The analysis of stability in queueing systems is the first step in studying suchmodels. The steady state solutions and performance characteristics of the sys-tem do not exist if it is not stable. The efficiency of a queueing system is relatedclosely to its stability and is considered as inefficient if it is unstable. Retrialqueues have the characteristic that an arriving customer who finds all waitingpositions and service zones occupied must join a group of ”blocked” customersin an additional queue called ”orbit” and reapplies for getting served after ran-dom time intervals according to a specific retrial policy. They arise in manypractical situations. The classical example can be found in telephone traffictheory where subscribers redial after receiving a busy signal. For computerand communication applications, peripherals in computer systems may make
Preprint submitted to Elsevier 22 September 2006 etrials to receive service from a central processor. Another example can beadopted from the aviation where an aircraft is directed into the waiting zone,if the runway is found busy, from which the demand of landing is repeated atrandom periods of time. Retrial queueing models are generally more compli-cated than traditional ones especially when dealing with general distributionfor retrial times. The existence of this supplementary flow from the orbit andthe random access to the server (as for the linear policies that depend on thenumber of customers in orbit) make the system more congested and difficultto model by simple random processes like Markovian ones, which have prop-erties that allow to derive easily conditions for stability, especially when we donot assume an exponential distribution (which has the memorylless property)allowing to obtain Markovian processes modeling the system. Furthermore,it has been observed in telecommunication systems that the exponential lawis not a good estimator for the distribution of retrial times (see Yang et al.,1994).The subject of this paper is to analyze the stability of single server retrialqueues under general distribution for retrial times and stationary ergodic ser-vice times (without independence assumption), for three main retrial policiesstudied in the literature: classical linear, constant and control policies. Stabil-ity results for such models with general retrial times are rare and generallyreduced to Markovian assumptions. For the linear retrial policy, Koba and Ko-valenko (2004) obtained a sufficient stability condition (arrival rate is less thanthe service rate) for an M/G/1 system with non-lattice distribution for retrialtimes satisfying an additional estimate condition, with i.i.d service times. Forthe constant retrial policy, Koba (2002) derived a stability condition for aGI/G/1 retrial system with a FIFO discipline for the access from the orbit tothe server and a general distribution for orbit time in latticed and non-latticedcases with i.i.d service times. For the control policy, Gomez-Corral (1999) stud-ied extensively an M/G/1 retrial queue with general retrial times where hederived the stability condition for i.i.d service times and a FIFO discipline.For non-independent service times, Altman and Borovkov (1997) obtained asufficient condition for the stability of a linear retrial queue under general sta-tionary ergodic service times and independent and exponentially distributedinterarrival and retrial times using the method of renovation events. Kernaneand A¨ıssani (2006) obtained sufficient conditions for the stability of variousretrial queues with versatile retrial policy which incorporates the constantand linear retrial policies under general stationary ergodic service times andindependent and exponentially distributed interarrival and retrial times.The main approach used in this paper is the method of renovation eventsoriginated in the work of Akhmarov and Leont’eva (1976) and developed byBorovkov (1984) in the stationary ergodic setting. In the following section,we derive stability and instability conditions for the classical linear retrialpolicy with general retrial times, stationary ergodic service times and Poisson2rrivals. In Section 3, we obtain a stability condition and an instability onefor the constant retrial policy system with general retrial times, stationaryergodic service times and Poisson arrivals. With the later assumptions, wederive in Section 4, stability and instability conditions for the control policyretrial model.
