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Heavy-traffic asymptotics of a priority polling system with threshold service policy
Computers & Operations Research 65 (2016) 19–28
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Heavy-traffic asymptotics of a priority polling system with threshold service policy Zaiming Liu, Yuqing Chu, Jinbiao Wu n School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, PR China
art ic l e i nf o
a b s t r a c t
Available online 29 June 2015
In this paper, by the singular-perturbation technique, we investigate the heavy-traffic behavior of a priority polling system with three queues under threshold policy. It turns out that the scaled queuelength of the critically loaded queue is exponentially distributed, independent of that of the stable queues, which possess the same distributions as a two-class priority queue with N-policy vacation. Further, we provide an approximation of the tail queue-length distribution of the stable queues, which shows that it has the same prefactors and decay rates as the classical two-class preemptive priority queue. Stochastic simulations are taken to support the results. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Polling system Heavy-traffic Singular-perturbation Tail queue-length distribution Stochastic simulation
1. Introduction The study of the priority polling system is motivated by its wide applications in computer and communication systems, such as ATM (Asynchronous Transfer Mode) switch systems and network standards like DQDB (Distributed Queue Dual Bus). ATM involves two different types of traffic: real time traffic (voice, video) and non-real time traffic (data), which also needs different types of QoS (Quality of Service) standard. By setting the threshold parameter, a higher priority is offered to real time traffic to shorten its delay and the delay of non-real time traffic is kept in a valid regime, which turns out to be a flexible way to control the operation of the whole system. Lee and Sengupta first investigated the threshold-based priority systems in [1]. Later, a special case of two-queue M/M/1 polling system with threshold policy was studied by Boxma et al. in [2,3]. The model was further extended with switch-over times by Deng et al. in [4,5] and with one more server by Feng in [6]. The motivation stems from [7], in which Landry and Stavrakakis proposed a third type of traffic called control traffic with Headof-Line (HoL) priority in the integrated ATM environment, which involves critical network control and reservation information. In this paper, we focus on the heavy-traffic limits when there is a single critically loaded queue. Using the singular-perturbation technique, we derive the lowest-order asymptotic of the joint queue-length distribution in
terms of a small positive parameter measuring the closeness of the system to instability. The perturbation technique is always utilized to compute the performance measures of queueing systems by giving the approximation solutions to certain equations and a survey in [8] can be referred. A main application of the singularperturbation technique in queueing systems is to investigate the heavy-traffic behavior (see [9,10]). With the singular-perturbation technique, we conclude that the queue lengths in the stable queues have the same joint distributions as a degraded model (see the definition in Remark 1) which is a two-class preemptive priority queue with N-policy vacation. In general, no closed-form expressions for the steadystate probabilities in this degraded model can be obtained. Using the Kernel method (see [11–13] for details), we present the exact tail asymptotics of queue lengths in the degraded model, which can further approximate the tail asymptotics of the stable queues. The remainder of this paper is organized as follows. In Section 2, the model and some notations are introduced. In Section 3, the singular-perturbation technique is applied to derive the heavytraffic limits and the detailed derivation is carried out in Section 4. In Section 5, we provide the exact tail asymptotics of queue lengths of the stable queues. In Section 6, some simulations are undertaken to evaluate the heavy-traffic asymptotics. We finally conclude the whole work in this paper and propose some topics for further research in Section 7.
2. Model description n
Corresponding author. E-mail addresses: [email protected] (Z. Liu), [email protected] (Y. Chu), [email protected] (J. Wu). http://dx.doi.org/10.1016/j.cor.2015.06.013 0305-0548/& 2015 Elsevier Ltd. All rights reserved.
We consider a polling model with single server consisting of three queues Q1, Q2, Q3. We refer to the customers queueing in Qi
20
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
as the type i customers, i ¼1,2,3. The buffer capacity of each queue is infinite. Customers arrive at Qi independently according to a Poisson process with rate λi. For type i customers, the service times are mutually independent and all follow an exponential distribution with rate μi. Q1 has the HoL priority and Q2 has a higher priority over Q3. In each queue customers are served according to the FIFO (First In First Out) discipline. We assume that the arrival processes and the service processes are independent. The service discipline is described as follows. 1. Q1 is served exhaustively, which means that the server serves the customers in Q1 until it is empty and then switches to Q2. 2. When the server is serving a customer in Q2, if a type 1 customer arrives, then the server switches to Q1 immediately, otherwise, it continues serving the customers in Q2 until Q2 becomes empty and then switches to Q3. 3. When the server is serving a customer in Q3, if a type 1 customer arrives, then the server switches to Q1 immediately, if the size of Q2 reaches a given threshold N and Q1 is empty, then the server switches to Q2 immediately, otherwise, it continues serving the customers in Q3 until Q3 becomes empty and then switches to Q2. It is assumed that all the switches are instantaneous. In addition, the switches caused by the threshold push the customer undergoing service to the head of the queue and the service of the interrupted customer resumes from the beginning, which means that the completed service time of the interrupted customer is invalid. The traffic load of Qi is denoted by ρi ¼ λi =μi , i¼ 1,2,3. We assume the ergodicity condition of the system ρ ¼ ρ1 þ ρ2 þ ρ3 o 1 is satisfied. Let Xi(t) be the number of customers in Qi at time t, and S(t) be the position of the server at time t with SðtÞ A f0; 1; 2; 3g, where SðtÞ ¼ 0 denotes that the server is idle at time t. The associated stochastic process fYðtÞ; t Z0g ¼ ðX 1 ðtÞ; X 2 ðtÞ; X 3 ðtÞ; SðtÞÞ; t Z 0 is an aperiodic and irreducible four-dimensional Markov process. Let Xi (i¼1,2,3) be the steady-state queue length of Qi and S be the steady-state position of the server. Define the stationary probabilities ps ðx1 ; x2 ; x3 Þ ¼ lim PrfYðtÞ ¼ ðx1 ; x2 ; x3 ; sÞg; t-1
s ¼ 0; 1; 2; 3:
We study the heavy-traffic limits of the joint queue-length distribution by increasing the arrival rate λ3 so that ρ-1 , while keeping λ1 a 0, λ2 a 0 and μ1 ; μ2 ; μ3 fixed. When ρ-1 , Q3 becomes critically loaded, whereas Q1 and Q2 remain stable since Q1 and Q2 have higher priorities over Q3. The single-perturbation technique is implemented here. We first apply a perturbation to λ3 in the balance equations, in which case Q3 is close to becoming critically loaded. Then we solve the lowest order terms in the balance equations to obtain the queuelength distributions of the stable queues Q1 and Q2. At last we solve the first-order and second-order terms to get a differential equation and compute the scaled number of customers in Q3. Applying the Markov property, we obtain the following balance equations when x3 Z 2 ðλ1 þ λ2 þ λ3 þ μ1 Þp1 ðx1 ; x2 ; x3 Þ
where δðÞ is Kronecker function. In the above equations, we have omitted the parts for x3 ¼ 0 and x3 ¼ 1 which do not play a role after the perturbation since X3 tends to infinity as Q3 becomes critically loaded and the probability of Q3 being empty or 1 goes to zero. Throughout the paper, we adopt the following standard notations:
Function F(x) is o(x) if FðxÞ=x-0 as x-0. Function F(x) is OðxÞ if there exists a c Z 0 such that FðxÞ=x-c as x-0 while Oð1Þ is a constant time complexity.
For functions f(n) and g(n) of nonnegative integers n, f ðnÞ gðnÞ f ðnÞ means limn-1 gðnÞ ¼ 1.
3. Perturbation From the stability condition, the system becomes unstable as
ρ3 -1 ρ1 ρ2 , i.e. λ3 -μ3 ð1 ρ1 ρ2 Þ. Therefore it is assumed that
λ3 ¼ μ3 ð1 ρ1 ρ2 Þ εω;
ω 4 0; 0 o ε⪡1:
ð4Þ
Let ζ ¼ εx3 , and ps ðx1 ; x2 ; x3 Þ ¼ ps ðx1 ; x2 ; ζ =εÞ ¼ εϕs;ðx1 ;x2 Þ ðζ ; εÞ;
0 o ζ ¼ Oð1Þ; s ¼ 1; 2; 3:
ð5Þ
Taking (4) and (5) into the balance equations (1)–(3) and then taking the Taylor expansion, we obtain ðλ1 þ λ2 þ μ1 Þϕ1;ðx1 ;x2 Þ ðζ ; εÞ
¼ λ1 ϕ1;ðx1 1;x2 Þ ðζ ; εÞδðx1 Z 2Þ þ λ2 ϕ1;ðx1 ;x2 1Þ ðζ ; εÞδðx2 Z 1Þ ∂ϕ1;ðx1 ;x2 Þ ðζ ; εÞ
!
ε2 ∂2 ϕ1;ðx1 ;x2 Þ ðζ ; εÞ 2 2 ∂ζ ∂ζ þ λ1 ϕ2;ð0;x2 Þ ðζ ; εÞδðx1 ¼ 1; x2 Z 1Þ þ μ1 ϕ1;ðx1 þ 1;x2 Þ ðζ ; εÞ þ λ1 ϕ3;ð0;x2 Þ ðζ ; εÞδðx1 ¼ 1; x2 o NÞ þ oðε2 Þ; x1 Z1; x2 Z 0; ð6Þ
ðμ3 ð1 ρ1 ρ2 Þ εωÞ ε
ðλ1 þ λ2 þ μ2 Þϕ2;ð0;x2 Þ ðζ ; εÞ ∂ϕ2;ð0;x2 Þ ðζ ; εÞ
!
ε2 ∂2 ϕ2;ð0;x2 Þ ðζ ; εÞ 2 2 ∂ζ ∂ζ þ λ2 ϕ2;ð0;x2 1Þ ðζ ; εÞδðx2 Z 2Þ þ μ1 ϕ1;ð1;x2 Þ ðζ ; εÞ þ λ2 ϕ3;ð0;N 1Þ ðζ ; εÞδðx2 ¼ NÞ þ μ2 ϕ2;ð0;x2 þ 1Þ ðζ ; εÞ þ oðε2 Þ; x2 Z 1;
¼ ðμ3 ð1 ρ1 ρ2 Þ εωÞ ε
þ
ð7Þ ðλ1 þ λ2 Þϕ3;ð0;x2 Þ ðζ ; εÞ ¼ ðμ3 ð1 ρ1 ρ2 Þ εωÞ ε ∂ϕ3;ð0;x2 Þ ðζ ; εÞ
þ μ3 ε
∂ζ
∂ϕ3;ð0;x2 Þ ðζ ; εÞ ∂ζ
ε2 ∂ ϕ3;ð0;x2 Þ ðζ ; εÞ þ 2 2 ∂ζ 2
þ
!
