Heavy-traffic Asymptotics of Priority Polling System with Threshold Service Policy
HHeavy-traffic Asymptotics of Priority Polling Systemwith Threshold Service Policy
Liu Zaiming, Chu Yuqing, Wu Jinbiao
Department of Mathematics and Statistics, Central South University, Changsha, Hunan410083, PR China
Abstract
In this paper, by the singular-perturbation technique, we investigate theheavy-traffic behavior of a priority polling system consisting of three
M/M/ N -policy vacation.Based on this fact, we provide the exact tail asymptotics of the vacationpolling system to approximate the tail distribution of the queue lengths of thestable queues, which shows that it has the same prefactors and decay ratesas the classical M/M/
Keywords:
Polling System, Heavy-traffic, Singular-perturbation, Tail Asymptotic,Stochastic Simulation
1. Introduction
The study of the two-queue priority polling system is motivated by itswide applications in computer and communication systems, such as ATM(Asynchronous Transfer Mode) switch systems and network standards likeDQDB(Distributed Queue Dual Bus). ATM involves two different types oftraffic: real time traffic(voice, video) and non-real time traffic (data), which
Email addresses: [email protected] (Liu Zaiming), [email protected] (Chu Yuqing),
Corresponding author: [email protected] (Wu Jinbiao) a r X i v : . [ m a t h . P R ] A ug lso need different types of QoS (Quality of Service) standard. By setting thethreshold parameter, a higher priority is offered to real time traffic to shortenits delay and the delay of non-real time traffic is kept in a valid regime, whichturns out to be a flexible way to control the operation of the whole system.Lee and Sengupta first investigated the threshold-based priority systemsin [1]. Later, a special case of two-queue M/M/ k -limited policies and presented theheavy-traffic behavior. It is noted that the singular-perturbation techniquecan be easily extended to a multi-queue system since it only needs the balanceequations.With the singular-perturbation technique, we conclude that the queuelengths in the stable queues have the same joint distribution as Model II, apreemptive priority polling system with N -policy vacation. In general, noclosed-form expressions for the steady-state probabilities in Model II can beobtained. Using the Kernel method, which is reported detailedly in [11, 12],we present the exact tail asymptotics of queue lengths in Model II, whichcan further approximate the tail asymptotics of the stable queues.The remainder of this paper is organized as follows. In Section 2, themodel and some notations are introduced. In Section 3, the singular-perturbationtechnique is applied to derive the heavy-traffic limits and the detailed deriva-tion is carried out in Section 4. In Section 5, we provide the exact tail asymp-totics of queue lengths in Model II to approximate the tail asymptotics ofthe stable queues. In Section 6, a simulation is undertaken to evaluate theheavy-traffic asymptotics. We finally conclude the whole procedure and pro-2ose some topics for further research in Section 7.
2. Model Description
We consider a polling model with single server consisting of three queues Q , Q , Q . We refer to the customers queueing in Q i as the type i customers, i = 1 , ,
3. The buffer capacity of each queue is infinite. Customers arriveat Q i independently according to a Poisson process with rate λ i . For type i customers, the service times are mutually independent and all follow anexponential distribution with rate µ i . Q has the HoL priority and Q hasa higher priority over Q . In each queue customers are served accordingto FCFS discipline. We assume that the arrival processes and the serviceprocesses are independent. The service discipline is described as follows.1. Q is served exhaustively, which means that the server serves the cus-tomers in Q until it is empty and then switches to Q ;2. When the server is serving a customer in Q , if a type 1 customer arrives,then the server switches to Q immediately, otherwise, it continues servingthe customers in Q until Q becomes empty and then switches to Q ;3. When the server is serving a customer in Q , if a type 1 customer arrives,then the server switches to Q immediately, if the size of Q reaches agiven threshold N and Q is empty, then the server switches to Q im-mediately, otherwise, it continues serving the customers in Q until Q becomes empty and then switches to Q .It is assumed that all the switches are instantaneous. In addition, theswitches caused by the threshold push the customer undergoing service tothe head of the queue and the service of the interrupted customer resumesfrom the beginning.The traffic load of Q i is denoted by ρ i = λ i /µ i , i = 1 , ,
3. We assumethe ergodicity condition of the system ρ = ρ + ρ + ρ < X i ( t ) be the number of customers in Q i at time t , and S ( t ) be the po-sition of the server at time t with S ( t ) ∈ { , , } . The associated stochasticprocess { Y ( t ) , t ≥ } = (cid:8)(cid:0) X ( t ) , X ( t ) , X ( t ) , S ( t ) (cid:1) , t ≥ (cid:9) is an aperiodicand irreducible four-dimensional Markov process. Let X i ( i = 1 , ,