We begin by considering the classical single server retrial system with linearretrial policy. Customers arrive from outside according to a Poisson processwith rate λ. If an arriving customer finds the server busy, he joins the orbitand repeats his attempt to get served after random time intervals. We considerthe linear retrial policy where each customer in orbit attempts to get servedindependently of other customers and we assume that the sequence of inter-retrial times of a single customer is an independent sequence with generaldistribution R ( · ), density function r ( · ) and Laplace transform r ∗ ( z ) , z > . The successive service times { σ n } are assumed to form a stationary (in thestrict sense) and ergodic (which essentially means that time averages convergeto constants a.s ) sequence with 0 < E σ n < ∞ . The inter-arrival, inter-retrialand service times are assumed to be mutually independent.Let Q ( t ) be the number of customers in orbit at time t and denote by s n the instant when the n th service time ends. Consider the embedded process Q n = Q ( s n +) of the number of customers in orbit just after the end of the n thservice duration. Denote by N λ ( t ) the counting Poisson process with parameter λ which counts the number of arriving customers during a time interval (0 , t ] . If Q n = k, then we denote by π ( n ) , ..., π k ( n ) the residual retrial times (forwardrecurrence times) of the customers in orbit just after the instant s n and by γ n the residual external arrival time at the same instant.It is easy to see that the process Q n satisfies the following recurrence relation: Q n +1 = ( Q n + ξ n ) + , (1)where x + = max[0 , x ] and ξ n = N λ ( σ n ) − I { min( π ( n ) , ..., π Q n ( n )) < γ n } , (2)We have then expressed Q n as a Stochastic Recursive Sequence (SRS) (for thedefinition see Borovkov, 1998).We introduce the σ − algebra F σn generated by the set of random variables { σ k : k ≤ n } and F σ generated by the entire sequence { σ n : −∞ < n < + ∞} and for which any independent sequence not depending on { σ n } is F σ -measurable3see Borovkov (1976) p.14). Let U be the measure preserving shift transforma-tion of F σ -measurable random variables, that is U σ k = σ k +1 , and if η ∈ F σ then the sequence { η n = U n η : −∞ < n < + ∞} is a stationary ergodicsequence where U n is the n th iteration of U and U − n is the inverse transfor-mation of U n n ∈ Z . We shall denote by T the corresponding transformationof events in F σ , that is for any F σ -measurable sequence η n : T { ω : ( η ( ω ) , ..., η k ( ω )) ∈ ( B , ..., B k ) } = { ω : ( η ( ω ) , ..., η k +1 ( ω )) ∈ ( B , ..., B k ) } , (3)where the events B i ∈ F σ , i = 0 , ..., k. An event A ∈ F ξn + m , m ≥
0, is a renovation event for the SRS { Q n } on thesegment [ n, n + m ] if there exists a measurable function g such that on the set A Q n + m +1 = g ( ξ n , ..., ξ n + m ) . (4)The sequence A n , A n ∈ F ξn + m , is a renovating sequence of events for the SRS { Q n } if there exists an integer n such that (4) holds true for n ≥ n with acommon function g for all n. We say that the SRS { Q n } is coupling convergent to a stationary sequence { Q n = U n Q } if lim n →∞ P n Q k = Q k ; ∀ k ≥ n o = 1 . (5)Set ν k = min { n ≥ − k : U − k Q n + k = Q n } and ν = sup k ≥ ν k . A SRS { Q n } is strong coupling convergent to a stationary sequence { Q n = U n Q } if ν < ∞ with probability 1. Theorem 1
Assume that λ E σ < . Then the process { Q n } is strong couplingconvergent to a unique stationary ergodic regime.If λ E σ > , then the process { Q n } converges in distribution to an improperlimiting sequence. PROOF.