ε2 ∂2 ϕ3;ð0;x2 Þ ðζ ; εÞ 2 2 ∂ζ
!
i þ μ1 ϕ1;ð1;0Þ ðζ ; εÞ þ μ2 ϕ2;ð0;1Þ ðζ ; εÞ δðx2 ¼ 0Þ
þ λ3 p1 ðx1 ; x2 ; x3 1Þ þ λ1 p2 ð0; x2 ; x3 Þδðx1 ¼ 1; x2 Z 1Þ þ μ1 p1 ðx1 þ 1; x2 ; x3 Þ þ λ1 p3 ð0; x2 ; x3 Þδðx1 ¼ 1; x2 o NÞ; x1 Z 1; x2 Z 0;
ð1Þ
¼ λ2 p2 ð0; x2 1; x3 Þδðx2 Z 2Þ þ λ3 p2 ð0; x2 ; x3 1Þ þ μ1 p1 ð1; x2 ; x3 Þ
ð2Þ
ðλ1 þ λ2 þ λ3 þ μ3 Þp3 ð0; x2 ; x3 Þ ¼ λ2 p3 ð0; x2 1; x3 Þδðx2 Z 1Þ þ λ3 p3 ð0; x2 ; x3 1Þ þ μ3 p3 ð0; x2 ; x3 þ 1Þ þ μ1 p1 ð1; 0; x3 Þ þ μ2 p2 ð0; 1; x3 Þ δðx2 ¼ 0Þ; 0 r x2 r N 1; ð3Þ
h
¼ λ1 p1 ðx1 1; x2 ; x3 Þδðx1 Z 2Þ þ λ2 p1 ðx1 ; x2 1; x3 Þδðx2 Z 1Þ
ðλ1 þ λ2 þ λ3 þ μ2 Þp2 ð0; x2 ; x3 Þ
þ λ2 p3 ð0; N 1; x3 Þδðx2 ¼ NÞ þ μ2 p2 ð0; x2 þ 1; x3 Þ; x2 Z 1;
þ λ2 ϕ3;ð0;x2 1Þ ðζ ; εÞδðx2 Z 1Þ þ oðε2 Þ;
0 r x2 r N 1:
ð8Þ
It is noted that λ3 only plays a role in equations for OðεÞ terms and higher. Throughout the paper, we do Taylor expansions of ϕs;ðx1 ;x2 Þ ðζ ; εÞ (s ¼ 1; 2; 3) in powers of ε as follows: ð1Þ 2 ϕs;ðx1 ;x2 Þ ðζ ; εÞ ¼ ϕð0Þ s;ðx1 ;x2 Þ ðζ Þ þ εϕs;ðx1 ;x2 Þ ðζ Þ þOðε Þ;
s ¼ 1; 2; 3:
ð9Þ
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
In the next section the lowest order terms of the resulting equations after Taylor expansions are equated to find expressions ð0Þ for ϕs;ðx1 ;x2 Þ ðζ Þ (s ¼ 1; 2; 3). Subsequently the first-order and secondorder terms are equated to find the scaled queue-length distribution of Q3. For convenience, we introduce the corresponding probability generating functions (PGFs) 1 1 X X
Q ðjÞ 1 ðx; y; ζ Þ ¼
x1 ¼ 1 x2 ¼ 0
1 X
Q ðjÞ 2 ðy; ζ Þ ¼
x2 ¼ 1
x1 1 x2 ϕðjÞ y ; 1;ðx1 ;x2 Þ ðζ Þx
x2 1 ϕðjÞ ; 2;ð0;x2 Þ ðζ Þy
1 1 X X
Q 1 ðx; y; ζ ; εÞ ¼
x1 ¼ 1 x2 ¼ 0 1 X
Q 2 ðy; ζ ; εÞ ¼
x2 ¼ 1
j ¼ 0; 1;
Q ðjÞ 3 ðy; ζ Þ ¼
N 1 X x2 ¼ 0
x2 ϕðjÞ 3;ð0;x2 Þ ðζ Þy ; j ¼ 0; 1;
ϕ1;ðx1 ;x2 Þ ðζ ; εÞxx1 1 yx2 ;
ϕ2;ð0;x2 Þ ðζ ; εÞyx2 1 ;
N 1 X x2 ¼ 0
ð0Þ
ðλ1 þ λ2 þ μ1 Þϕ1;ðx1 ;x2 Þ ðζ Þ ð0Þ
x1 Z1; x2 Z0;
μ ϕ
ζδ
ζ
μ ϕ
ð0Þ
ð0Þ
ð0Þ 3;ð0;0Þ ð
ð0Þ 1;ð1;0Þ ð
ðλ1 þ λ2 Þϕ3;ð0;x2 Þ ðζ Þ ¼ λ2 ϕ3;ð0;x2 1Þ ðζ Þ;
ζ Þ þ μ2 ϕ
ζ Þ;
x2 Z 1;
1 r x2 r N 1; ð0Þ 2;ð0;1Þ ð
ζ Þ:
ð11Þ ð12Þ ð13Þ
ð0Þ such that We introduce P 0 ðζ Þ and π s;ðx 1 ;x2 Þ ð0Þ ϕs;ðx ðζ Þ ¼ π ð0Þ s;ðx1 ;x2 Þ P 0 ðζ Þ; 1 ;x2 Þ
x1 ¼ 1 x2 ¼ 0
ð0Þ π 1;ðx þ 1 ;x2 Þ
1 X x2 ¼ 1
s ¼ 1; 2; 3;
π ð0Þ 2;ð0;x2 Þ þ
N 1 X
L2ð0Þ ðyÞ ¼ L3ð0Þ ðyÞ ¼
1 1 X X
x2 ¼ 0
ð0Þ π 3;ð0;x ¼ 1: 2Þ
ð0Þ x1 1 x2 y ; 1;ðx1 ;x2 Þ x x1 ¼ 1 x2 ¼ 0 1 X ð0Þ x2 1 ; 2;ð0;x2 Þ y x2 ¼ 1
π
π
N 1 X x2 ¼ 0
x2 π ð0Þ 3;ð0;x2 Þ y :
Then it is clear that ð0Þ Q ð0Þ 1 ðx; y; ζ Þ ¼ L1 ðx; yÞP 0 ðζ Þ;
ð14Þ
ð0Þ Q ð0Þ s ðy; ζ Þ ¼ Ls ðyÞP 0 ðζ Þ;
ð15Þ
s ¼ 2; 3:
From (12), we get L3ð0Þ ðyÞ ¼
N 1 X x2 ¼ 0
ð20Þ
μ
2 Lð0Þ ðyÞ: xKðx; yÞ 1 αðyÞ 3 μ2 λ2 y λ1 y 1 y λ1 ½x αðyÞ
ð21Þ
ð0Þ ð0Þ ðr 2 yÞx2 π ð0Þ 3;ð0;0Þ ¼ HðyÞπ 3;ð0;0Þ ¼ β ðyÞL3 ð1Þ;
N HðyÞ where r 2 ¼ λ λþ2 λ , HðyÞ ¼ ðrr22yÞy 1 1 and β ðyÞ ¼ Hð1Þ . 1 2
rule,
we
obtain
ρ1 ð0Þ Lð0Þ 1 ð1; 1Þ ¼ 1 ρ ρ L3 ð1Þ 1
2
ρ2 ð0Þ Lð0Þ 2 ð1Þ ¼ 1 ρ1 ρ2 L3 ð1Þ. By the normalizing condition, it is ð0Þ get Lð0Þ 3 ð1Þ ¼ 1 1 2 . Therefore, we have L1 ð1; 1Þ ¼ ð0Þ L2 ð1Þ ¼ 2 . Moreover,
ρ
ρ
and
easy to
ρ1 and
ρ
Lð0Þ 3 ðyÞ ¼ ð1 ρ1 ρ2 Þβ ðyÞ:
ð22Þ
We will determine the unknown expression of P 0 ðζ Þ in the rest of this section. Remark 1. Compared to the balance equations (1)–(3), Eqs. (10)– (13) have only considered type 1 and type 2 customers, which can be seen as the balance equations of a degraded model. In the view point of an arbitrary type 1 or type 2 customer, Q3 behaves like an N-policy vacation in heavy traffic which starts when Q1 and Q2 become empty and terminates once a type 1 customer arrives or there are N type 2 customers queueing. Hence the degraded model is actually a two-class priority queue with N-policy vacation in which the queue length PGF's are Eqs. (20)–(22). To be more intact, we give a detailed description of the degraded model as follows:
Define L1ð0Þ ðx; yÞ ¼
1 αðyÞ 1 y Lð0Þ ðyÞ: 1 αðyÞ 3 μ 2 λ2 y λ1 y 1y
λ2 þ λ1
L'Hôpital's
ð0Þ ðx2 Z 2Þ þ 1 1;ð1;x2 Þ ð Þ ð0Þ Þ ðx2 ¼ NÞ þ 2 2;ð0;x2 þ 1Þ ð
ζδ
1 1 X X
Lð0Þ 2 ðyÞ ¼
Letting y-1 and then letting x-1 in (20) and (21), with
ð0Þ
ζ Þ ¼ μ1 ϕ
1 Kðx; yÞ ¼ λ1 ð1 xÞ þ λ2 ð1 yÞ þ μ1 1 ; x 1 aðyÞ ¼ λ1 þ λ2 ð1 yÞ þ μ2 1 : y
ð10Þ
ðλ1 þ λ2 þ μ2 Þϕ2;ð0;x2 Þ ðζ Þ
ðλ1 þ λ2 Þϕ
where
Lð0Þ 1 ðx; yÞ ¼
ð0Þ
þ λ1 ϕ2;ð0;x2 Þ ðζ Þδðx1 ¼ 1; x2 Z 1Þ þ μ1 ϕ1;ðx1 þ 1;x2 Þ ðζ Þ
λ ϕ
ð18Þ
Substituting (20) into (19) leads to
ð0Þ
¼ λ1 ϕ1;ðx1 1;x2 Þ ðζ Þδðx1 Z 2Þ þ λ2 ϕ1;ðx1 ;x2 1Þ ðζ Þδðx2 Z 1Þ
λ ϕ
ð0Þ yaðyÞL2ð0Þ ðyÞ ¼ μ1 Lð0Þ 1 ð0; yÞ ½λ1 þ λ2 ð1 yÞL3 ðyÞ;
Clearly, for every j yj r 1, the kernel xKðx; yÞ has a unique zero x ¼ αðyÞ with j xj r1. Since Lð0Þ 1 ðx; yÞ is analytic in j xj o1 for every j yj o 1 and continuous up to j xj r 1 and j yj r 1, the right-hand side of Eq. (19) must vanish when x ¼ αðyÞ, which yields
Equating the lowest-order terms of the resulting equation after the Taylor expansions of (6)–(8), we obtain
ð0Þ ¼ 2 2;ð0;x2 1Þ ð Þ ð0Þ þ 2 3;ð0;N 1Þ ð
ð17Þ
ð19Þ
4.1. Equating the lowest-order terms
ð0Þ
ð0Þ ð0Þ xKðx; yÞL1ð0Þ ðx; yÞ ¼ λ1 x½yLð0Þ 2 ðyÞ þ L3 ðyÞ μ1 L1 ð0; yÞ;
ð0Þ xKðx; yÞL1ð0Þ ðx; yÞ ¼ y½λ1 x aðyÞLð0Þ 2 ðyÞ þ ½λ1 ðx 1Þ þ λ2 ðy 1ÞL3 ðyÞ:
ϕ3;ð0;x2 Þ ðζ ; εÞyx2 :
4. Model analysis
þ λ1 ϕ3;ð0;x2 Þ ðζ Þδðx1 ¼ 1; x2 o NÞ;
Using the PGFs to rewrite the balance equations (10) and (11) leads to
Adding (18) to (17), we have
Q 3 ðy; ζ ; εÞ ¼
ð0Þ
21
ð16Þ
(i) There are two classes of customers in the system, the highand low-priority customers, arriving independently according to two Poisson processes with rates λ1 and λ2, respectively. (ii) Each class of customer is served according to the FIFO discipline. (iii) The server takes a vacation once the system empties and goes back to work once the size of the low-priority customers reaches N or there is a high-priority customer's arrival. (iv) The high-priority customers have preemptive priorities over the low-priority customers just like in the classical two-class preemptive priority queue. (v) Both classes of customers require an exponential amount of service times with rates μ1 and μ2, respectively. All service
22
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
times are independent and also independent of the arrival processes.
Taking the summation over all x1 and x2 of (6)–(8) gives
μ1
1 X
ϕ1;ð1;x2 Þ ðζ ; εÞ
x2 ¼ 0
4.2. Equating the first-order terms
1 X
¼ λ1
x2 ¼ 1
"
In this subsection, after the Taylor expansion of the perturbed balance equations (6)–(8), equating the first-order terms, we present an equation in Proposition 1.