3) be thesteady-state queue length of Q i and S be the steady-state position of theserver. Define the stationary probabilities: p s ( x , x , x ) = lim t →∞ P r { Y ( t ) = ( x , x , x , s ) } , s = 1 , , .
3e study the heavy-traffic limits of the joint queue-length distribution byincreasing the arrival rate λ so as to ρ → − , while keeping λ (cid:54) = 0, λ (cid:54) = 0and µ , µ , µ fixed. When ρ → − , Q becomes critically loaded, whereas Q and Q remain stable since Q and Q have higher priorities over Q .The single-perturbation technique is implemented here. We first applya perturbation to λ in the balance equations, in which case Q is close tobecoming critically loaded. Then we solve the lowest order terms in the bal-ance equations to obtain the queue-length distributions of the stable queues Q and Q . At last we solve the first-order and second-order terms to get adifferential equation and compute the scaled number of customers in Q .Applying the Markov property, we obtain the following balance equationswhen x ≥ λ + λ + λ + µ ) p ( x , x , x )= λ p ( x − , x , x ) δ ( x ≥
2) + λ p ( x , x − , x ) δ ( x ≥ λ p ( x , x , x −
1) + λ p (0 , x , x ) δ ( x = 1 , x ≥ µ p ( x + 1 , x , x ) + λ p (0 , x , x ) δ ( x = 1 , x < N ) , x ≥ , x ≥ , (1)( λ + λ + λ + µ ) p (0 , x , x )= λ p (0 , x − , x ) δ ( x ≥
2) + λ p (0 , x , x −
1) + µ p (1 , x , x )+ λ p (0 , N − , x ) δ ( x = N ) + µ p (0 , x + 1 , x ) , x ≥ , (2)( λ + λ + λ + µ ) p (0 , x , x )= λ p (0 , x − , x ) δ ( x ≥
1) + λ p (0 , x , x −
1) + µ p (0 , x , x + 1)+ [ µ p (1 , , x ) + µ p (0 , , x )] δ ( x = 0) , ≤ x ≤ N − , (3)where δ ( · ) is Kronecker function.In the above equations, we have omitted the parts for x = 0 and x = 1which do not play a role after the perturbation since X tends to infinity as Q becomes critically loaded and the probability of Q being empty or 1 goesto zero.Throughout the paper, we adopt the standard notations: a function F ( x )is o ( x ) if F ( x ) /x → x →
0; a function F ( x ) is O ( x ) if there exists a c ≥ F ( x ) /x → c as x → O (1) is a constant timecomplexity; functions f ( n ) and g ( n ) of nonnegative integers n , f ( n ) ∼ g ( n )means lim n →∞ f ( n ) g ( n ) = 1. 4 . Perturbation From the stability condition the system becomes unstable as ρ → − ρ − ρ , i.e. λ → µ (1 − ρ − ρ ). Therefore it is assumed that λ = µ (1 − ρ − ρ ) − εω, ω > , < ε (cid:28) . (4)Let ζ = εx , and p s ( x , x , x ) = p s ( x , x , ζ/ε ) = εφ s, ( x ,x ) ( ζ, ε ) , < ζ = O (1) , s = 1 , , . (5)Taking (4) and (5) into the balance equations (1)-(3) and then taking theTaylor expansion, we obtain( λ + λ + µ ) φ , ( x ,x ) ( ζ, ε )= λ φ , ( x − ,x ) ( ζ, ε ) δ ( x ≥
2) + λ φ , ( x ,x − ( ζ, ε ) δ ( x ≥ − ( µ (1 − ρ − ρ ) − εω ) (cid:18) ε ∂φ , ( x ,x ) ( ζ, ε ) ∂ζ − ε ∂ φ , ( x ,x ) ( ζ, ε ) ∂ζ (cid:19) + λ φ , (0 ,x ) ( ζ, ε ) δ ( x = 1 , x ≥
1) + µ φ , ( x +1 ,x ) ( ζ, ε )+ λ φ , (0 ,x ) ( ζ, ε ) δ ( x = 1 , x < N ) + o ( ε ) , x ≥ , x ≥ , (6)( λ + λ + µ ) φ , (0 ,x ) ( ζ, ε )= ( µ (1 − ρ − ρ ) − εω ) (cid:18) − ε ∂φ , (0 ,x ) ( ζ, ε ) ∂ζ + ε ∂ φ , (0 ,x ) ( ζ, ε ) ∂ζ (cid:19) + λ φ , (0 ,x − ( ζ, ε ) δ ( x ≥
2) + µ φ , (1 ,x ) ( ζ, ε )+ λ φ , (0 ,N − ( ζ, ε ) δ ( x = N ) + µ φ , (0 ,x +1) ( ζ, ε ) + o ( ε ) , x ≥ , (7)( λ + λ ) φ , (0 ,x ) ( ζ, ε )= ( µ (1 − ρ − ρ ) − εω ) (cid:18) − ε ∂φ , (0 ,x ) ( ζ, ε ) ∂ζ + ε ∂ φ , (0 ,x ) ( ζ, ε ) ∂ζ (cid:19) + µ (cid:18) ε ∂φ , (0 ,x ) ( ζ, ε ) ∂ζ + ε ∂ φ , (0 ,x ) ( ζ, ε ) ∂ζ (cid:19) + (cid:2) µ φ , (1 , ( ζ, ε ) + µ φ , (0 , ( ζ, ε ) (cid:3) δ ( x = 0)+ λ φ , (0 ,x − ( ζ, ε ) δ ( x ≥
1) + o ( ε ) , ≤ x ≤ N − . (8)It is noted that λ only plays a role in equations for O ( ε ) terms and higher.Throughout the paper, we do Taylor expansions of φ s, ( x ,x ) ( ζ, ε ) ( s = 1 , , ε as follows φ s, ( x ,x ) ( ζ, ε ) = φ (0) s, ( x ,x ) ( ζ ) + εφ (1) s, ( x ,x ) ( ζ ) + o ( ε ) , s = 1 , , . (9)5n the next section the lowest order terms of the resulting equations afterTaylor expansions are equated to find expressions for φ (0) s, ( x ,x ) ( ζ ) ( s = 1 , , Q .For convenience, we introduce the corresponding probability generatingfunctions(PGFs): Q ( j )1 ( x, y, ζ ) = ∞ (cid:88) x =1 ∞ (cid:88) x =0 φ ( j )1 , ( x ,x ) ( ζ ) x x − y x , j = 0 , ,Q ( j )2 ( y, ζ ) = ∞ (cid:88) x =1 φ ( j )2 , (0 ,x ) ( ζ ) y x − , j = 0 , ,Q ( j )3 ( y, ζ ) = N − (cid:88) x =0 φ ( j )3 , (0 ,x ) ( ζ ) y x , j = 0 , ,Q ( x, y, ζ, ε ) = ∞ (cid:88) x =1 ∞ (cid:88) x =0 φ , ( x ,x ) ( ζ, ε ) x x − y x ,Q ( y, ζ, ε ) = ∞ (cid:88) x =1 φ , (0 ,x ) ( ζ, ε ) y x − ,Q ( y, ζ, ε ) = N − (cid:88) x =0 φ , (0 ,x ) ( ζ, ε ) y x .
4. Model analysis
Equating the lowest-order terms of the resulting equation after the Taylorexpansions of (6)-(8), we obtain( λ + λ + µ ) φ (0)1 , ( x ,x ) ( ζ )= λ φ (0)1 , ( x − ,x ) ( ζ ) δ ( x ≥
2) + λ φ (0)1 , ( x ,x − ( ζ ) δ ( x ≥ λ φ (0)2 , (0 ,x ) ( ζ ) δ ( x = 1 , x ≥
1) + µ φ (0)1 , ( x +1 ,x ) ( ζ )+ λ φ (0)3 , (0 ,x ) ( ζ ) δ ( x = 1 , x < N ) , x ≥ , x ≥ , (10)6 λ + λ + µ ) φ (0)2 , (0 ,x ) ( ζ )= λ φ (0)2 , (0 ,x − ( ζ ) δ ( x ≥