Since the driving sequence ξ n depend on Q n , we will proceed firstby considering an auxiliary sequence Q ∗ n which majorizes Q n and having adriving sequence ξ ∗ n independent of Q n and it has the following form: Q ∗ = Q , Q ∗ n +1 = max( C, Q ∗ n + ξ ∗ n ) , (6)where ξ ∗ n = N λ ( σ n ) − I { min( π ( n ) , ..., π C ( n )) < γ n } . (7)The constant integer C will be chosen later appropriately, and if Q ∗ n > C the C customers for which we consider the forward recurrence times π ( n ) , ..., π C ( n )are chosen randomly by an urn scheme without repetition. Following the pro-cedure used in Altman and Borovkov (1997) and later in Kernane and A¨ıssani42006), we will construct stationary renovation events with strictly positiveprobability for Q n from those of Q ∗ n , and applying an ergodic theorem (The-orem 11.4 in Borovkov, 1998) which states that an SRS is strong couplingconvergent to a unique stationary regime, satisfying the same recursion, ifthere exist stationary renovating events of strictly positive probability.The stationarity and ergodicity of ξ ∗ n follows from the fact that ξ ∗ n is F σ -measurable (for more details on the ergodicity and stationarity of ξ ∗ n see Ker-nane and A¨ıssani, 2006). We have E ξ ∗ n = λ E σ − P (min( π ( n ) , ..., π C ( n )) < γ n ) (8)= λ E σ − [1 − P ( π ( n ) ≥ γ n , ..., π C ( n ) ≥ γ n )] (9)= λ E σ − − ∞ Z λe − λt P ( π ( n ) ≥ t, ..., π C ( n ) ≥ t ) dt (10)= λ E σ − − ∞ Z λe − λt C Y i =1 P ( π i ( n ) ≥ t ) dt . (11)Since lim C →∞ C Q i =1 P ( π i ( n ) ≥ t ) = 0 , then by dominated convergence theoremlim C →∞ ∞ Z λe − λt C Y i =1 P ( π i ( n ) ≥ t ) dt = 0 . (12)If the condition λ E σ < C suchthat E ξ ∗ n < . It follows from example 11.1 in Borovkov (1998) that thereexists a stationary renovating sequence of events with positive probability for Q ∗ n , from which we deduce those of Q n (see Altman and Borovkov, 1997).Applying the ergodic theorem (Theorem 11.4 in Borovkov, 1998) we obtainthat the sequence Q n is strong coupling convergent to a unique stationaryprocess e Q n = U n e Q , with e Q F σ -measurable and since e Q n is an U − shifted F σ -measurable sequence then it is ergodic.For the instability condition, consider the auxiliary process Q Sn correspondingto a simple single server queue without retrials, that is Q S = Q , Q Sn +1 = ( Q Sn + ξ Sn ) + , (13)where ξ Sn = N λ ( σ n ) − . (14)Clearly Q Sn ≤ st Q n and it is well known that if λ E σ > n →∞ Q Sn =+ ∞ a.s. (see Theorem 1.7 in Borovkov, 1976). Thus, the process { Q n } con-verges in distribution to an improper limiting sequence.5 Constant Retrial Policy
Consider now a single server retrial queue governed by the constant retrialpolicy which is described as follows. After a random time generally distributed(which we will call the orbit retrial time), one customer from the orbit (at thehead of the queue or a randomly chosen one if any) take his service if theserver is free, so an orbit time can be in progress even though the server isbusy, this may happen in system where the orbit has no information aboutthe state of the server. The sequence of orbit cycle times { r i } is assumed to bei.i.d, having R ( · ) as cdf, r ( · ) as density function with mean E r and Laplacetransform r ∗ ( z ) , z > . Let π ( n ) be the forward recurrence time of the orbitretrial time after the end of the n th service time. Then the process Q n hasnow the following representation as a SRS: Q n +1 = ( Q n + ξ n ) + , (15)where ξ n = N λ ( σ n ) − I { π ( n ) < γ n } . (16) Theorem 2 If R is nonlattice and λ E σ < [1 − r ∗ ( λ )] λ E r , (17) then the process { Q n } is strong coupling convergent to a unique stationaryergodic regime.If λ E σ > (1 − r ∗ ( λ )) / ( λ E r ) . Then the process Q n converges in distributionto an improper limiting sequence. PROOF.