λ1
Proposition 1. h i ð1Þ ð1Þ ð1Þ ð1 ρ1 ρ2 Þ Q ð1Þ 1 ð1; 1; ζ Þ þQ 2 ð1; ζ Þ þ Q 3 ð1; ζ Þ Q 3 ð1; ζ Þ μ μ ¼ 3 ρ1 þ 3 ρ2 P 00 ðζ Þ:
μ1
ζÞ ¼ λ
μ3 xð1 ρ
ð1Þ Þ 1 Q ð1Þ 1 ð0; y; Þ þ 1 xQ 3 ðy; ð0Þ 0 1 2 ÞL1 ðx; yÞP 0 ð Þ;
ζ
μ
ρ
ζ λ ζ
ð24Þ ð1Þ
ð1Þ N ½λ1 þ λ2 ð1 yÞQ ð1Þ 3 ðy; ζ Þ ¼ λ2 y ϕ3;ð0;x2 1Þ ðζ Þ þ μ1 Q 1 ð0; 0; ζ Þ
þ μ2 Q 2ð1Þ ð0; ζ Þ þ μ3 ðρ1 þ ρ2 ÞL3ð0Þ ðyÞP 00 ðζ Þ: ð25Þ Applying the same method in solving (17) and (18) to Eqs. (23)– (25), after some elementary calculations, we get
λ1 ½x αðyÞ
ð1Þ ½yQ ð1Þ 2 ðy; ζ Þ þ Q 3 ðy; ζ Þ xKðx; yÞ μ ð1 ρ1 ρ2 Þ ð0Þ 0 ½xL1 ðx; yÞ αðyÞLð0Þ 3 1 ðαðyÞ; yÞP 0 ðζ Þ: xKðx; yÞ
ð26Þ ð1Þ ½yaðyÞ λ1 αðyÞyQ ð1Þ 2 ðy; ζ Þ þ ½λ1 ð1 αðyÞÞ þ λ2 ð1 yÞQ 3 ðy; ζ Þ n h io ð0Þ ð0Þ ¼ μ3 ðρ1 þ ρ2 ÞL3 ðyÞ ð1 ρ1 ρ2 Þ yL2 ðyÞ þ αðyÞLð0Þ P 00 ðζ Þ: 1 ðαðyÞ; yÞ
ð27Þ Letting y-1 and then letting x-1 in (26), with L'Hôpital's rule, we obtain
ρ1 μ3 ρ1 ð1 ρ1 ρ2 Þ2 0 ð1Þ ½Q ð1Þ P 0 ðζ Þ: 2 ð1; ζ Þ þQ 3 ð1; ζ Þ 1 ρ1 μ1 ð1 ρ1 Þ2 ð28Þ
Similarly, letting y-1 in (27) leads to i 1 h ð1 ρ1 ρ2 ÞQ 2ð1Þ ð1; ζ Þ ρ2 Q 3ð1Þ ð1; ζ Þ 1 ρ1 ( " # ) μ3 ρ1 ρ2 ρ1 ρ2 ð1 ρ1 ρ2 Þ μ3 ¼ þ ρ2 P 00 ðζ Þ: μ1 1 ρ1 μ2 ð1 ρ1 Þ2
ð29Þ
Combining (28) and (29), we have
∂Q 1 ð1; 1; ζ ; εÞ ∂ζ
ð30Þ
∂Q 2 ð1; ζ ; εÞ ∂ζ
ϕ1;ð1;x2 Þ ðζ ; εÞ þ λ2 ϕ3;ð0;N 1Þ ðζ ; εÞ μ3 ð1 ρ1 ρ2 Þε
ð31Þ
ϕ3;ð0;x2 Þ ðζ ; εÞ þ λ2 ϕ3;ð0;N 1Þ ðζ ; εÞ ∂Q 3 ð1; ζ ; εÞ ∂Q ð1; ζ ; εÞ þ μ3 3 ∂ζ ∂ζ #
¼ μ1 ϕ1;ð1;0Þ ðζ ; εÞ þ μ2 ϕ2;ð0;1Þ ðζ ; εÞ μ3 ð1 ρ1 ρ2 Þ þ ω
ð0Þ 0 þ μ1 Q ð1Þ 1 ð0; y; ζ Þ μ3 yð1 ρ1 ρ2 ÞL2 ðyÞP 0 ðζ Þ;
Q ð1Þ 1 ð1; 1; ζ Þ ¼
1 X
"
ð1Þ
Q ð1Þ 1 ðx; y; ζ Þ ¼
N 1 X x2 ¼ 0
ð1Þ yaðyÞQ 2ð1Þ ðy; ζ Þ ¼ λ2 yN ϕ3;ð0;x2 1Þ ðζ Þ μ1 Q ð1Þ 1 ð0; 0; ζ Þ μ2 Q 2 ð0; ζ Þ
ϕ3;ð0;x2 Þ ðζ ; εÞ μ3 ð1 ρ1 ρ2 Þε
" # ∂Q ð1; ζ ; εÞ μ3 ð1 ρ1 ρ2 Þ ∂2 Q 2 ð1; ζ ; εÞ 2 þ ω 2 þ ε þ Oðε3 Þ; 2 2 ∂ζ ∂ζ
λ1
ð23Þ
x2 ¼ 0
# ∂Q 1 ð1; 1; ζ ; εÞ μ3 ð1 ρ1 ρ2 Þ ∂2 Q 1 ð1; 1; ζ ; εÞ 2 þ ε þ Oðε3 Þ; 2 2 ∂ζ ∂ζ
x2 ¼ 1
ζÞ
N 1 X
ϕ2;ð0;x2 Þ ðζ ; εÞ þ μ2 ϕ2;ð0;1Þ ðζ ; εÞ
¼ μ1
Proof. Taking the PGF of the first-order terms of the resulting equations after the Taylor expansion of (6)–(8), we have ð1Þ 1 xyQ 2 ðy;
1 X x2 ¼ 1
μ2
xKðx; yÞQ 1ð1Þ ðx; y;
ω
þ
ϕ2;ð0;x2 Þ ðζ ; εÞ þ λ1
∂Q 3 ð1; ζ ; εÞ μ3 ð2 ρ1 ρ2 Þ ∂2 Q 3 ð1; ζ ; εÞ 2 þ ε þOðε3 Þ: 2 2 ∂ζ ∂ζ
ð32Þ
Summing over (30)–(32), we obtain ∂Q 1 ð1; 1; ζ ; εÞ ∂Q 2 ð1; ζ ; εÞ ∂Q 3 ð1; ζ ; εÞ þ þ 0 ¼ μ3 ð1 ρ1 ρ2 Þ ∂ζ ∂ζ ∂ζ " 2 ∂Q ð1; ζ ; εÞ μ ð1 ρ1 ρ2 Þ ∂ Q 1 ð1; 1; ζ ; εÞ εþ 3 þ μ3 3 2 2 ∂ζ ∂ζ ! ∂2 Q 2 ð1; ζ ; εÞ ∂2 Q 3 ð1; ζ ; εÞ ∂Q 1 ð1; 1; ζ ; εÞ þ þ þω 2 2 ∂ζ ∂ζ ∂ζ # 2 ∂Q ð1; ζ ; εÞ ∂Q 3 ð1; ζ ; εÞ μ ∂ Q 3 ð1; ζ ; εÞ 2 þ 2 þ ε þ Oðε3 Þ: ð33Þ þ 3 2 2 ∂ζ ∂ζ ∂ζ Now taking the Taylor expansion of (33) with respect to ε like (9) gives h 0 ¼ μ3 ð1 ρ1 ρ2 ÞP ″0 ðζ Þ þ ωP 00 ðζ Þ þ μ3 Q 30ð1Þ ð1; ζ Þ μ3 ð1 ρ1 ρ2 Þ
0 i 0 0 Q 1ð1Þ ð1; 1; ζ Þ þ Q 2ð1Þ ð1; ζ Þ þ Q 3ð1Þ ð1; ζ Þ ε2 þ Oðε3 Þ μ μ ¼ μ3 1 ρ1 ρ2 þ 3 ρ1 þ 3 ρ2 P ″0 ðζ Þ þ ωP 00 ðζ Þ ε2 þ Oðε3 Þ:
μ1
μ2
In the above equation, the first equation follows from (14) and the second equation follows from Proposition 1. From the above derivation procedure, we can conclude the following Proposition. Proposition 2. After taking the summation over all x1 and x2 of the Taylor series of all perturbed balance equations (6)–(8), the Oð1Þ and OðεÞ terms cancel and, moreover, equating the Oðε2 Þ terms yields the following differential equation for P 0 ðζ Þ: μ μ ωP 00 ðζ Þ ¼ ð1 ρ1 ρ2 Þ þ 3 ρ1 þ 3 ρ2 μ3 P ″0 ðζ Þ:
μ1
μ2
h i ð1Þ ð1Þ ð1Þ ð1 ρ1 ρ2 Þ Q ð1Þ 1 ð1; 1; ζ Þ þ Q 2 ð1; ζ Þ þ Q 3 ð1; ζ Þ Q 3 ð1; ζ Þ i μ ρ ð1 ρ ρ Þ2 1 h ð1Þ 3 1 1 2 ð1 ρ1 ρ2 ÞQ ð1Þ P 00 ðζ Þ 2 ð1; ζ Þ ρ2 Q 3 ð1; ζ Þ μ 1 ρ1 ð1 ρ1 Þ2 1 μ μ ¼ 3 ρ1 þ 3 ρ2 P 00 ðζ Þ:□
¼
4.4. The scaled number of customers in the critically loaded queue
4.3. Equating the second-order terms
Now we can finally present the density of the scaled number of customers in Q3, i.e. P 0 ðζ Þ. By combining the differential equation R1 in Proposition 2 with 0 P 0 ðζ Þdζ ¼ 1, we obtain
In this subsection, we consider the sum of all Oðε2 Þ terms in (6)–(8) to determine P 0 ðζ Þ.
P 0 ðζ Þ ¼ ηe ηζ ; h i with ωη ¼ 1 ρ1 ρ2 þ μμ3 ρ1 þ μμ3 ρ2 μ3 .
μ1
μ2
1
2
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
As a special case, we may take ω ¼ μ3 , which gives
ζ ¼ ð1 ρÞX 3 , then 1 μ μ ¼ 1 ρ1 ρ2 þ 3 ρ1 þ 3 ρ2 : η μ1 μ2
By applying the multiclass distributional law of Bertsimas and Mourtzinou [16], we get that the scaled waiting time at Q3 follows an exponential distribution with parameter μ3 η. 4.5. Main result Theorem 1. For λ3 ¼ μ3 ð1 ρ1 ρ2 Þ εω, we have limPfX 1 r x1 ; X 2 r x2 ; εX 3 r ζ g ¼ Lðx1 ; x2 Þð1 e ηζ Þ; ε↓0
where Lð; Þ is the joint cumulative distribution function (cdf) of the queue lengths of the degraded model described in Remark 1 with PGFs of (20)–(22). The main result stated in Theorem 1 can be interpreted as follows: in the heavy-traffic regime, R1. The queue lengths in the stable queues have the same distribution as a two-class priority queue with N-policy vacation. R2. The scaled number of customers in the critically loaded queue is exponentially distributed with parameter η. R3. The queue lengths in the stable queues and the (scaled) number of customers in the critically loaded queue are independent. We now elaborate on these interpretations. For R1, since Q3 is critically loaded, Q3 would be visited during each cycle. From the perspective of type 1 or type 2 customers, Q3 behaves like an Npolicy vacation in heavy traffic which starts when Q1 and Q2 become empty and terminates once a type 1 customer arrives or there are N type 2 customers queueing. For R2, we note that the total workload in the system equals the amount of workload in an M/G/1 queue with arrival rate λ1 þ λ2 þ λ3 and hyperexponentially distributed service times, i.e. the service time is exponentially distributed with parameter μi with probability λ þ λλi þ λ , i ¼ 1; 2; 3. Based on the heavy-traffic 1 2 3 results for the M/G/1 queue (see [16]), the distribution of the scaled total workload converges to an exponential distribution with mean ρE½R, where R is a residual service time and 1 E½R ¼
μ1
ρ1 þ
1
1
ρ þ ρ μ2 2 μ3 3 : ρ
In the heavy traffic, since almost all customers are located in Q3, the total number of customers at this queue is also exponentially distributed with mean μ3 μ1 ρ1 þ μ1 ρ2 þ μ1 ρ3 . Since λ3 ↑μ3 1 2 3 ð1 ρ1 ρ2 Þ, the scaled number of customers in Q3 is exponentially distributed with parameter η. Finally, R3 follows from the time-scale separation in the heavy traffic which implies that the dynamics of the stable queues evolve at a much faster time scale than the dynamics of the critically loaded queue. Since the amount of “memory” of the stable queues asymptotically vanishes compared to that of the critically loaded queue, the queue lengths in the stable queues are independent of the (scaled) number of customers in the critically loaded queue in the limit. Remark 2. For the degraded model, we provide a stochastic decomposition of the queue length distribution in Q2, which ðlÞ explains N appears only via Lð0Þ 3 ðyÞ. Let L ðÞ be the PGF of the number of type 2 customers.