2) + µ φ (0)1 , (1 ,x ) ( ζ )+ λ φ (0)3 , (0 ,N − ( ζ ) δ ( x = N ) + µ φ (0)2 , (0 ,x +1) ( ζ ) , x ≥ , (11)( λ + λ ) φ (0)3 , (0 ,x ) ( ζ ) = λ φ (0)3 , (0 ,x − ( ζ ) , ≤ x ≤ N − , (12)( λ + λ ) φ (0)3 , (0 , ( ζ ) = µ φ (0)1 , (1 , ( ζ ) + µ φ (0)2 , (0 , ( ζ ) . (13)We introduce P ( ζ ) and π (0) s, ( x ,x ) such that φ (0) s, ( x ,x ) ( ζ ) = π (0) s, ( x ,x ) P ( ζ ) , s = 1 , , , ∞ (cid:88) x =1 ∞ (cid:88) x =0 π (0)1 , ( x ,x ) + ∞ (cid:88) x =1 π (0)2 , (0 ,x ) + N − (cid:88) x =0 π (0)3 , (0 ,x ) = 1 . Define L (0)1 ( x, y ) = ∞ (cid:88) x =1 ∞ (cid:88) x =0 π (0)1 , ( x ,x ) x x − y x , L (0)2 ( y ) = ∞ (cid:88) x =1 π (0)2 , (0 ,x ) y x − ,L (0)3 ( y ) = N − (cid:88) x =0 π (0)3 , (0 ,x ) y x . Then it is clear that Q (0)1 ( x, y, ζ ) = L (0)1 ( x, y ) P ( ζ ) , (14) Q (0)2 ( y, ζ ) = L (0)2 ( y ) P ( ζ ) , (15) Q (0)3 ( y, ζ ) = L (0)3 ( y ) P ( ζ ) . (16)From (12), we get L (0)3 ( y ) = N − (cid:88) x =0 ( r y ) x π (0)3 , (0 , = H ( y ) π (0)3 , (0 , = β ( y ) L (0)3 (1) , (17)where r = λ λ + λ , H ( y ) = ( r y ) N − r y − and β ( y ) = H ( y ) H (1) .Using the PGFs to rewrite the balance equations (10) and (11) leads to xK ( x, y ) L (0)1 ( x, y ) = λ x [ yL (0)2 ( y ) + L (0)3 ( y )] − µ L (0)1 (0 , y ) , (18)7 a ( y ) L (0)2 ( y ) = µ L (0)1 (0 , y ) − [ λ + λ (1 − y )] L (0)3 ( y ) , (19)where K ( x, y ) = λ (1 − x ) + λ (1 − y ) + µ (cid:18) − x (cid:19) ,a ( y ) = λ + λ (1 − y ) + µ (cid:18) − y (cid:19) . Clearly, for every | y | ≤
1, the kernel xK ( x, y ) has a unique zero: x = α ( y ).Applying the Kernel method to (18) and (19), it is easy to get L (0)1 ( x, y ) = λ µ [ x − α ( y )]( y − xK ( x, y )[ ya ( y ) − λ yα ( y )] L (0)3 ( y ) , (20) L (0)2 ( y ) = λ ( α ( y ) −
1) + λ ( y − ya ( y ) − λ yα ( y ) L (0)3 ( y ) , (21)Letting y → x → L (0)1 (1 ,
1) = ρ − ρ − ρ L (0)3 (1) and L (0)2 (1) = ρ − ρ − ρ L (0)3 (1). By thenormalizing condition, it is easy to get L (0)3 (1) = 1 − ρ − ρ . Therefore, wehave L (0)1 (1 ,
1) = ρ and L (0)2 (1) = ρ . Moreover, L (0)3 ( y ) = β ( y )(1 − ρ − ρ ) . (22)It is not hard to see that equations (20)-(22) actually state an M/M/ N -policy vacation, denoted as ModelII for short, described as follows:There are two classes of customers in the system, the high- and low-priority customers, arriving independently according to two Poisson processeswith rates λ and λ , respectively. Each class of customer is served accordingto the FCFS discipline. The server takes a vacation once the system emptiesand goes back to work once the size of the low-priority customers reaches N orthere is a high-priority customer’s arrival. The high-priority customers havepreemptive priorities over the low-priority customers just like in the classicaltwo-queue preemptive priority queueing system. Both classes of customersrequire an exponential amount of service times and are served with servicerates µ and µ , respectively. All service times are independent and alsoindependent of the arrival processes.We determine the unkown expression of P ( ζ ) in the rest of this section.8 .2. Equating the first-order terms In this subsection, by equating the first-order terms of the resulting equa-tions after the Taylor expansion of the perturbed balance equations (6)-(8),we present an equation in Proposition 1.
Proposition 1. (1 − ρ − ρ ) (cid:104) Q (1)1 (1 , , ζ ) + Q (1)2 (1 , ζ ) + Q (1)3 (1 , ζ ) (cid:105) − Q (1)3 (1 , ζ )= − (cid:20) µ µ ρ + µ µ ρ (cid:21) P (cid:48) ( ζ ) . Proof.
Taking the PGF of the first-order terms of the resulting equationsafter the Taylor expansion of (6)-(8), we have xK ( x, y ) Q (1)1 ( x, y, ζ ) = λ xyQ (1)2 ( y, ζ ) − µ Q (1)1 (0 , y, ζ ) + λ xQ (1)3 ( y, ζ ) − µ x (1 − ρ − ρ ) L (0)1 ( x, y ) P (cid:48) ( ζ ) , (23) ya ( y ) Q (1)2 ( y, ζ ) = λ y N φ (1)3 , (0 ,x − ( ζ ) − µ Q (1)1 (0 , , ζ ) − µ Q (1)2 (0 , ζ )+ µ Q (1)1 (0 , y, ζ ) − µ y (1 − ρ − ρ ) L (0)2 ( y ) P (cid:48) ( ζ ) , (24)[ λ + λ (1 − y )] Q (1)3 ( y, ζ ) = − λ y N φ (1)3 , (0 ,x − ( ζ ) + µ Q (1)1 (0 , , ζ )+ µ Q (1)2 (0 , ζ ) + µ ( ρ + ρ ) L (0)3 ( y ) P (cid:48) ( ζ ) . (25)Applying the Kernel method to (23)-(25), after some elementary calculations,we get Q (1)1 ( x, y, ζ ) = λ [ x − α ( y )] xK ( x, y ) [ yQ (1)2 ( y, ζ ) + Q (1)3 ( y, ζ )] − µ (1 − ρ − ρ ) xK ( x, y ) [ xL (0)1 ( x, y ) − α ( y ) L (0)1 ( α ( y ) , y )] P (cid:48) ( ζ ) . (26)[ ya ( y ) − λ α ( y ) y ] Q (1)2 ( y, ζ ) + [ λ (1 − α ( y )) + λ (1 − y )] Q (1)3 ( y, ζ )= µ (cid:110) ( ρ + ρ ) L (0)3 ( y ) − (1 − ρ − ρ ) (cid:104) yL (0)2 ( y ) + α ( y ) L (0)1 ( α ( y ) , y ) (cid:105)(cid:111) P (cid:48) ( ζ ) . (27)Letting y → x → Q (1)1 (1 , , ζ ) = ρ − ρ [ Q (1)2 (1 , ζ )+ Q (1)3 (1 , ζ )] − µ µ ρ (1 − ρ − ρ ) (1 − ρ ) P (cid:48) ( ζ ) . (28)9etting y → − ρ (cid:104) (1 − ρ − ρ ) Q (1)2 (1 , ζ ) − ρ Q (1)3 (1 , ζ ) (cid:105) = (cid:26) − µ µ (cid:20) ρ ρ − ρ + ρ ρ (1 − ρ − ρ )(1 − ρ ) (cid:21) − µ µ ρ (cid:27) P (cid:48) ( ζ ) . (29)From (28) and (29), we have(1 − ρ − ρ ) (cid:104) Q (1)1 (1 , , ζ ) + Q (1)2 (1 , ζ ) + Q (1)3 (1 , ζ ) (cid:105) − Q (1)3 (1 , ζ )= 11 − ρ (cid:104) (1 − ρ − ρ ) Q (1)2 (1 , ζ ) − ρ Q (1)3 (1 , ζ ) (cid:105) − µ µ ρ (1 − ρ − ρ ) (1 − ρ ) P (cid:48) ( ζ )= − (cid:20) µ µ ρ + µ µ ρ (cid:21) P (cid:48) ( ζ ) . (cid:3) In this subsection we consider the sum of all O ( ε ) terms in equations(6)-(8) to determine P ( ζ ).Taking the summation