We have E ξ n = λ E σ − P ( π ( n ) < γ n ) . (18)Since the interarrival times are exponentially distributed then so is the residualarrival time γ n , hence P ( π ( n ) < γ n ) = + ∞ Z P ( π ( n ) < t ) λ e − λt dt. (19)Since we are interesting on steady state behaviour of the system and by as-suming a nonlattice (also called non-arithmetic) distribution R ( t ) for orbitretrial times, then from a well known result in renewal theory (see Cox, 1962)6e have the following asymptotic distribution for the forward recurrence time P ( π ( n ) < t ) = 1 E r t Z [1 − R ( x )] dx. (20)The formula (19) becomes P ( π ( n ) < γ n ) = 1 E r ∞ Z [1 − R ( x )] + ∞ Z x λe − λt dt dx (21)= 1 E r ∞ Z [1 − R ( x )] e − λx dx = 1 E r " − r ∗ ( λ ) λ . (22)Now if condition (17) is satisfied then E ξ n < . Since ξ n is F σ -measurable(generated by σ n ) then it is a stationary ergodic sequence. From this andexample 11.1 in Borovkov (1998), there exists a stationary sequence of reno-vation events with positive probability for { Q n } . Hence, using Theorem 11.4 ofBorovkov (1998), the sequence { Q n } is strong coupling convergent to a uniquestationary sequence e Q n obeying the equation e Q n +1 = ( e Q n + ξ n ) + , the ergodic-ity of e Q n follows from the fact that e Q n is an U − shifted sequence ( e Q n = U n e Q , with e Q F σ -measurable) generated by the stationary and ergodic sequence ξ n .The instability condition λ E σ > [1 − r ∗ ( λ )] /λ E r yields to E ξ n > , and itis well known that for SRS of the form Q n +1 = ( Q n + ξ n ) + this implies theconvergence of the process Q n to an improper limiting sequence (see Theorem1.7 of Borovkov (1976)). By assuming an exponential distribution with parameter θ for retrial times,that is R ( x ) = 1 − e − θx , it is well known that r ∗ ( s ) = s/ ( s + θ ) , and E r = 1 /θ. The condition (17) will read up, after some algebra, as follows λ E σ < θλ + θ . (23)Which is the condition obtained in the paper of Kernane and A¨ıssani (2006),in exponential retrial context. Consider a single server retrial queue with a control retrial policy. Primarycustomers enter from the outside according to a Poisson process with rate λ.
7f a primary customer finds the server busy upon arrival it joins the orbitto connect later according to the control retrial policy, which is described asfollows. Just after the end of a service time a generally distributed retrial timebegins to find the server free. If the retrial time finishes before an externalarrival, then one customer from the orbit (at the head of the queue or arandomly chosen one if any) receives its service and leaves the system. Weassume that the sequence of retrial times { r n } is an i.i.d sequence having r ( · )as pdf , R ( · ) as cdf and Laplace transform r ∗ ( · ), with finite mean E r . The n thservice duration of a call is σ n and we assume that the sequence of servicetimes { σ n } is stationary and ergodic with 0 < E σ < ∞ .The process { Q n } has the following representation as a stochastic recursivesequence SRS: Q n +1 = ( Q n + ξ n ) + , (24)where ξ n = N λ ( σ n ) − I { r n < γ n } , (25)where γ n is the residual arrival time of an external call at the end of the n thservice period. Theorem 3
Assume that λ E σ < r ∗ ( λ ) . (26) Then the process Q n is strong coupling convergent to a unique stationary er-godic regime.If λ E σ > r ∗ ( λ ) . Then the process Q n converges in distribution to an improperlimiting sequence. PROOF.