23
Kella and Yechiali introduced the concept of the delay-cycle in [14] and succeeded in deriving the waiting times in an M/G/1 queue with priorities and vacations. Here we do some modifications and apply it to our degraded model. Since FIFO or LIFO (Last In First Out) discipline lead to the same distribution of the queue length in the system, without loss of generality, we assume a nonpreemptive LIFO queueing discipline (among the same class customers) here. Define vacation customers to be customers who arrive during the vacation time. According to the argument in [14], we divide the whole time into two delaycycles: T1-cycle and TU-cycle (see Fig. 1). T1-cycle starts with the service time B1 of the type 1 vacation customer and the customers served thereafter are all his descendants only. TU-cycle is the vacation time U, plus the service time of all the type 2 vacation customers and their descendants. For type 2 customers, each delay-cycle behaves like an M=G=1 queue with a vacation. We consider a type 2 customer and his type 1 descendants as a generalized type 2 customer, denoted by a type 20 customer. Hence, type 20 customers arrive with intensity λ2 and have service time B20 with LST and mean B20 ðsÞ ¼
μ2 ; μ2 þ s þ λ1 ð1 θ1 ðsÞÞ
EB20 ¼
1
μ2 ð1 ρ1 Þ
:
ð34Þ
Since B20 is the time between the service initiations of two consecutive type 2 customers, the number of type 2 customers at an arbitrary time is equal to that of type 20 customers. It is noted that any point in time is either within T1-cycle or TU1 cycle. The length of a T1-cycle equals 1 EB ρ1 ρ2 . Hence, the fraction 1 of the time that the system is in a T1-cycle equals λ1 ρ3 1 EB ρ1 ρ2 ¼ ρ1 and the fraction of the time in a TU-cycle equals 1 ρ1 . By (16), the PGF of the number of type 2 customers present at a random point during the vacation time equals βðÞ. Due to the LIFO discipline, if there exist type 2 vacation customers when T1-cycle begins, they will wait until the T1-cycle ends and then get service. By applying the Fuhrmann–Cooper Decomposition Theorem (see [15]), the number of type 2 customers at a random point in a T1cycle equals the number of type 2 customers present when B1 begins, plus the number of type 2 customers at a random point during B1 and the number of type 2 customers in an M/G/1 queue. That is B 0 ðλ2 ð1 yÞÞð1 ρ20 Þð1 yÞ 1 θ1 ðλ2 ð1 yÞÞ LðlÞ ðyÞ ¼ 2 βðyÞ ρ1 þ ð1 ρ1 Þ ; B20 ðλ2 ð1 yÞÞ y λ2 ð1 zÞEθ1 ð35Þ ρ
where ρ20 ¼ ð1 2ρ Þ is the traffic load of type 20 customers. Sub1 stituting (34) into the above equation leads to 1 αðyÞ 1y Lð0Þ ðyÞ: L ðyÞ ¼ 1 αðyÞ 3 ρ2 μ2 λ2 y λ1 y 1y ðlÞ
1
λ2 þ λ1
Since the vacation time depends on the arriving process in the degraded model, N appears only via β ðÞ in (35). Remark 3. Since the singular-perturbation technique only depends on the balance equations, by using the supplementary variable method, it also applies to the system with general services. However there would be one more parameter. Thus the derivation will be much more tedious. In order to make the
Fig. 1. Diagram of T1-cycle and TU-cycle.
24
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
methodology easier to handle and get a better understanding of the conclusions, we only consider the polling system with exponential services. Remark 4. In a K-queue polling system with threshold priority, because of the priority policy, there would be a single critically loaded queue (QK for example) in heavy traffic. Along the same derivation procedure, the stable queues are transferred into a multi-class priority polling system with N-policy vacations and the scaled number of customers in the critically loaded queue is exponentially distributed with parameter η such that PK 1 P K 1 μK 1 i ¼ 1 ρi þ i ¼ 1 μ ρi . η ¼ 1
Proof. Rewriting (20), we get Lð0Þ 2 ðyÞ ¼
6 1 ¼ ð1 ρ1 ρ2 Þ6 4 y
5. Exact tail asymptotics in the degraded model
5.1. Preliminary First we introduce some necessary notations. The marginal distributions for the high- and low-priority customers are denoted by π iðhÞ and π ðlÞ j , respectively. The joint stationary distribution is ð0Þ ð0Þ ð0Þ ð0Þ denoted by π ð0Þ ði;jÞ such that π ð0;jÞ ¼ π 2;ð0;jÞ and π ði;jÞ ¼ π 1;ði;jÞ (i Z1). The distribution of the total number of customers is denoted by π ðTÞ n . Let λ ¼ λ1 þ λ2 and ρ 1 ¼ λ=μ1 . Without loss of generality, throughout this section we assume that λ1 þ λ2 þ μ1 þ μ2 ¼ 1. To completely derive the exact tail asymptotics, we also need the following notations:
λ2 λ2 b1 ¼ pffiffiffiffiffiffi pffiffiffiffiffi ; b2 ¼ pffiffiffiffiffiffi pffiffiffiffiffi ; λ2 þ ð μ1 λ1 Þ2 λ2 þ ð μ1 þ λ1 Þ2 ΔðyÞ ¼ ðλ þ μ1 λ2 yÞ2 4λ1 μ1 ¼ λ22 ð1 b1 yÞð1 b2 yÞ=b1 b2 ; pffiffiffiffiffiffiffiffiffiffi ðλ þ μ1 Þ Δð0Þ λ2 c 0 c0 ¼ ; c1 ¼ pffiffiffiffiffiffiffiffiffiffi; 2μ 1 Δð0Þ x1 ðyÞ ¼
ðλ þ μ1 λ2 yÞ 2λ1
pffiffiffiffiffiffiffiffiffiffi ΔðyÞ
¼ αðyÞ;
x1 ¼ x1 ð0Þ ¼
c0
; pffiffiffiffiffiffiffiffiffiffi ðλ þ μ1 λ2 yÞ þ ΔðyÞ 1 ; x2 ¼ x2 ð0Þ ¼ ; x2 ðyÞ ¼ c0 2λ1 xKðx; yÞ ¼ λ1 x2 þ ðλ þ μ1 λ2 yÞx μ1 ¼ λ1 ðx x1 ðyÞÞðx x2 ðyÞÞ;
ρ1
FðyÞ ¼ λ2 y2 ð1 2μ1 þ μ2 Þyþ 2μ2 ; pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi T n ðyÞ ¼ FðyÞ þy ΔðyÞ; TðyÞ ¼ FðyÞ y ΔðyÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2μ1 Þ þ ð1 2μ1 Þ2 þ4ðμ1 μ2 Þλ2 η1 ¼ ; 2μ2 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2μ1 Þ ð1 2μ1 Þ2 þ4ðμ1 μ2 Þλ2 η2 ¼ ; 2μ2 TðyÞT n ðyÞ ¼ 4μ22 ð1 yÞð1 η1 yÞð1 η2 yÞ; 1 ρ1 ρ2 η1 1 ρ1 ρ2 η2 ; b¼ ; a¼ 2μ 2 η1 η2 2μ 2 η η1 qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 2 D ¼ ðλ þ μ1 2 λ1 μ1 Þðμ1 μ2 λ1 μ1 Þ þ λ2 μ2 : Using the above notations, we rewrite (20) as the following Proposition. Proposition 3. n 1 aT n ðyÞ bT ðyÞ ð0Þ L2 ðyÞ ¼ þ ð1 ρ1 ρ2 Þ βðyÞ: y 1 η1 y 1 η2 y
3
μ2
7 17 5β ðyÞ 1 αðyÞ μ2 λ2 y λ1 y 1y n 1 2T ðyÞμ2 ð1 yÞ ¼ ð1 ρ1 ρ2 Þ 1 β ðyÞ n y TðyÞT ðyÞ 1 1 ρ1 ρ2 T n ðyÞ ð1 ρ1 ρ2 Þ β ðyÞ ¼ y ð1 η1 yÞð1 η2 yÞ 2μ2 n n 1 aT ðyÞ bT ðyÞ þ ð1 ρ1 ρ2 Þ β ðyÞ:□ ¼ y 1 η1 y 1 η2 y
i
In Section 4, we have derived the PGFs of the queue-length distributions in the degraded model expressed by (20)–(22) to approximate the stable queue lengths in the primary model in heavy traffic. In this section, to give an approximation of the distributions of the stable queues, we carry out a detailed analysis on the exact tail asymptotics for the stationary distributions of the degraded model.
1 αðyÞ 1y Lð0Þ ðyÞ 1 αðyÞ 3 μ2 λ2 y λ1 y 1y 2
λ2 þ λ1
5.2. Analysis of singularities and asymptotic expansions Along the same idea used for the classical priority model in [11], asymptotics of the coefficients are obtained by using the following Tauberian-like theorem, which is Corollary 2 given in [17]. For a function f(y) that is analytic at y ¼0, we denote the coefficient of yk in the Taylor expression of f(y) by C k ½f ðyÞ. Define
Δ~ ðϕ; ε; aÞ ¼ fz : j azj r 1 þ ε; j Argðaz 1Þj Z ϕ for 0 o a o 1; ε 4 0 and 0 o ϕ o π =2g f1=ag: Lemma 1 (Flajolet and Odlyzko). Assume that f(z) is analytic in Δðϕ; ε; 1Þ and f ðzÞ Kð1 zÞs
as z-1
in Δðϕ; εÞ:
Then as n-1: 1. If s2 = f0; 1; 2; …g, K n s 1: C n ½f ðzÞ Γ ð sÞ 2. If s is a nonnegative integer, then C n ½f ðzÞ ¼ oðn s 1 Þ:
From Lemma 1, the key goal is to locate the dominant singularity and to characterize the nature of the dominant singularity, which determines the decay, the prefactor and the singularity coefficient. For this goal, we also need the following Lemmas and Propositions. The proofs can be referred to [11]. Lemma 2. For the non-unit zeros 1=η1 and 1=η2 , we have Both 1=η1 and 1=η2 are real. η1 4 0. η1 4 η2 , and η1 o j η2 j implies η2 o0. η2 o 0, η2 ¼ 0 or η2 4 0 if and only if μ2 o μ1 , μ2 ¼ μ1 or μ2 4 μ1 , respectively. 5. η2 a b1 , and either T nð1=η2 Þ ¼ 0 or j η2 j ob1 .
1. 2. 3. 4.
Lemma 3 (Key Lemma). There are three cases for the dominant singularity of L2ð0Þ ðyÞ: 1. If D 4 0, then 1 o 1=η1 o 1=b1 and 1=η1 is a zero of T(y) (but not T n ðyÞ), and therefore 1=η1 is the dominant singularity of Lð0Þ 2 ðyÞ, which is a simple pole.
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
2. If D¼0, then 1 o 1=η1 ¼ 1=b1 and 1=η1 is a zero of T(y) and T n ðyÞ, and therefore 1=b1 is the dominant singularity of Lð0Þ 2 ðyÞ, which is both a branch point and a simple pole. 3. If D o 0, then 1 o 1=η1 o 1=b1 and 1=η1 is a zero of T n ðyÞ (but not T(y)), and therefore 1=b1 is the dominant singularity of L2ð0Þ ðyÞ, which is a branch point.