over all x and x of (6)-(8), we get µ ∞ (cid:88) x =0 φ , (1 ,x ) ( ζ, ε )= λ ∞ (cid:88) x =1 φ , (0 ,x ) ( ζ, ε ) + λ N − (cid:88) x =0 φ , (0 ,x ) ( ζ, ε ) − µ (1 − ρ − ρ ) ε ∂Q (1 , , ζ, ε ) ∂ζ + (cid:20) ω ∂Q (1 , , ζ, ε ) ∂ζ + µ (1 − ρ − ρ )2 ∂ Q (1 , , ζ, ε ) ∂ζ (cid:21) ε + O ( ε ) , (30) λ ∞ (cid:88) x =1 φ , (0 ,x ) ( ζ, ε ) + µ φ , (0 , ( ζ, ε )= µ ∞ (cid:88) x =1 φ , (1 ,x ) ( ζ, ε ) + λ φ , (0 ,N − ( ζ, ε ) − µ (1 − ρ − ρ ) ε ∂Q (1 , ζ, ε ) ∂ζ + (cid:20) ω ∂Q (1 , ζ, ε ) ∂ζ + µ (1 − ρ − ρ )2 ∂ Q (1 , ζ, ε ) ∂ζ (cid:21) ε + O ( ε ) , (31)10 N − (cid:88) x =0 φ , (0 ,x ) ( ζ, ε ) + λ φ , (0 ,N − ( ζ, ε )= µ φ , (1 , ( ζ, ε ) + µ φ , (0 , ( ζ, ε ) − µ (1 − ρ − ρ ) ∂Q (1 , ζ, ε ) ∂ζ + µ ∂Q (1 , ζ, ε ) ∂ζ + (cid:20) ω ∂Q (1 , ζ, ε ) ∂ζ + µ (2 − ρ − ρ )2 ∂ Q (1 , ζ, ε ) ∂ζ (cid:21) ε + O ( ε ); (32)Summing over (30)-(32), we obtain0 = (cid:34) − µ (1 − ρ − ρ ) (cid:18) ∂Q (1 , , ζ, ε ) ∂ζ + ∂Q (1 , ζ, ε ) ∂ζ + ∂Q (1 , ζ, ε ) ∂ζ (cid:19) + µ ∂Q (1 , ζ, ε ) ∂ζ (cid:35) ε + (cid:34) µ (1 − ρ − ρ )2 (cid:16) ∂ Q (1 , , ζ, ε ) ∂ζ + ∂ Q (1 , ζ, ε ) ∂ζ + ∂ Q (1 , ζ, ε ) ∂ζ (cid:17) + ω (cid:16) ∂Q (1 , , ζ, ε ) ∂ζ + ∂Q (1 , ζ, ε ) ∂ζ + ∂Q (1 , ζ, ε ) ∂ζ (cid:17) + µ ∂ Q (1 , ζ, ε ) ∂ζ (cid:35) ε + O ( ε ) , (33)Now taking the Taylor expansion (9) of equation (33), we obtain0 = (cid:104) µ (1 − ρ − ρ ) P (cid:48)(cid:48) ( ζ ) + ωP (cid:48) ( ζ ) + µ Q (cid:48) (1)3 (1 , ζ ) − µ (1 − ρ − ρ ) × (cid:16) Q (cid:48) (1)1 (1 , , ζ ) + Q (cid:48) (1)2 (1 , ζ ) + Q (cid:48) (1)3 (1 , ζ ) (cid:17) (cid:105) ε + O ( ε )= (cid:20) µ (cid:18) − ρ − ρ + µ µ ρ + µ µ ρ (cid:19) P (cid:48)(cid:48) ( ζ ) + ωP (cid:48) ( ζ ) (cid:21) ε + O ( ε ) . (34)In (34), the first equation follows from (14)-(16) and the second equationfollows from Proposition 1.From the above derivation procedure, we can conclude the followingProposition. Proposition 2.
After taking the summation over all x and x of the Taylorseries of all perturbed balance equations (6) - (8) , the O (1) and O ( ε ) terms ancel and, moreover, equating the O ( ε ) terms yields the following differen-tial equation for P ( ζ ) : ωP (cid:48) ( ζ ) = − (cid:20) (1 − ρ − ρ ) + µ µ ρ + µ µ ρ (cid:21) µ P (cid:48)(cid:48) ( ζ ) . Now we can finally present the density of the scaled number of customersin Q , i.e. P ( ζ ). It can be obtained by combining the differential equationin Proposition 2 with (cid:82) ∞ P ( ζ ) dζ = 1 that P ( ζ ) = η e − ηζ , with ωη = (cid:104) − ρ − ρ + µ µ ρ + µ µ ρ (cid:105) µ .As a special case, we may take ω = µ , which gives ζ = (1 − ρ ) X , then1 η = 1 − ρ − ρ + µ µ ρ + µ µ ρ . By applying the multiclass distributional law of Bertsimas and Mourtzinou[13] it directly follows that the scaled waiting time at Q follows an exponen-tial distribution with parameter µ η . Theorem 1.
For λ = µ (1 − ρ − ρ ) − εω , we have lim ε ↓ P { X ≤ x , X ≤ x , εX ≤ ζ } = L ( x , x )(1 − e − ηζ ) , where L ( · , · ) is the joint cumulative distribution function(cdf ) of the queuelengths of a preemptive priority polling system with N-policy vacation de-scribed in subsection 4.1. The main result stated in Theorem 1 can be interpreted as follows: in theheavy-traffic regime,R1. The queue lengths in the stable queues have the same distribution asthat of a preemptive priority polling system with N -policy vacation.R2. The scaled number of customers in the critically loaded queue is expo-nentially distributed with parameter η .123. The queue lengths in the stable queues and the (scaled) number of cus-tomers in the critically loaded queue are independent.For R1, since Q is critically loaded, Q would be visited during eachcycle. From the perspective of Q and Q , the server goes on a vacation oncethe server goes to Q when Q and Q are empty, and goes back to workonce a type 1 customer arrives or there are N type 2 customers queueing,which actually is an N -policy vacation.For R2, we note that the total workload in the system equals the amountof workload in an M/G/1 queue with arrival rate λ + λ + λ and hyper-exponentially distributed service times, i.e. the service time is exponentiallydistributed with parameter µ i with probability λ i λ + λ + λ , i = 1 , ,
3. Basedon the heavy-traffic results for the M/G/1 queue (see [13]), the distributionof the scaled total workload converges to an exponential distribution withmean ρ E [ R ], where R is a residual service time and E [ R ] = µ ρ + µ ρ + µ ρ ρ . In the heavy traffic, since almost all customers are located in Q , the totalnumber of customers at this queue is also exponentially distributed withmean µ (cid:16) µ ρ + µ ρ + µ ρ (cid:17) . Since λ ↑ µ (1 − ρ − ρ ), the scaled numberof customers in Q is exponentially distributed with parameter η .Finally, R3 follows from the time-scale separation in the heavy trafficwhich implies that the dynamics of the stable queues evolve at a much fastertime scale than the dynamics of the critically loaded queue. Since the amountof “memory” of the stable queues asymptotically vanish compared to thatof the critically loaded queue, the queue lengths in the stable queues areindependent of the (scaled) number of customers in the critically loadedqueue in the limit. Remark 1.