The proof is similar to that of Theorem 2, by noting that E ξ n = λ E σ − r ∗ ( λ ) . Assume that the retrial times are exponentially distributed with mean 1 /θ, then r ∗ ( λ ) = θ/ ( λ + θ ) and the stability condition (26) becomes: λ E σ < θλ + θ . (27)This condition is quite evident since it can be obtained from the constantpolicy from the memorylless property of the exponential distribution.8 .2 Hyperexponential distribution for retrial times Assume now that the retrial times follow the hyperexponential distributionwith density r ( x ) = pθ exp( − θx ) + (1 − p ) θ exp( − θ x ) , ≤ p < . Then r ∗ ( λ ) = θ [ λ ( p + (1 − p ) θ ) + θ ] / ( λ + θ ) ( λ + θ ) , and the stability condition(26) in this case is λ E σ < θ [ λ ( p + (1 − p ) θ ) + θ ]( λ + θ ) ( λ + θ ) . (28) The Erlang distribution has been found useful for describing random variablesin queueing applications. The density of an
Erlang ( n, µ ) distribution is givenby r ( x ) = µ n exp( − µx ) x n − / ( n − , x > n ∈ N ∗ . Its Laplace transformis r ∗ ( s ) = µ n / ( s + µ ) n . Then the control policy model will be stable if λ E σ < µλ + µ ! n . (29) Remark 4
It should be noted that the assumption E ξ n = 0 , and weak depen-dence among the ξ n , does not preclude the possibility that the process { Q n } converges to a proper stationary regime. Remark 5
Conditions for the stability of modified models with general retrialtimes, such as allowing breakdowns of the server, two types of arrivals, neg-ative arrivals and batch arrivals models may be obtained easily following theprocedure used in Kernane and A¨ıssani (2006). The conditions of the stabilitywill be written by replacing the left hand side of the classical linear policy bythe left hand sides of the case of a linear versatile policy obtained in Kernaneand A¨ıssani (2006). For the constant policy, we have to make the appropri-ate changes to the driving sequences in the SRS modeling the dynamics of themodified models in Kernane and A¨ıssani (2006) by considering the residualorbit time as shown here in Section 3, the conditions of stability will followdirectly after some algebra.
Remark 6
We may also consider the versatile retrial policy by incorporatingthe residual orbit retrial time π ( n ) in the equation 2 and considering the wholeretrial times of the customers in orbit r ( n ) , ..., r Q n ( n ) as follows ξ n = N λ ( σ n ) − I { min( π ( n ) + r ( n ) , ..., π ( n ) + r Q n ( n )) < γ n } , (30) the condition of Theorem 1 still holds for this versatile retrial policy, by noting hat in the proof we have to consider E ξ ∗ n = λ E σ − − ∞ Z t Z λe − λt P ( r ( n ) ≥ t − s, ..., r C ( n ) ≥ t − s ) dG ( s ) dt , (31) where G ( s ) is the cdf of the residual orbit retrial time satisfying G ( s ) = 1 E α s Z [1 − A ( x )] dx, (32) with E α the mean of the orbit retrial time and A ( x ) its cdf. References [1] Akhmarov, I., Leont’eva, N.P. (1976), Conditions for convergence to limitprocesses and the strong law of large numbers for queueing systems,
Teor.Veroyatnost i ee Primenen
21 (in Russian) pp. 559-570, MR 58
QueueingSystems.
26, 343-363.[3] Borovkov, A.A., (1976),
Stochastic Processes in Queueing Theory (Springer-Verlag)[4] Borovkov, A.A., (1984)
Asymptotic Methods in Queueing Theory (John Wiley& Sons).[5] Borovkov, A.A., (1998),
Ergodicity and Stability of Stochastic Processes (JohnWiley & Sons) .[6] Cox, D.R., (1962)
Renewal theory , Methuen, London.[7] Gomez-Corral, A. (1999), Stochastic analysis of a single server retrial queuewith general retrial times,
Naval Research Logistics . 46, 561-581.[8] Kernane, T., A¨ıssani, A. (2006), Stability of retrial queues with versatile retrialpolicy,
Journal of Applied Mathematics and Stochastic Analysis,
Volume 2006,Article ID 54359, Pages 1–16.[9] Koba, O.V. (2002), On a
GI/G/
Theory of Stochastic Processes . 8, 201-207 .[10] Koba, O.V., Kovalenko, I.M. (2004), The ergodicity condition for a retrialsystem with a non-latticed distribution of a cycle on an orbit,
Dop. NANUkrainy . 8, 70-77.[11] Yang, T., Posner, M. J. M., Templeton J. G. C., Li, H. (1994), An approximationmethod for the M/G/1 retrial queue with general retrial times,
EuropeanJournal of Operational Research . 76: 552-562.. 76: 552-562.