In this subsection, we provide a complete exact tail asymptotic of the stationary distributions (the joint and marginal queue lengths and the total number of customers) by using the Lemma 1 to the related generating functions. Theorem 2. The exact tail distribution in the marginal stationary distribution of the high-priority queue is given by
π nðhÞ ¼ ð1 ρ1 Þρn1 :
which readily yields the conclusion.□
Proof. Taking the Taylor expansion of (21) with respect to x gives 0 1
μ2
C n1 Lð0Þ ðyÞC Ax 1 αðyÞ 3 μ2 λ2 y λ1 y 1y 2 3 1 1 1 X X X ð0Þ n1 j5 n 1 4 ¼ f n ðyÞx ¼ π 1;ðn;jÞ y x ;
1 ½x2 ðyÞn
μ2
Lð0Þ 3 ðyÞ;
n ¼ 1; 2; …
ð37Þ
ðjÞ
f n ð0Þ ; j!
n ¼ 0; 1; 2; …:
ð38Þ
By elementary calculation, we have 2 j 6 d 1 ðjÞ lim f ð0Þ ¼ lim j 6 4 n-1 n n-1dy ½x2 ðyÞn y¼0
3. (Geometric decay with prefactor n 3/2) In the region of D o0, ffi ð0Þ ~ pffiffiffiffiffi π ði;nÞ C l;3 ð1 þ iBÞð ρ1 Þi n 3=2 bn1 :
1
λ1 μ1
2
1 μ2 λ2 η uðηÞ ¼ 2μ1
pffiffiffiffiffiffiffiffiffiffi ΔðηÞ :
Proof. Case 1: We consider i¼0. In this case, we have P1 ð0Þ n yLð0Þ n ¼ 1 π ð0;nÞ y . 2 ðyÞ ¼ i. if D 40, then Tðη1 Þ ¼ 0. Hence, 1 " # yLð0Þ 2 ðyÞ ¼ a lim T n ðyÞβ ðyÞ lim η1 y-1 ð1 η yÞ 1 η1 y-1 1 ¼ aT n ð1=η1 Þβ ð1=η1 Þ ¼ 2aFð1=η1 Þη1 β ð1=η1 Þ: ~ ðϕ; ε; η Þ. By Lemma 1, we obtain Clearly, yLð0Þ ðyÞ is analytic in Δ 1
ii. if D ¼0, then Tðb11 Þ ¼ Tðη1 Þ ¼ 0, hence, 1 pffiffiffiffiffiffiffiffiffiffi T n ðyÞ FðyÞ Fð1=b1 Þ y ΔðyÞ βðyÞ ¼ βðyÞ þ βðyÞ 1 η1 y 1 b1 y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b1 y 0 F ð1=b1 Þβð1=b1 Þ λ2 1 b2 =b1 βð1=b1 Þ þ pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 b1 b2 1 b1 y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 1 b2 =b1 βð1=b1 Þ pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b1 b2 1 b1 y and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n T ðyÞ 1 b1 y βðyÞ ¼ 0: 1 η2 y b1 y-1 lim
3 7 μ 7 5 1 αðyÞ μ2 λ2 y λ1 y 1 y y¼0
Applying Lemma 1 again, we get
ð0Þ 2 L3 ðyÞ
¼ lim ð 1Þj nj x2 ð0Þ n j x02 ð0Þj Lð0Þ 3 ð0Þ: n-1
2. (Geometric decay with prefactor n 1/2) In the region of D¼0, pffiffiffiffiffiffi n ð0Þ π ði;nÞ C l;2 ð ρ1 Þi n 1=2 b1 :
ð36Þ
By applying Taylor expansion to y, we get
π ð0Þ 1;ðn;jÞ ¼
1. (Exact geometric decay) In the region of D 4 0, ð0Þ π ði;nÞ C l;1 ½uðη1 Þi ηn1 :
2
j¼0
1 αðyÞ μ2 λ2 y λ1 y 1y
ð41Þ
ð0Þ π 2;ð0;nÞ C 2;l;1 ηn1 :
where f n ðyÞ ¼
where the last equation follows from the definition of c1 and x2 ð0Þ ¼ c10 . Taking (40) into (39), we get
l;3
Theorem 3. The exact tail asymptotics in the joint stationary distribution along the high-priority queue is characterized by: for a fixed number jZ 0 of low-priority customers, ! cj1 j n j ð0Þ π ðn;jÞ βð0Þð1 ρ1 ρ2 Þ n c0 : j!
n¼1
ð40Þ
Here C l;1 , C l;2 , C l;3 , uðηÞ and B~ are given below: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aλ2 1 b2 =b1 C l;1 ¼ 2aFð1=η1 Þβð1=η1 Þ; C l;2 ¼ pffiffiffiffi pffiffiffiffiffiffiffiffiffiffi βð1=b1 Þ; π b1 b1 b2 pffiffiffiffiffiffiffiffiffiffiffi μ μ1 μ2 b1 þ λ1 μ1 pffiffiffiffiffiffiffiffiffiffiffi ; C ¼ ½aσ ðη Þ þ bσ ðη Þβ ð1=b1 Þ; B~ ¼ 2
Proof. Letting y-1 in (19) leads to
1 X ρ 1 ρ1 ; π ðnhÞ xn 1 ¼ 1 L1ð0Þ ðx; 1Þ ¼ 1 ρ1 x n¼1
n¼1
λ2 x2 ð0Þ λ2 x2 ð0Þ c1 ¼ pffiffiffiffiffiffiffiffiffiffi ¼ 2 ; 2λ1 x2 ð0Þ þ λ þ μ1 Δð0Þ c0
Theorem 4. The exact tail asymptotics in the joint stationary distribution along the low-priority queue is characterized by: for a fixed number i Z 0 of high-priority customers,
b1 b2 π ðη b1 Þ
1 B X B 1 @½x ðyÞn 2 n¼1
x02 ð0Þ ¼
Combining (41) with (38) yields the conclusion.□
5.3. Main results of exact tail asymptotics
L1ð0Þ ðx; yÞ ¼
Differentiating xKðx; yÞ ¼ 0 yields
ðjÞ ð0Þnj cn0 j cj1 : lim f ð0Þ ¼ lim Lð0Þ n-1 n n-1 3
Proposition 4. If D o0, then for η ¼ ηi , i¼1,2, n T ðyÞ n b1 σ ðηÞn 3=2 b1 ; Cn 1 ηy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ 1 b =b with σ ðηÞ ¼ p2ffiffiffiffiffiffiffipffiffiffi 2 1 . 2
25
n ð0Þ π 2;ð0;nÞ C 2;l;2 n 1=2 b1 :
ð39Þ
iii. if D o 0, the conclusion is a direct consequence of Proposition 4.
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Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
Case 2: We consider i 40. Rewriting Eq. (37), we get 1 μ2 Lð0Þ 3 ðyÞ; ½x2 ðyÞi μ λ y λ y1 αðyÞ 2 1 2 1y n n T ðyÞ T ðyÞ þb ½uðyÞi βðyÞ; i ¼ 1; 2; … ¼ a 1 η1 y 1 η2 y
Table 1 The exact tail asymptotics of total number of customers when μ1 a μ2 .
f i ðyÞ ¼
Tail asymptotic
D40
(i) ρ 1 Z 1; or (ii) ρ 1 o 1 and ρ 1 o η1
π Tn C t;1a ηn1
ρ 1 o 1 and ρ 1 4 η1
π Tn C t;1b ðρ 1 Þn
ρ 1 o 1 and ρ 1 ¼ η1 D ¼0
π Tn C t;1c nηn1
ρ1 Z 1
n
π Tn C t;2a n 1=2 b1 π Tn C t;2b ðρ 1 Þn
ρ1 o 1
1 x1 ðyÞ ¼ ¼ ρ1 x1 ðyÞ ¼ uðyÞ: x2 ðyÞ x1 ðyÞx2 ðyÞ
Do0
i. if D 4 0, applying the same methodology in Case 1-i leads to the conclusion. pffiffiffiffiffiffi ii if D ¼0, we have uðb11 Þ ¼ ρ1 . The proof is similar to Case 1-ii. iii if D o 0, then for η ¼ ηi , i¼1,2, we have 6 lim 6 b1 y-14
Case
ð42Þ
where the last equation follows from:
2
Region
3 d T n ðyÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1 λ2 1 b2 =b1 dy 1 ηy 7 7¼ u 1 p ffiffiffiffiffiffiffiffiffiffi ; β 5 b1 b1 2 b1 b2 ð1 η=b1 Þ ð1 b1 yÞ 1=2
½uðyÞi β ðyÞ
ð43Þ
ρ1 Z 1
n
pffiffiffiffiffi ρ 1 o 1 and ρ 1 a ρ1 pffiffiffiffiffi ρ 1 o 1 and ρ 1 ¼ ρ1 , (ρ 1 ¼ b1 a η1 )
π Tn C t;3a n 3=2 b1 π Tn C t;3b ðρ 1 Þn π Tn C t;3c ðρ 1 Þn
C t;1b ¼ C t;2b ¼ C t;3b ¼ C t;3c are given below: ðμ μ2 Þη1 ðμ μ2 Þ C ; C t;1c ¼ 1 C 2;l;1 ; C t;1a ¼ 1 μ1 ðη1 ρ 1 Þ 2;l;1 μ1
μ2 μ1 b1 C ; μ1 ρ1 b1 2;l;2
C t;3a ¼
C t;2a ¼ C t;1b ¼
μ2 μ1 b1 C ; μ1 ρ1 b1 2;l;3
μ2 ð1 ρ1 ρ2 Þ μ1 μ2 T n ð1=ρ 1 Þ T n ð1=ρ 1 Þ þ a þb 1 η 1 =ρ 1 1 η 2 =ρ 1 μ1 μ1
β
1
ρ1
:
and 3 1 T n ðyÞ d
i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i b1 F 61 ηy dy ½uðyÞ βðyÞ 7 1 1 λ 1 b =b b1 7¼i u p2ffiffiffiffiffiffiffiffiffiffi 2 1 : lim 6 β 5 4 1=2 1 b b b1 y-1 ð1 b1 yÞ 1 1 2 b1 b2 ð1 η=b1 Þ 2 μ1 u b1 2
ð44Þ In addition, it can be deduced that 1 b1 F pffiffiffiffiffiffi b 1 ; uðb1 Þ ¼ ρ1 : B~ ¼ 1 2μ1 u b1
ð45Þ
Combining (43)–(45) yields 3 2 n d T ðyÞ ½uðyÞi β ðyÞ 6dy 1 ηy 7 pffiffiffiffi 1 ffii ~ pffiffiffiffiffi 7 ¼ ð1 þ iBÞð Þ lim 6 b1 π σ ðηÞ; ρ β 1 4 5 1=2 b b1 y-1 ð1 b yÞ 1 1
where σ ðηÞ is defined in 4. Applying Corollary 4.1 in [12], we get n T ðyÞ 1 ffi ~ pffiffiffiffiffi ½uðyÞi βðyÞ ð1 þ iBÞð ρ1 Þi β σ ðηÞn ð3=2Þ bn1 ; Cn 1 ηy b1 ð46Þ Taking (46) into (42) leads to the conclusion.□ Theorem 5. The exact tail asymptotics in the marginal stationary distribution of the low-priority queue is given by
π ðlÞ n ¼
μ2 π : λ2 2;ð0;n þ 1Þ
Proof. It is ¼ μλ 2 Lð0Þ 2 ðyÞ.□
clear
since
LðlÞ ðyÞ ¼ Lð0Þ 1 ð1; yÞ þ
yL2ð0Þ ðyÞ þLð0Þ 3 ðyÞ
2
Theorem 6. For the degraded model, 1. If μ1 ¼μ2 , then 1 π Tn ¼ β ð1 ρ 1 Þðρ 1 Þn ; 1 ρ1
n ¼ 0; 1; 2; …:
2. If μ1 a μ2 , then the exact tail asymptotics in the stationary distribution of total number of customers in the system is characterized in Table 1, where C t;1a , C t;1c , C t;2a , C t;3a and
Proof. Let LðTÞ ðyÞ ¼
P1
n¼0
n π ðTÞ n y . Then we have
ð0Þ ð0Þ LðTÞ ðyÞ ¼ yLð0Þ 1 ðy; yÞ þyL2 ðyÞ þ L3 ðyÞ 1 αðyÞ μ2 μ1 y λ2 þ λ1 1y Lð0Þ ðyÞ ¼ 1 αðyÞ 3 μ1 ð1 ρ 1 yÞ μ2 λ2 y λ1 y 1y 2
6 1 6μ þ ðμ μ Þ ¼ 1 2 μ1 ð1 ρ 1 yÞ4 2 ¼
3
μ2
7 ð0Þ 7L ðyÞ 1 αðyÞ 5 3 μ2 λ 2 y λ 1 y 1y
1 T n ðyÞ T n ðyÞ μ2 ð1 ρ1 ρ2 Þ þ ðμ1 μ2 Þ a βðyÞ; þb μ1 ð1 ρ 1 yÞ 1 η1 y 1 η2 y
where the last equation follows the same idea used in Proposition 3. 1. If μ1 ¼ μ2 , then LðTÞ ðyÞ ¼ 1 1ρ yL3ð0Þ ðyÞ and ρ 1 ¼ ρ1 þ ρ2 . Hence, the 1 conclusion is true. 2. If μ1 a μ2 , the proof is similar to Theorem 4. It is noted that two cases exist inpffiffiffiffi the region of D ¼0: ρ 1 Z 1 and ρ 1 o 1. Since 2 pffiffiffiffiffiffi ρ 1 b1 ¼ ρð1ρþ1 1 2ρp1 Þffiffiffiffi Z 0, we have ρ 1 ¼ b1 iff ρ 1 ¼ ρ1 and ρp 1 ffiffiffiffiffiffi ρ 1 4 b1 if ρ 1 a ρ1 . It is assumed that ρ 1 ¼ b1 . Then pffiffiffiffiffiffi D ¼ μ1 ð1 ρ1 Þ2 ðμ1 μ2 Þ a 0, which contradicts with D¼ 0. Hence, ρ 1 4 b1 .□ Remark 5. When N ¼1, the conclusions in tail asymptotics are consistent with [11]. Theorems 3 and 4 when i4 0 can also be proved using the mathematical induction based on the balance equations (see Theorems 3.1 and 3.2 in [11]). However, our proofs based on the PGFs could illustrate the value of parameters c1, B~ and u(y) much better. Remark 6. From Theorems 3–6, the tail asymptotics of the joint and marginal queue lengths and the total number of customers in the degraded model have the same decay rates and prefactors as the classical two-class preemptive priority queue in [11] and the coefficients in the degraded model are β ðÞ times of that in the second model. According to the stochastic decomposition in [15], the PGF of the number of customers at an arbitrary time in a vacation system is the product of two parts: one is the PGF of the number of customers in a classical M/G/1 queue and the other is the PGF of the number of customers present in the system at a random point
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
in the vacation. It is noted that the second part is analytic. Hence, the decay rates and the prefactors just depend on the M/G/1 queue. From this point of view, for all queueing systems with vacations, the vacation policy would not change the decay rates and prefactors in the tail asymptotics. They only appear via adding a factor in the coefficients.