From the above procedure, it is easy to see that, when there isa single critically loaded queue in the heavy traffic, the stable queues withthreshold policies can always be transferred into a priority polling systemwith N -policy vacation.
5. Exact tail asymptotics in Model II
In Section 4, we have derived the PGFs of the queue-length distribu-tions of the stable queues, which have the same distributions as Model II. As13nown, no closed-form expressions for the steady-state queue-length proba-bilities can be obtained. In this section, we carry out a detailed analysis onthe exact tail asymptotics for the stationary distributions in Model II, whichprovides us an approximation of the stable queues.
First we introduce some necessary notations. The marginal distributionsfor the high- and low-priority customers are denoted by π ( h ) i and π ( l ) j , respec-tively. When j >
0, we write π ( l ) j = π ( l )1 ,j + π ( l )2 ,j , where π ( l ) s,j is the marginal dis-tribution of the low-priority customers when the server is visiting Q s , s = 1 , π ( T ) n . Let λ = λ + λ and ρ = λ/µ . Without loss of generality, throughout thissection we assume that λ + λ + µ + µ = 1. To completely derive the exacttail asymptotics, we first introduce the following notations: b = λ λ + ( √ µ − √ λ ) , b = λ λ + ( √ µ + √ λ ) , ∆( y ) = ( λ + µ − λ y ) − λ µ = λ (1 − b y )(1 − b y ) /b b ,x ( y ) = ( λ + µ − λ y ) − (cid:112) ∆( y )2 λ = α ( y ) ,x ( y ) = ( λ + µ − λ y ) + (cid:112) ∆( y )2 λ ,xK ( x, y ) = − λ x + ( λ + µ − λ y ) x − µ = − λ ( x − x ( y ))( x − x ( y )) ,c = ( λ + µ ) − (cid:112) ( λ + µ ) − λ µ µ , c = λ c (cid:112) ( λ + µ ) − λ µ ,x = x (0) = c ρ , x = x (0) = 1 c ,F ( y ) = λ y − (1 − µ + µ ) y + 2 µ ,T ∗ ( y ) = F ( y ) + y (cid:112) ∆( y ) , T ( y ) = F ( y ) − y (cid:112) ∆( y ) ,η = (1 − µ ) + (cid:112) (1 − µ ) + 4( µ − µ ) λ µ ,η = (1 − µ ) − (cid:112) (1 − µ ) + 4( µ − µ ) λ µ ,T ( y ) T ∗ ( y ) = 4 µ (1 − y )(1 − η y )(1 − η y ) , = 1 − ρ − ρ µ η η − η , b = 1 − ρ − ρ µ η η − η ,D = ( λ + µ − (cid:112) λ µ )( µ − µ − (cid:112) λ µ ) + λ µ . Define the following PGFs of the stationary queue-length distributions: ψ (0) j ( x ) = ∞ (cid:88) i =1 π (0)1 , ( i,j ) x i − , j = 0 , , , . . . ,L ( l ) ( y ) = ∞ (cid:88) n =0 π ( l ) n y n , L ( T ) ( y ) = ∞ (cid:88) n =0 π ( T ) n y n . Now we present some Propositions to give the exact expressions of thePGFs defined above.
Proposition 3. L (0)1 ( x,
1) = ρ (1 − ρ )1 − ρ x , (35) L (0)1 (1 , y ) = µ − λ yλ L (0)2 ( y ) − L (0)3 ( y ) , (36) L (0)1 ( y, y ) = λy − µ µ (1 − ¯ ρ y ) L (0)2 ( y ) + λµ (1 − ¯ ρ y ) L (0)3 ( y ) , (37) L (0)1 ( x,
0) = c − c x L (0)3 (0) , (38) where L (0)1 ( x, y ) and L (0)2 ( y ) are expressed in (20) and (21) respectively. Proof.
Adding (19) to (18), we have xK ( x, y ) L (0)1 ( x, y ) = y [ λ x − a ( y )] L (0)2 ( y )+[ λ ( x − λ ( y − L (0)3 ( y ) . (39)Then, letting y → x → x → y and y →
0, respectively, we get equations(35)-(38). (cid:3)
Proposition 4. ψ (0)0 ( x ) = c − c x L (0)3 (0) , (40)15 (0) j ( x ) = a j − c x + λ c xλ (1 − c x ) ψ (0) j − ( x ) − ψ (0) j − ( x ) x − x , j = 1 , , . . . (41) where a j = c λ (cid:104) λ ψ (0) j − ( x ) + λ (cid:16) π (0)2 , (0 ,j ) + π (0)3 , (0 ,j ) δ ( j < N ) (cid:17)(cid:105) . Proof.
Equation (40) is obvious since ψ (0)0 ( x ) = L (0)1 ( x, ψ (0) j ( x ) = λ xψ (0) j − ( x ) + λ x (cid:16) π (0)2 , (0 ,j ) + π (0)3 , (0 ,j ) δ ( j < N ) (cid:17) − µ ψ (0) j (0) − λ ( x − x )( x − x ) . (42)Note that x < ψ (0) j ( x ) is analytic inside the unit circle, which impliesthat x is also a zero of the numerator of the righthand side of (42). Therefore, λ x ψ (0) j − ( x ) + λ x (cid:16) π (0)2 , (0 ,j ) + π (0)3 , (0 ,j ) δ ( j < N ) (cid:17) = µ ψ (0) j (0) . (43)Taking (43) into the numerator of the right hand side of (42) yields ψ (0) j ( x ) = (cid:104) λ ψ (0) j − ( x ) + λ (cid:16) π (0)2 , (0 ,j ) + π (0)3 , (0 ,j ) δ ( j < N ) (cid:17)(cid:105) ( x − x ) − λ ( x − x )( x − x )+ λ x (cid:16) ψ (0) j − ( x ) − ψ (0) j − ( x ) (cid:17) − λ ( x − x )( x − x ) . Since x = c , (41) can be obtained by simplifying the above equation. (cid:3) Proposition 5. L (0)2 ( y ) = (cid:20) aT ∗ ( y )1 − η y + bT ∗ ( y )1 − η y (cid:21) ι ( y ) β ( y ) , with ι ( y ) = µ − λ + λ y − √ ∆( y )2 µ ( y − . roof. Simplifying (21), we get L (0)2 ( y ) = λ ( x ( y ) −
1) + λ ( y − ya ( y ) − λ yx ( y ) L (0)3 ( y )= (1 − ρ − ρ ) 2 T ∗ ( y ) µ (1 − y ) T ( y ) T ∗ ( y ) λ ( x ( y ) −
1) + λ ( y − µ ( y − β ( y )= 1 − ρ − ρ µ T ∗ ( y )(1 − η y )(1 − η y ) ι ( y ) β ( y )= (cid:20) aT ∗ ( y )1 − η y + bT ∗ ( y )1 − η y (cid:21) ι ( y ) β ( y ) . (cid:3) Proposition 6. L ( T ) ( y ) = (cid:20) a T ∗ ( y )1 − η y + b T ∗ ( y )1 − η y (cid:21) κ ( y ) β ( y ) , with κ ( y ) = µ (1 − y ) − (1 − µ y ) T ( y )2 µ µ y (1 − y )(1 − ¯ ρ y ) . Proof.