6. Stochastic simulation In this section, some simulations are undertaken to test the validity of our main results in Theorem 1 and the tail asymptotic results in Theorem 5. We consider the model with fixed parameters λ1 ¼ 1, λ2 ¼ 3, μ1 ¼ 5, μ2 ¼ 10, μ3 ¼ 15 and N ¼ 7. We take ρ ¼ 0:7; 0:8; 0:9; 0:95 to describe the procedure of ρ-1 and λ3 can be determined by λ3 ¼ μ3 ðρ ρ1 ρ2 Þ. With the help of Matlab, we undertake
1 0.9 0.8
0.6 0.5 0.4 0.3
heavy−traffic ρ=0.7 ρ=0.8 ρ=0.9 ρ=0.95
0.2 0.1 0
2
4
6
8
simulations under different traffic loads. Due to the discrete property of the queue lengths which accumulates too many datum to one isolated point and definitely increases the errors, the relative error of the pth percentile is not a good criterion here. Hence we use the cdf (cumulative distribution function) of the queue lengths to observe the validity of the asymptotics. Each simulation runs until at least 5 105 customers are served and the first 1000 customers are discarded due to the warm-up phase. For this model, the scaled queue-length in the critically loaded queue is exponentially distributed with parameter η in the heavytraffic scenario. In Fig. 2, the cdf of ð1 ρÞX 3 is plotted. It is shown that the approximation performs well either for ρ↑1 or for moderate ρ. Meanwhile, we take down the waiting times of an arbitrary type 1 customer and type 2 customer under different traffic load and plot the cdfs in Fig. 3. In the heavy-traffic regime, the queue lengths in the stable queues have the same distributions as a twoclass priority queue with vacations (see the cdf with the legend of heavy-traffic in Fig. 3). Because of the preemptive priority, the waiting time of an arbitrary type 1 customer is independent of the traffic load ρ. When ρ is very close to 1 (when ρ ¼ 0:9 and 0.95 in Fig. 3), the empirical cdf of the waiting time of an arbitrary type 2 customer is very close to that in the degraded model. Since the degraded model is independent of ρ3, the asymptotic illustrates that the queue lengths in the stable queues are independent of the queue length in the critically loaded queue in heavy traffic. In order to illustrate the accuracy of the tail asymptotics in Section 5, another three simulations are taken here. We take ρ ¼ 0:99 to denote the heavy traffic regime and take the tail asymptotic of the marginal queue length of Q2 (see conclusions Table 2 Parameter values in three cases of tail asymptotics.
10
x
Cases
λ2
D
Tail asymptotic
Case 1 Case 2 Case 3
λ2 ¼ 4 λ2 ¼ 3 λ2 ¼ 2
D¼ 0.0042 D¼ 0 D ¼ 0:0051
Exact geometric decay Geometric decay with prefactor n ð1=2Þ Geometric decay with prefactor n ð3=2Þ
Fig. 2. Empirical cdf of ð1 ρÞX 3 for different loads.
1
1
0.98
0.9 0.8
0.96
0.7
0.94 P{W2 ≤ x}
0
P{W1 ≤ x}
P{(1−ρ)X3 ≤ x}
0.7
27
0.92 0.9
0.86 0.84
0
0.5
x
1
0.5 0.4
heavy−traffic ρ=0.7 ρ=0.8 ρ=0.9 ρ=0.95
0.88
0.6
heavy−traffic ρ=0.7 ρ=0.8 ρ=0.9 ρ=0.95
0.3 0.2 1.5
0.1
0
2
4 x
Fig. 3. Empirical cdf of waiting times in Q1 (left) and Q2 (right) for different loads.
6
8
28
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
0.05 P{X2=n}
simulation tail asymptotic 0
5
10
15
20 25 30 n (case1: D=0.0042)
35
40
45
0.05 P{X2=n}
simulation tail asymptotic 0
4
6
8
10
12 14 16 n (case2: D=0)
18
20
22
24
0.05 P{X2=n}
simulation tail asymptotic 0
4
6
8
10 12 n (case3: D=−0.0051)
14
16
18
Fig. 4. Tail asymptotic of the marginal queue length of Q2 in heavy traffic.
in Theorem 5) as an example. Since the tail asymptotic depends on the parameter D, we fix λ1 ¼ 2, μ1 ¼ 5, μ2 ¼ 10, μ3 ¼ 15 and N ¼7 while taking three different λ2 such that the following three cases in Table 2 show up. The corresponding marginal queue lengths of Q2 in heavy traffic are shown in Fig. 4. It is shown that the methodology that uses the result obtained by singular perturbation technique to get the tail asymptotics in the stable queues is effective.
7. Conclusions In this paper, using the singular-perturbation technique, we first derived the exact heavy-traffic limits of a three-queue priority polling system with threshold service policy. Then based on the result obtained, we provided an approximation of the tail asymptotics of the stable queues, which describes the heavy-traffic behaviors more distinctly. The singular-perturbation technique is based on the balance equations and hence can be easily extended to polling systems with multiple queues and with general services. It can also be used to analyze the heavy-traffic limits of polling systems without multitype branching properties [18]. However, if we apply the singular technique to the models with more than one critically loaded queues, then, initially, we need to know which queue becomes unstable first. This can be referred to the concepts of relative stability and, further, the degree of stability in [19]. In addition, when ρ is moderate, the approximation seems not so accurate. Hence, it is necessary to seek for more efficient approximation techniques. Acknowledgments This research is partially supported by the National Natural Science Foundation of China (11271373, 11201489, 11371374) and Independent Innovation Project of Graduate School of Central South University (2014zzts009). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.
References [1] Lee DS, Sengupta B. Queueing analysis of a threshold based priority scheme for ATM networks. IEEE/ACM Trans Netw (TON) 1993;1(6):709–17. [2] Boxma OJ, Koole GM, Mitrani I. A two-queue polling model with a threshold service policy. In: Proceedings of the third international workshop on ieee modeling, analysis, and simulation of computer and telecommunication systems, 1995. MASCOTS'95; 1995. p. 84–8. [3] Boxma O, Koole G, Mitrani I. Polling models with threshold switching. In: Quantitative Methods in Parallel Systems. Springer; 1995. p. 129–40. [4] Deng Y, Tan J. Priority queueing model with changeover times and switching threshold. J Appl Probab 2001;38:263–73. [5] Deng Y, Song S, Tan J. Non-preemptive priority queueing model with change over times and switching threshold. Commun Appl Math Comput 2001;15:28–40. [6] Wei F, Kowada M, Adachi K. Performance analysis of a two-queue model with an (M, N)-threshold service schedule. J Oper Res Soc Jpn-Keiei Kagaku 2001;44 (2):101–24. [7] Landry R, Stavrakakis I. Queueing study of a 3-priority policy with distinct service strategies. IEEE/ACM Trans Netw (TON) 1993;1(5):576–89. [8] Knessl C, Tier C. Applications of singular perturbation methods in queueing. Advances in queueing theory, methods, and open problems, probability and stochastics series 1995:311–36. [9] Morrison JA, Borst SC. Interacting queues in heavy traffic. Queueing Syst 2010;65(2):135–56. [10] Boon MAA, Winands EMM. Heavy-traffic analysis of k-limited polling systems. Probab Eng Inf Sci 2014;28(4):451–71. [11] Li H, Zhao YQ. Exact tail asymptotics in a priority queue-characterizations of the preemptive model. Queueing Syst 2009;63(1–4):355–81. [12] Li H, Zhao YQ. Exact tail asymptotics in a priority queue-characterizations of the non-preemptive model. Queueing Systems 2011;68(2):165–92. [13] Song Y, Liu Z, Dai H. Exact tail asymptotics for a discrete-time preemptive priority queue. Acta Math Appl Sin Engl Ser 2015;31(1):43–58. [14] Kella O, Yechiali U. Priorities in M/G/1 queue with server vacations. Naval Res Logist 1988;35(1):23–34. [15] Fuhrmann S, Cooper RB. Stochastic decompositions in the M/G/1 queue with generalized vacations. Oper Res 1985;33(5):1117–29. [16] Bertsimas D, Mourtzinou G. Multiclass queueing systems in heavy traffic: an asymptotic approach based on distributional and conservation laws. Oper Res 1997;45(3):470–87. [17] Flajolet P, Odlyzko A. Singularity analysis of generating functions. SIAM J Discrete Math 1990;3(2):216–40. [18] Resing JAC. Polling systems and multitype branching processes. Queueing Syst 1993;13(4):409–26. [19] Sum L, Chang RK, Xie Y. Relative stability analysis of multiple queues. In: Proceedings of the 1st international conference on performance evaluation methodologies and tools. ACM New York; 2006. p. 65.
Contents lists available at ScienceDirect
Computers & Operations Research journal homepage: www.elsevier.com/locate/caor
Heavy-traffic asymptotics of a priority polling system with threshold service policy Zaiming Liu, Yuqing Chu, Jinbiao Wu n School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, PR China
art ic l e i nf o
a b s t r a c t
Available online 29 June 2015
In this paper, by the singular-perturbation technique, we investigate the heavy-traffic behavior of a priority polling system with three queues under threshold policy. It turns out that the scaled queuelength of the critically loaded queue is exponentially distributed, independent of that of the stable queues, which possess the same distributions as a two-class priority queue with N-policy vacation. Further, we provide an approximation of the tail queue-length distribution of the stable queues, which shows that it has the same prefactors and decay rates as the classical two-class preemptive priority queue. Stochastic simulations are taken to support the results. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Polling system Heavy-traffic Singular-perturbation Tail queue-length distribution Stochastic simulation
1. Introduction The study of the priority polling system is motivated by its wide applications in computer and communication systems, such as ATM (Asynchronous Transfer Mode) switch systems and network standards like DQDB (Distributed Queue Dual Bus). ATM involves two different types of traffic: real time traffic (voice, video) and non-real time traffic (data), which also needs different types of QoS (Quality of Service) standard. By setting the threshold parameter, a higher priority is offered to real time traffic to shorten its delay and the delay of non-real time traffic is kept in a valid regime, which turns out to be a flexible way to control the operation of the whole system. Lee and Sengupta first investigated the threshold-based priority systems in [1]. Later, a special case of two-queue M/M/1 polling system with threshold policy was studied by Boxma et al. in [2,3]. The model was further extended with switch-over times by Deng et al. in [4,5] and with one more server by Feng in [6]. The motivation stems from [7], in which Landry and Stavrakakis proposed a third type of traffic called control traffic with Headof-Line (HoL) priority in the integrated ATM environment, which involves critical network control and reservation information. In this paper, we focus on the heavy-traffic limits when there is a single critically loaded queue. Using the singular-perturbation technique, we derive the lowest-order asymptotic of the joint queue-length distribution in
terms of a small positive parameter measuring the closeness of the system to instability. The perturbation technique is always utilized to compute the performance measures of queueing systems by giving the approximation solutions to certain equations and a survey in [8] can be referred. A main application of the singularperturbation technique in queueing systems is to investigate the heavy-traffic behavior (see [9,10]). With the singular-perturbation technique, we conclude that the queue lengths in the stable queues have the same joint distributions as a degraded model (see the definition in Remark 1) which is a two-class preemptive priority queue with N-policy vacation. In general, no closed-form expressions for the steadystate probabilities in this degraded model can be obtained. Using the Kernel method (see [11–13] for details), we present the exact tail asymptotics of queue lengths in the degraded model, which can further approximate the tail asymptotics of the stable queues. The remainder of this paper is organized as follows. In Section 2, the model and some notations are introduced. In Section 3, the singular-perturbation technique is applied to derive the heavytraffic limits and the detailed derivation is carried out in Section 4. In Section 5, we provide the exact tail asymptotics of queue lengths of the stable queues. In Section 6, some simulations are undertaken to evaluate the heavy-traffic asymptotics. We finally conclude the whole work in this paper and propose some topics for further research in Section 7.