By the definition of L ( T ) ( y ), we have L ( T ) ( y ) = yL (0)1 ( y, y ) + yL (0)2 ( y ) + L (0)3 ( y )= 11 − ¯ ρ y L (0)3 ( y ) + ( µ − µ ) yµ (1 − ¯ ρ y ) L (0)2 ( y )= (cid:20) a T ∗ ( y )1 − η y + b T ∗ ( y )1 − η y (cid:21) κ ( y ) β ( y ) , where the second equation follows from the expression (37) and the lastfollows the same idea used in Proposition 5. (cid:3) Along the same idea used for the classical priority model in [11], asymp-totics of the coeffients are obtained using the following Tauberian-like theo-rem, which is Corollary 2 given in [14]. For a function f ( y ) that is analyticat y = 0, we denote the coefficient of y k in the Taylor expression of f ( y ) by C k [ f ( y )].For the compactness, we omit all the proofs in this subsection, which canbe referred to [11]. 17 emma 1 (Flayolet and Odlyzko) . Assume that f ( z ) is analytic in ∆( φ, ε ) = { z : | z | ≤ ε, | Arg ( z − | ≥ φ for ε > and < φ < π/ } except at z = 1 and f ( z ) ∼ K (1 − z ) s as z → in ∆( φ, ε ) . Then as n → ∞ : If s / ∈ { , , , . . . } , f n ∼ K Γ( − s ) n − s − . If s is a nonnegative integer, then f n = o ( n − s − ) . The key goal is to locate the dominant singularity, which determinesthe decay and to characterize the nature of the dominant singularity, whichdetermines the prefactor and the singularity coefficient.Define˜∆( φ, ε, a ) = { z : | az | ≤ ε, | Arg ( az − | ≥ φ for 0 < a < ,ε > < φ < π/ } − { /a } . Lemma 2.
For the non-unit zeros /η and /η , we have Both /η and /η are real. η > . η > η , and η < | η | implies η < . η < , η = 0 or η > if and only if µ < µ , µ = µ or µ > µ ,respectively. η (cid:54) = b , and either T ∗ (1 /η ) = 0 or | η | < b . Lemma 3 (Key Lemma) . There are three cases for the dominant singularityof L (0)2 ( y ) : If D > , then < /η < /b and /η is a zero of T ( y ) (but not T ∗ ( y ) ), and therefore /η is the dominant singularity of L (0)2 ( y ) , which isa simple pole. If D = 0 , then < /η = 1 /b and /η is a zero of T ( y ) and T ∗ ( y ) ,and therefore /b is the dominant singularity of L (0)2 ( y ) , which is both abranch point and a simple pole. If D < , then < /η < /b and /η is a zero of T ∗ ( y ) (but not T ( y ) ), and therefore /b is the dominant singularity of L (0)2 ( y ) , which isa branch point. Proposition 7. If η satisfies: (i) η (cid:54) = 0 ; (ii) η (cid:54) = b ; (iii) | η | < b or T ∗ ( η ) = 0 , then for η = η i , i = 1 , , C n (cid:20) T ∗ ( y )1 − ηy ι ( y ) β ( y ) (cid:21) ∼ b β (1 /b ) σ ( η ) n − / b n , with σ ( η ) = K ( η ) b √ π and K ( η ) = λ b √ − b /b √ b b ( η − b ) . Proposition 8. If ¯ ρ ≥ and η satisfy: (i) η (cid:54) = 0 ; (ii) η (cid:54) = b ; (iii) | η | < b or T ∗ ( η ) = 0 , then for η = η i , i = 1 , , C n (cid:20) T ∗ ( y )1 − ηy κ ( y ) β ( y ) (cid:21) ∼ β (1 /b ) σ ( η ) n − / b n , with σ ( η ) = K ( η ) b √ π and K ( η ) = λ b √ − b /b (cid:2) (1 − µ /b ) (cid:0) ( F (1 /b )+1 (cid:1) − µ (1 − /b ) (cid:3) µ µ √ b b (1 − η/b )(1 − /b )(1 − ¯ ρ /b ) .5.4. Main results of exact tail asymptotics In this subsection, we provide a complete exact tail asymptotics of thestationary distributions(the joint and marginal queue lengths and the totalnumber of customers ) by using the Tauberian-like Theorem to the relatedgenerating functions.
Theorem 2.
The exact tail asymptotics in the marginal stationary distribu-tion π ( h ) n of the high-priority queue is given by π ( h ) n ∼ (1 − ρ ) ρ n . The decay rate in the marginal distribution for the high-priority queue is ρ . Proof.
It is a direct consequence of the Taylor expansion of (35). (cid:3)
Theorem 3.
The exact tail asymptotics in the joint stationary distributionalong the high-priority queue is characterized by: for a fixed number j ≥ oflow-priority customers, π (0)1 , ( n,j ) ∼ β (0)(1 − ρ − ρ ) (cid:32) c j j ! (cid:33) n j c n − j . roof. First, by the induction, we prove ψ (0) j ( x ) ∼ c β (0)(1 − ρ − ρ ) (cid:18) c c (cid:19) j − c x ) j +1 , j ≥ , as c x → . (44)It is true for j = 0 since ψ (0)0 ( x ) = c − c x L (0)3 (0). Assume that (44) is true for j = k , we then show it is true for j = k + 1. Rewrite equation (41) as ψ (0) k +1 ( x ) = a k +1 − c x + λ c xλ (1 − c x ) ψ (0) k ( x ) − ψ (0) k ( x ) x − x , where a k +1 is a constant. Note that λ λ c − x c = c c . Hence,lim c x → ψ (0) k +1 ( x )(1 − c x ) − ( k +2) = c β (0)(1 − ρ − ρ ) (cid:18) c c (cid:19) k +1 , which is equivalent to (44). Therefore, (44) is true for all j ≥ C n [ ψ (0) j ( x )] c n ∼ c L (0)3 (0) (cid:18) c c (cid:19) j n ( j +1) − Γ( j + 1) = c L (0)3 (0) (cid:18) c c (cid:19) j n j j ! , j ≥ , that is π (0)1 , ( n +1 ,j ) ∼ L (0)3 (0) (cid:32) c j j ! (cid:33) n j c n +1 − j , j ≥ , which completes the proof. (cid:3) Theorem 4.