2. Model description n
Corresponding author. E-mail addresses: [email protected] (Z. Liu), [email protected] (Y. Chu), [email protected] (J. Wu). http://dx.doi.org/10.1016/j.cor.2015.06.013 0305-0548/& 2015 Elsevier Ltd. All rights reserved.
We consider a polling model with single server consisting of three queues Q1, Q2, Q3. We refer to the customers queueing in Qi
20
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
as the type i customers, i ¼1,2,3. The buffer capacity of each queue is infinite. Customers arrive at Qi independently according to a Poisson process with rate λi. For type i customers, the service times are mutually independent and all follow an exponential distribution with rate μi. Q1 has the HoL priority and Q2 has a higher priority over Q3. In each queue customers are served according to the FIFO (First In First Out) discipline. We assume that the arrival processes and the service processes are independent. The service discipline is described as follows. 1. Q1 is served exhaustively, which means that the server serves the customers in Q1 until it is empty and then switches to Q2. 2. When the server is serving a customer in Q2, if a type 1 customer arrives, then the server switches to Q1 immediately, otherwise, it continues serving the customers in Q2 until Q2 becomes empty and then switches to Q3. 3. When the server is serving a customer in Q3, if a type 1 customer arrives, then the server switches to Q1 immediately, if the size of Q2 reaches a given threshold N and Q1 is empty, then the server switches to Q2 immediately, otherwise, it continues serving the customers in Q3 until Q3 becomes empty and then switches to Q2. It is assumed that all the switches are instantaneous. In addition, the switches caused by the threshold push the customer undergoing service to the head of the queue and the service of the interrupted customer resumes from the beginning, which means that the completed service time of the interrupted customer is invalid. The traffic load of Qi is denoted by ρi ¼ λi =μi , i¼ 1,2,3. We assume the ergodicity condition of the system ρ ¼ ρ1 þ ρ2 þ ρ3 o 1 is satisfied. Let Xi(t) be the number of customers in Qi at time t, and S(t) be the position of the server at time t with SðtÞ A f0; 1; 2; 3g, where SðtÞ ¼ 0 denotes that the server is idle at time t. The associated stochastic process fYðtÞ; t Z0g ¼ ðX 1 ðtÞ; X 2 ðtÞ; X 3 ðtÞ; SðtÞÞ; t Z 0 is an aperiodic and irreducible four-dimensional Markov process. Let Xi (i¼1,2,3) be the steady-state queue length of Qi and S be the steady-state position of the server. Define the stationary probabilities ps ðx1 ; x2 ; x3 Þ ¼ lim PrfYðtÞ ¼ ðx1 ; x2 ; x3 ; sÞg; t-1
s ¼ 0; 1; 2; 3:
We study the heavy-traffic limits of the joint queue-length distribution by increasing the arrival rate λ3 so that ρ-1 , while keeping λ1 a 0, λ2 a 0 and μ1 ; μ2 ; μ3 fixed. When ρ-1 , Q3 becomes critically loaded, whereas Q1 and Q2 remain stable since Q1 and Q2 have higher priorities over Q3. The single-perturbation technique is implemented here. We first apply a perturbation to λ3 in the balance equations, in which case Q3 is close to becoming critically loaded. Then we solve the lowest order terms in the balance equations to obtain the queuelength distributions of the stable queues Q1 and Q2. At last we solve the first-order and second-order terms to get a differential equation and compute the scaled number of customers in Q3. Applying the Markov property, we obtain the following balance equations when x3 Z 2 ðλ1 þ λ2 þ λ3 þ μ1 Þp1 ðx1 ; x2 ; x3 Þ
where δðÞ is Kronecker function. In the above equations, we have omitted the parts for x3 ¼ 0 and x3 ¼ 1 which do not play a role after the perturbation since X3 tends to infinity as Q3 becomes critically loaded and the probability of Q3 being empty or 1 goes to zero. Throughout the paper, we adopt the following standard notations:
Function F(x) is o(x) if FðxÞ=x-0 as x-0. Function F(x) is OðxÞ if there exists a c Z 0 such that FðxÞ=x-c as x-0 while Oð1Þ is a constant time complexity.
For functions f(n) and g(n) of nonnegative integers n, f ðnÞ gðnÞ f ðnÞ means limn-1 gðnÞ ¼ 1.
3. Perturbation From the stability condition, the system becomes unstable as
ρ3 -1 ρ1 ρ2 , i.e. λ3 -μ3 ð1 ρ1 ρ2 Þ. Therefore it is assumed that
λ3 ¼ μ3 ð1 ρ1 ρ2 Þ εω;
ω 4 0; 0 o ε⪡1:
ð4Þ
Let ζ ¼ εx3 , and ps ðx1 ; x2 ; x3 Þ ¼ ps ðx1 ; x2 ; ζ =εÞ ¼ εϕs;ðx1 ;x2 Þ ðζ ; εÞ;
0 o ζ ¼ Oð1Þ; s ¼ 1; 2; 3:
ð5Þ
Taking (4) and (5) into the balance equations (1)–(3) and then taking the Taylor expansion, we obtain ðλ1 þ λ2 þ μ1 Þϕ1;ðx1 ;x2 Þ ðζ ; εÞ
¼ λ1 ϕ1;ðx1 1;x2 Þ ðζ ; εÞδðx1 Z 2Þ þ λ2 ϕ1;ðx1 ;x2 1Þ ðζ ; εÞδðx2 Z 1Þ ∂ϕ1;ðx1 ;x2 Þ ðζ ; εÞ
!
ε2 ∂2 ϕ1;ðx1 ;x2 Þ ðζ ; εÞ 2 2 ∂ζ ∂ζ þ λ1 ϕ2;ð0;x2 Þ ðζ ; εÞδðx1 ¼ 1; x2 Z 1Þ þ μ1 ϕ1;ðx1 þ 1;x2 Þ ðζ ; εÞ þ λ1 ϕ3;ð0;x2 Þ ðζ ; εÞδðx1 ¼ 1; x2 o NÞ þ oðε2 Þ; x1 Z1; x2 Z 0; ð6Þ
ðμ3 ð1 ρ1 ρ2 Þ εωÞ ε
ðλ1 þ λ2 þ μ2 Þϕ2;ð0;x2 Þ ðζ ; εÞ ∂ϕ2;ð0;x2 Þ ðζ ; εÞ
!
ε2 ∂2 ϕ2;ð0;x2 Þ ðζ ; εÞ 2 2 ∂ζ ∂ζ þ λ2 ϕ2;ð0;x2 1Þ ðζ ; εÞδðx2 Z 2Þ þ μ1 ϕ1;ð1;x2 Þ ðζ ; εÞ þ λ2 ϕ3;ð0;N 1Þ ðζ ; εÞδðx2 ¼ NÞ þ μ2 ϕ2;ð0;x2 þ 1Þ ðζ ; εÞ þ oðε2 Þ; x2 Z 1;
¼ ðμ3 ð1 ρ1 ρ2 Þ εωÞ ε
þ
ð7Þ ðλ1 þ λ2 Þϕ3;ð0;x2 Þ ðζ ; εÞ ¼ ðμ3 ð1 ρ1 ρ2 Þ εωÞ ε ∂ϕ3;ð0;x2 Þ ðζ ; εÞ
þ μ3 ε
∂ζ
∂ϕ3;ð0;x2 Þ ðζ ; εÞ ∂ζ
ε2 ∂ ϕ3;ð0;x2 Þ ðζ ; εÞ þ 2 2 ∂ζ 2
þ
!
ε2 ∂2 ϕ3;ð0;x2 Þ ðζ ; εÞ 2 2 ∂ζ
!
i þ μ1 ϕ1;ð1;0Þ ðζ ; εÞ þ μ2 ϕ2;ð0;1Þ ðζ ; εÞ δðx2 ¼ 0Þ
þ λ3 p1 ðx1 ; x2 ; x3 1Þ þ λ1 p2 ð0; x2 ; x3 Þδðx1 ¼ 1; x2 Z 1Þ þ μ1 p1 ðx1 þ 1; x2 ; x3 Þ þ λ1 p3 ð0; x2 ; x3 Þδðx1 ¼ 1; x2 o NÞ; x1 Z 1; x2 Z 0;
ð1Þ
¼ λ2 p2 ð0; x2 1; x3 Þδðx2 Z 2Þ þ λ3 p2 ð0; x2 ; x3 1Þ þ μ1 p1 ð1; x2 ; x3 Þ
ð2Þ
ðλ1 þ λ2 þ λ3 þ μ3 Þp3 ð0; x2 ; x3 Þ ¼ λ2 p3 ð0; x2 1; x3 Þδðx2 Z 1Þ þ λ3 p3 ð0; x2 ; x3 1Þ þ μ3 p3 ð0; x2 ; x3 þ 1Þ þ μ1 p1 ð1; 0; x3 Þ þ μ2 p2 ð0; 1; x3 Þ δðx2 ¼ 0Þ; 0 r x2 r N 1; ð3Þ
h
¼ λ1 p1 ðx1 1; x2 ; x3 Þδðx1 Z 2Þ þ λ2 p1 ðx1 ; x2 1; x3 Þδðx2 Z 1Þ
ðλ1 þ λ2 þ λ3 þ μ2 Þp2 ð0; x2 ; x3 Þ
þ λ2 p3 ð0; N 1; x3 Þδðx2 ¼ NÞ þ μ2 p2 ð0; x2 þ 1; x3 Þ; x2 Z 1;
þ λ2 ϕ3;ð0;x2 1Þ ðζ ; εÞδðx2 Z 1Þ þ oðε2 Þ;
0 r x2 r N 1:
ð8Þ
It is noted that λ3 only plays a role in equations for OðεÞ terms and higher. Throughout the paper, we do Taylor expansions of ϕs;ðx1 ;x2 Þ ðζ ; εÞ (s ¼ 1; 2; 3) in powers of ε as follows: ð1Þ 2 ϕs;ðx1 ;x2 Þ ðζ ; εÞ ¼ ϕð0Þ s;ðx1 ;x2 Þ ðζ Þ þ εϕs;ðx1 ;x2 Þ ðζ Þ þOðε Þ;
s ¼ 1; 2; 3:
ð9Þ
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