The exact tail asymptotics in the joint stationary distributionalong the low-priority queue is characterized by: for a fixed number i ≥ ofhigh-priority customers, (Exact geometric decay) In the region of D > , π (0)2 , ( i,n ) ∼ C ,l, [ u ( η )] i η n . (Geometric decay with prefactor n − / ) In the region of D = 0 , π (0)2 , ( i,n ) ∼ C ,l, ( √ ρ ) i n − / b n . (Geometric decay with prefactor n − / ) In the region of D < , π (0)2 , ( i,n ) ∼ C ,l, (1 + i (cid:101) B )( √ ρ ) i n − / b n . Here C ,l, , C ,l, , C ,l, , u ( η ) and (cid:101) B are given below: C ,l, = 2 aF (1 /η ) β (1 /η ) ,C ,l, = aλ (cid:112) − b /b √ πb √ b b β (1 /b ) ,C ,l, = [ aσ ( η ) + bσ ( η )] β (1 /b ) ,u ( η ) = 1 − µ − ( λ /η ) − (cid:113) [1 − µ − ( λ /η )] − λ µ µ , (cid:101) B = µ − µ − µ b + √ λ µ √ λ µ . Proof.
In the case of i = 0,1. if D >
0, then T ( η ) = 0, and we can prove ι ( η ) = η . Hence,lim η y → (cid:34) L (0)2 ( y )(1 − η y ) − / (cid:35) = a lim η y → T ∗ ( y ) ι ( y ) β ( y ) + b lim η y → (cid:20) (1 − η y ) T ∗ ( y )1 − η y ι ( y ) β ( y ) (cid:21) = aT ∗ (1 /η ) ι (1 /η ) β (1 /η ) = 2 aF (1 /η ) η β (1 /η ) . Clearly, L (0)2 ( y ) is analytic in ˜∆( φ, ε, η ). By Lemma 1, we obtain π (0)2 , (0 ,n +1) ∼ C ,l, η n +11 .
2. if D = 0, then T ( b ) = T ( η ) = 0, hence, ι ( η ) = b and T ∗ ( y )1 − η y ι ( y ) β ( y )= F ( y ) − F (1 /b )1 − b y ι ( y ) β ( y ) + y (cid:112) ∆( y )1 − b y ι ( y ) β ( y ) ∼ ρ F (cid:48) (1 /b ) (cid:112) − b /b − b ) √ b b (cid:112) − b y + λ (cid:112) − b /b β (1 /b ) √ b b √ − b y ∼ λ (cid:112) − b /b β (1 /b ) √ b b √ − b y . T ∗ ( y )1 − η y ι ( y ) β ( y ) is analytic in ˜∆( φ, ε, b ), applying Lemma 1, we get C n (cid:20) T ∗ ( y )1 − η y ι ( y ) β ( y ) (cid:21) ∼ λ (cid:112) − b /b β (1 /b ) √ b b √ π n − / b n +11 . While with Proposition 7, we have C n (cid:20) T ∗ ( y )1 − η y ι ( y ) β ( y ) (cid:21) ∼ β (1 /b ) σ ( η ) n − / b n +11 . Combining the above two asymptotics gives π (0)2 , (0 ,n +1) ∼ C ,l, n − / b n +11 .
3. if
D <
0, the conclusion is a direct consequence of Proposition 7.In the case of i >
0, the theorem can be proved by induction on i .1. if D >
0, for i = 1, the balance equation is µ π (0)1 , (1 ,n ) η n = ( λ + λ + µ ) π (0)2 , (0 ,n ) η n − λ η π (0)2 , (0 ,n − η n − − µ η π (0)2 , (0 ,n +1) η n +11 . It is easy to see that u ( η ) is the root of the equation with smallermodule: µ [ t ( η )] − [1 − µ − λ /η ] t ( η ) + λ = 0. Since T ( η ) = 0, wehave u ( η ) = − µ − µ η − λ /ηµ . Therefore, we obtain π (0)1 , (1 ,n ) ∼ C ,l, A η n , where A = u ( η ). Assume that for i ≤ k , π (0)1 , ( i,n ) ∼ C ,l, A i η n . Based on the balance equation µ π (0)1 , (2 ,n ) = ( λ + λ + µ ) π (0)1 , (1 ,n ) − λ π (0)1 , (1 ,n − − λ π (0)2 , (0 ,n ) ,µ π (0)1 , ( k +1 ,n ) = ( λ + λ + µ ) π (0)1 , ( k,n ) − λ π (0)1 , ( k,n − − λ π (0)1 , ( k − ,n ) , and the inductive assumption π (0)1 , ( k +1 ,n ) η n → C ,l, A k +1 , we have µ A k +1 = ( λ + λ + µ − λ η ) A k − λ A k − , k = 1 , , , . . . A = 1 and A = u ( η ). Solving this difference equation leads to A k = [ u ( η )] k , k = 0 , , , . . . which gives the conclusion.2. if D = 0, the proof is similar to that for case 1.3. if D <
0, then u ( b ) = √ ρ . Along the same idea in the proof of case1, we get a difference equation A k +1 = 2 √ ρ A k − ρ A k − , k = 1 , , , . . . with A = 1 and A = u ( b ). Solving the equation yields the conclusion. (cid:3) Theorem 5.
The exact tail asymptotics in the marginal stationary distribu-tion π ( l ) n of the low-priority queue is given by π ( l ) n = µ λ π , (0 ,n +1) . Proof.
It is clear since L ( l ) ( y ) = L (0)1 (1 , y )+ yL (0)2 ( y ) = µ λ L (0)2 ( y ) − L (0)3 ( y ). (cid:3) Theorem 6.
The exact tail asymptotics in the stationary distribution π Tn oftotal number of customers in the system is characterized below:If µ = µ , then π Tn = β (cid:18) − ρ − ρ (cid:19) (1 − ρ − ρ )( ρ + ρ ) n , n = 0 , , , . . . . If µ (cid:54) = µ , then In the region of
D > , three cases exist: a) If (i) ¯ ρ ≥ ; or (ii) ¯ ρ < and ¯ ρ < η , then π Tn ∼ C t, a η n . b) If ¯ ρ < and ¯ ρ > η , then π Tn ∼ C t, b ( ¯ ρ ) n . If ¯ ρ < and ¯ ρ = η , then π Tn ∼ C t, c nη n . In the region of D = 0 , two cases exist: a) If ¯ ρ ≥ , then π Tn ∼ C t, a n − / b n . b) If ¯ ρ < , then π Tn ∼ C t, b ( ¯ ρ ) n . In the region of
D < , three cases exist: a) If ¯ ρ ≥ , then π Tn ∼ C t, a n − / b n . b) If ¯ ρ < and ¯ ρ (cid:54) = √ ρ ,then π Tn ∼ C t, b ( ¯ ρ ) n . c) If ¯ ρ < and ¯ ρ = √ ρ ,then ¯ ρ = b (cid:54) = η and π Tn ∼ C t, c ( ¯ ρ ) n . Here C t, a , C t, b , C t, c , C t, a , C t, b , C t, a , C t, b and C t, c are given below: C t, a = ( µ − µ ) η µ ( η − ¯ ρ ) C ,l, ,C t, b = C t, b = C t, b = C t, c = ( µ − µ ) µ ρ L (0)2 ( 1¯ ρ ) + L (0)3 ( 1¯ ρ ) ,C t, c = ( µ − µ ) µ C ,l, ,C t, a = κ (1 /b ) b C ,l, ,C t, a = [ aσ ( η ) + bσ ( η )] β (1 /b ) . Proof. If µ = µ , then L ( T ) ( y ) = − ¯ ρ y L (0)3 ( y ) and ¯ ρ = ρ + ρ . Hence,the conclusion is true. Now we consider the case µ (cid:54) = µ .1. In the region of D >
0, 24) If (i) ¯ ρ ≥
1; or (ii) ¯ ρ < ρ < η , thenlim η y → (cid:20) L ( T ) ( y )(1 − η y ) − (cid:21) = a lim η y → ( µ − µ ) yµ (1 − ¯ ρ y ) T ∗ ( y ) ι ( y ) β ( y )= ( µ − µ ) η µ ( η − ¯ ρ ) C ,l, . Since L ( T ) ( y ) is analytic in ˜∆( φ, ε, η ), applying Lemma 1, we get π ( T ) n ∼ C t, a η n . b) If ¯ ρ < ρ > η , then L ( T ) ( y ) is analytic in ˜∆( φ, ε, ¯ ρ ) andlim ¯ ρ y → (cid:20) L ( T ) ( y )(1 − ¯ ρ y ) − (cid:21) = C t, b . By Lemma 1, we have π ( T ) n ∼ C t, b ( ¯ ρ ) n . c) If ¯ ρ < ρ = η , then L ( T ) ( y ) is analytic in ˜∆( φ, ε, η ) andlim η y → (cid:20) L ( T ) ( y )(1 − η y ) − (cid:21) = ( µ − µ ) µ C ,l, . By Lemma 1, we obtain π ( T ) n ∼ C t, c nη n .
2. In the region of D = 0, two cases exist:a) If ¯ ρ ≥
1, by Proposition 6, L ( T ) ( y ) = (cid:20) a T ∗ ( y )1 − η y + b T ∗ ( y )1 − η y (cid:21) κ ( y ) β ( y ) . Similarly to the case of D = 0 in Theorem 5, we have T ∗ ( y )1 − η y κ ( y ) β ( y ) ∼ λ (cid:112) − b /b κ (1 /b ) β (1 /b ) √ b b √ − b y .
25n addition, T ∗ ( y )1 − η y κ ( y ) β ( y ) is analytic in ˜∆( φ, ε, b ). Hence, C n (cid:20) T ∗ ( y )1 − η y κ ( y ) β ( y ) (cid:21) ∼ λ (cid:112) − b /b κ (1 /b ) β (1 /b ) √ b b √ π n − / b n . While with Proposition 8, we have C n (cid:20) T ∗ ( y )1 − η y κ ( y ) β ( y ) (cid:21) ∼ β (1 /b ) σ ( η ) n − / b n +11 . Combining the above two asymptotics leads to π Tn ∼ C t, a n − / b n . b) If ¯ ρ <
1, then ¯ ρ > b . This can be proved by contradiction: if¯ ρ = b , then ¯ ρ = √ ρ , which follows from ¯ ρ − b = − (¯ ρ −√ ρ ) ¯ ρ +1 − √ ρ .After some manipulations, we get D = µ (1 − √ ρ ) ( µ − µ ) (cid:54) = 0,which is contradict with D = 0. Hence, ¯ ρ > b . The remainder ofthe proof follows the same idea in the case 1-b).3. In the region of D <
0, three cases exist:a) If ¯ ρ ≥
1, then the conclusion follows from Proposition 8.b) If ¯ ρ < ρ (cid:54) = √ ρ , then ¯ ρ > b , the rest of the proof is similarto the case 1-b).c) If ¯ ρ < ρ = √ ρ , then ¯ ρ = b (cid:54) = η , we havelim ¯ ρ y → (cid:20) L ( T ) ( y )(1 − ¯ ρ y ) − (cid:21) = L (0)3 (1 / ¯ ρ ) + µ − µ µ ¯ ρ L (0)2 (1 / ¯ ρ ) . In addition, L ( T ) ( y ) is analytic in ˜∆( φ, ε, ¯ ρ ). By applying Lemma1, we get π Tn ∼ C t, c ( ¯ ρ ) n . (cid:3)
6. Stochastic simulation
This section tests our main results in Theorem 1 by comparing the ratioerror of the waiting times and the cdfs of the queue lengths and waitingtimes. The ratio error was defined in [15] byRatio error = Estimated value − Simulated valueSimulated value × , ab. 1 The ratio error of (1 − ρ ) W for different loads ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . E -95.6000 -91.1732 -81.4272 -72.7356 -20.8699 Std -93.4831 -80.9886 -68.5507 -39.3165 32.0129where the Estimated value is the result in Theorem 1 and the Simulatedvalue is obtained by simulating under different traffic loads.We consider a model with fixed parameters λ = 0 . λ = 0 . µ = 0 . µ = 1, µ = 1 . N = 10. We let ρ = 0 . , . , . , . , .
99 to describethe procedure of ρ → λ can be determined by λ = µ ( ρ − ρ − ρ ). Weuse Matlab to undertake simulations under different traffic loads and eachsimulation runs until at least 10000 customers are served. P { ( − ρ ) W ≤ x } ρ =0.8 ρ =0.9 ρ =0.95 ρ =0.975 ρ =0.99heavy−traffic Fig. 1 The cdf of (1 − ρ ) W for different loads For this model, the scaled queue-length and waiting-time in the criticallyloaded queue are exponential distributed with parameter η and µ η respec-tively in the heavy-traffic scenario. Fig.1 shows the cdf of (1 − ρ ) W and27ab.1 presents the ratio error of E (1 − ρ ) W and Std (1 − ρ ) W , where EX means the expectation of X and StdX means the standard deviation of X .It is showed that the approximation performs well when ρ is very close to1. However when ρ is moderate, the approximation seems not so accurate.This may own to the error of the simulation technique and the approximationtheory since we only take the lowest order terms in the Taylor expansion.Fortunately, the higher-order terms can be obtained in the same procedure. ) F ( x ) ρ =0.8 ρ =0.9 ρ =0.99heavy−traffic 0 20 40 60 8000.10.20.30.40.50.60.70.80.91 x (Q ) F ( x ) ρ =0.8 ρ =0.9 ρ =0.99heavy−traffic Fig. 2 Empirical cdf of waiting times in Q and Q for different loads In the heavy-traffic regime, the queue lengths in the stable queues havethe same distributions as that of a preemptive priority polling system withvacation, which is showed in Fig.2. From Fig.2, the distributions remain soclosely whatever the traffic load ρ is, which can be explained by the pre-emptive priority service policy. The queue lengths in the stable queues areindependent of the value of ρ , which may illustrate the conclusion that thequeue lengths in the stable queues and the queue length in the criticallyloaded queue are independent. This can be showed more exactly in non-preemptive policy systems.
7. Conclusions