In the next section the lowest order terms of the resulting equations after Taylor expansions are equated to find expressions ð0Þ for ϕs;ðx1 ;x2 Þ ðζ Þ (s ¼ 1; 2; 3). Subsequently the first-order and secondorder terms are equated to find the scaled queue-length distribution of Q3. For convenience, we introduce the corresponding probability generating functions (PGFs) 1 1 X X
Q ðjÞ 1 ðx; y; ζ Þ ¼
x1 ¼ 1 x2 ¼ 0
1 X
Q ðjÞ 2 ðy; ζ Þ ¼
x2 ¼ 1
x1 1 x2 ϕðjÞ y ; 1;ðx1 ;x2 Þ ðζ Þx
x2 1 ϕðjÞ ; 2;ð0;x2 Þ ðζ Þy
1 1 X X
Q 1 ðx; y; ζ ; εÞ ¼
x1 ¼ 1 x2 ¼ 0 1 X
Q 2 ðy; ζ ; εÞ ¼
x2 ¼ 1
j ¼ 0; 1;
Q ðjÞ 3 ðy; ζ Þ ¼
N 1 X x2 ¼ 0
x2 ϕðjÞ 3;ð0;x2 Þ ðζ Þy ; j ¼ 0; 1;
ϕ1;ðx1 ;x2 Þ ðζ ; εÞxx1 1 yx2 ;
ϕ2;ð0;x2 Þ ðζ ; εÞyx2 1 ;
N 1 X x2 ¼ 0
ð0Þ
ðλ1 þ λ2 þ μ1 Þϕ1;ðx1 ;x2 Þ ðζ Þ ð0Þ
x1 Z1; x2 Z0;
μ ϕ
ζδ
ζ
μ ϕ
ð0Þ
ð0Þ
ð0Þ 3;ð0;0Þ ð
ð0Þ 1;ð1;0Þ ð
ðλ1 þ λ2 Þϕ3;ð0;x2 Þ ðζ Þ ¼ λ2 ϕ3;ð0;x2 1Þ ðζ Þ;
ζ Þ þ μ2 ϕ
ζ Þ;
x2 Z 1;
1 r x2 r N 1; ð0Þ 2;ð0;1Þ ð
ζ Þ:
ð11Þ ð12Þ ð13Þ
ð0Þ such that We introduce P 0 ðζ Þ and π s;ðx 1 ;x2 Þ ð0Þ ϕs;ðx ðζ Þ ¼ π ð0Þ s;ðx1 ;x2 Þ P 0 ðζ Þ; 1 ;x2 Þ
x1 ¼ 1 x2 ¼ 0
ð0Þ π 1;ðx þ 1 ;x2 Þ
1 X x2 ¼ 1
s ¼ 1; 2; 3;
π ð0Þ 2;ð0;x2 Þ þ
N 1 X
L2ð0Þ ðyÞ ¼ L3ð0Þ ðyÞ ¼
1 1 X X
x2 ¼ 0
ð0Þ π 3;ð0;x ¼ 1: 2Þ
ð0Þ x1 1 x2 y ; 1;ðx1 ;x2 Þ x x1 ¼ 1 x2 ¼ 0 1 X ð0Þ x2 1 ; 2;ð0;x2 Þ y x2 ¼ 1
π
π
N 1 X x2 ¼ 0
x2 π ð0Þ 3;ð0;x2 Þ y :
Then it is clear that ð0Þ Q ð0Þ 1 ðx; y; ζ Þ ¼ L1 ðx; yÞP 0 ðζ Þ;
ð14Þ
ð0Þ Q ð0Þ s ðy; ζ Þ ¼ Ls ðyÞP 0 ðζ Þ;
ð15Þ
s ¼ 2; 3:
From (12), we get L3ð0Þ ðyÞ ¼
N 1 X x2 ¼ 0
ð20Þ
μ
2 Lð0Þ ðyÞ: xKðx; yÞ 1 αðyÞ 3 μ2 λ2 y λ1 y 1 y λ1 ½x αðyÞ
ð21Þ
ð0Þ ð0Þ ðr 2 yÞx2 π ð0Þ 3;ð0;0Þ ¼ HðyÞπ 3;ð0;0Þ ¼ β ðyÞL3 ð1Þ;
N HðyÞ where r 2 ¼ λ λþ2 λ , HðyÞ ¼ ðrr22yÞy 1 1 and β ðyÞ ¼ Hð1Þ . 1 2
rule,
we
obtain
ρ1 ð0Þ Lð0Þ 1 ð1; 1Þ ¼ 1 ρ ρ L3 ð1Þ 1
2
ρ2 ð0Þ Lð0Þ 2 ð1Þ ¼ 1 ρ1 ρ2 L3 ð1Þ. By the normalizing condition, it is ð0Þ get Lð0Þ 3 ð1Þ ¼ 1 1 2 . Therefore, we have L1 ð1; 1Þ ¼ ð0Þ L2 ð1Þ ¼ 2 . Moreover,
ρ
ρ
and
easy to
ρ1 and
ρ
Lð0Þ 3 ðyÞ ¼ ð1 ρ1 ρ2 Þβ ðyÞ:
ð22Þ
We will determine the unknown expression of P 0 ðζ Þ in the rest of this section. Remark 1. Compared to the balance equations (1)–(3), Eqs. (10)– (13) have only considered type 1 and type 2 customers, which can be seen as the balance equations of a degraded model. In the view point of an arbitrary type 1 or type 2 customer, Q3 behaves like an N-policy vacation in heavy traffic which starts when Q1 and Q2 become empty and terminates once a type 1 customer arrives or there are N type 2 customers queueing. Hence the degraded model is actually a two-class priority queue with N-policy vacation in which the queue length PGF's are Eqs. (20)–(22). To be more intact, we give a detailed description of the degraded model as follows:
Define L1ð0Þ ðx; yÞ ¼
1 αðyÞ 1 y Lð0Þ ðyÞ: 1 αðyÞ 3 μ 2 λ2 y λ1 y 1y
λ2 þ λ1
L'Hôpital's
ð0Þ ðx2 Z 2Þ þ 1 1;ð1;x2 Þ ð Þ ð0Þ Þ ðx2 ¼ NÞ þ 2 2;ð0;x2 þ 1Þ ð
ζδ
1 1 X X
Lð0Þ 2 ðyÞ ¼
Letting y-1 and then letting x-1 in (20) and (21), with
ð0Þ
ζ Þ ¼ μ1 ϕ
1 Kðx; yÞ ¼ λ1 ð1 xÞ þ λ2 ð1 yÞ þ μ1 1 ; x 1 aðyÞ ¼ λ1 þ λ2 ð1 yÞ þ μ2 1 : y
ð10Þ
ðλ1 þ λ2 þ μ2 Þϕ2;ð0;x2 Þ ðζ Þ
ðλ1 þ λ2 Þϕ
where
Lð0Þ 1 ðx; yÞ ¼
ð0Þ
þ λ1 ϕ2;ð0;x2 Þ ðζ Þδðx1 ¼ 1; x2 Z 1Þ þ μ1 ϕ1;ðx1 þ 1;x2 Þ ðζ Þ
λ ϕ
ð18Þ
Substituting (20) into (19) leads to
ð0Þ
¼ λ1 ϕ1;ðx1 1;x2 Þ ðζ Þδðx1 Z 2Þ þ λ2 ϕ1;ðx1 ;x2 1Þ ðζ Þδðx2 Z 1Þ
λ ϕ
ð0Þ yaðyÞL2ð0Þ ðyÞ ¼ μ1 Lð0Þ 1 ð0; yÞ ½λ1 þ λ2 ð1 yÞL3 ðyÞ;
Clearly, for every j yj r 1, the kernel xKðx; yÞ has a unique zero x ¼ αðyÞ with j xj r1. Since Lð0Þ 1 ðx; yÞ is analytic in j xj o1 for every j yj o 1 and continuous up to j xj r 1 and j yj r 1, the right-hand side of Eq. (19) must vanish when x ¼ αðyÞ, which yields
Equating the lowest-order terms of the resulting equation after the Taylor expansions of (6)–(8), we obtain
ð0Þ ¼ 2 2;ð0;x2 1Þ ð Þ ð0Þ þ 2 3;ð0;N 1Þ ð
ð17Þ
ð19Þ
4.1. Equating the lowest-order terms
ð0Þ
ð0Þ ð0Þ xKðx; yÞL1ð0Þ ðx; yÞ ¼ λ1 x½yLð0Þ 2 ðyÞ þ L3 ðyÞ μ1 L1 ð0; yÞ;
ð0Þ xKðx; yÞL1ð0Þ ðx; yÞ ¼ y½λ1 x aðyÞLð0Þ 2 ðyÞ þ ½λ1 ðx 1Þ þ λ2 ðy 1ÞL3 ðyÞ:
ϕ3;ð0;x2 Þ ðζ ; εÞyx2 :
4. Model analysis
þ λ1 ϕ3;ð0;x2 Þ ðζ Þδðx1 ¼ 1; x2 o NÞ;
Using the PGFs to rewrite the balance equations (10) and (11) leads to
Adding (18) to (17), we have
Q 3 ðy; ζ ; εÞ ¼
ð0Þ
21
ð16Þ
(i) There are two classes of customers in the system, the highand low-priority customers, arriving independently according to two Poisson processes with rates λ1 and λ2, respectively. (ii) Each class of customer is served according to the FIFO discipline. (iii) The server takes a vacation once the system empties and goes back to work once the size of the low-priority customers reaches N or there is a high-priority customer's arrival. (iv) The high-priority customers have preemptive priorities over the low-priority customers just like in the classical two-class preemptive priority queue. (v) Both classes of customers require an exponential amount of service times with rates μ1 and μ2, respectively. All service
22
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
times are independent and also independent of the arrival processes.
Taking the summation over all x1 and x2 of (6)–(8) gives
μ1
1 X
ϕ1;ð1;x2 Þ ðζ ; εÞ
x2 ¼ 0
4.2. Equating the first-order terms
1 X
¼ λ1
x2 ¼ 1
"
In this subsection, after the Taylor expansion of the perturbed balance equations (6)–(8), equating the first-order terms, we present an equation in Proposition 1.
λ1
Proposition 1. h i ð1Þ ð1Þ ð1Þ ð1 ρ1 ρ2 Þ Q ð1Þ 1 ð1; 1; ζ Þ þQ 2 ð1; ζ Þ þ Q 3 ð1; ζ Þ Q 3 ð1; ζ Þ μ μ ¼ 3 ρ1 þ 3 ρ2 P 00 ðζ Þ:
μ1
ζÞ ¼ λ
μ3 xð1 ρ
ð1Þ Þ 1 Q ð1Þ 1 ð0; y; Þ þ 1 xQ 3 ðy; ð0Þ 0 1 2 ÞL1 ðx; yÞP 0 ð Þ;
ζ
μ
ρ
ζ λ ζ
ð24Þ ð1Þ
ð1Þ N ½λ1 þ λ2 ð1 yÞQ ð1Þ 3 ðy; ζ Þ ¼ λ2 y ϕ3;ð0;x2 1Þ ðζ Þ þ μ1 Q 1 ð0; 0; ζ Þ
þ μ2 Q 2ð1Þ ð0; ζ Þ þ μ3 ðρ1 þ ρ2 ÞL3ð0Þ ðyÞP 00 ðζ Þ: ð25Þ Applying the same method in solving (17) and (18) to Eqs. (23)– (25), after some elementary calculations, we get
λ1 ½x αðyÞ
ð1Þ ½yQ ð1Þ 2 ðy; ζ Þ þ Q 3 ðy; ζ Þ xKðx; yÞ μ ð1 ρ1 ρ2 Þ ð0Þ 0 ½xL1 ðx; yÞ αðyÞLð0Þ 3 1 ðαðyÞ; yÞP 0 ðζ Þ: xKðx; yÞ
ð26Þ ð1Þ ½yaðyÞ λ1 αðyÞyQ ð1Þ 2 ðy; ζ Þ þ ½λ1 ð1 αðyÞÞ þ λ2 ð1 yÞQ 3 ðy; ζ Þ n h io ð0Þ ð0Þ ¼ μ3 ðρ1 þ ρ2 ÞL3 ðyÞ ð1 ρ1 ρ2 Þ yL2 ðyÞ þ αðyÞLð0Þ P 00 ðζ Þ: 1 ðαðyÞ; yÞ
ð27Þ Letting y-1 and then letting x-1 in (26), with L'Hôpital's rule, we obtain
ρ1 μ3 ρ1 ð1 ρ1 ρ2 Þ2 0 ð1Þ ½Q ð1Þ P 0 ðζ Þ: 2 ð1; ζ Þ þQ 3 ð1; ζ Þ 1 ρ1 μ1 ð1 ρ1 Þ2 ð28Þ
Similarly, letting y-1 in (27) leads to i 1 h ð1 ρ1 ρ2 ÞQ 2ð1Þ ð1; ζ Þ ρ2 Q 3ð1Þ ð1; ζ Þ 1 ρ1 ( " # ) μ3 ρ1 ρ2 ρ1 ρ2 ð1 ρ1 ρ2 Þ μ3 ¼ þ ρ2 P 00 ðζ Þ: μ1 1 ρ1 μ2 ð1 ρ1 Þ2
ð29Þ
Combining (28) and (29), we have
∂Q 1 ð1; 1; ζ ; εÞ ∂ζ
ð30Þ
∂Q 2 ð1; ζ ; εÞ ∂ζ
ϕ1;ð1;x2 Þ ðζ ; εÞ þ λ2 ϕ3;ð0;N 1Þ ðζ ; εÞ μ3 ð1 ρ1 ρ2 Þε
ð31Þ
ϕ3;ð0;x2 Þ ðζ ; εÞ þ λ2 ϕ3;ð0;N 1Þ ðζ ; εÞ ∂Q 3 ð1; ζ ; εÞ ∂Q ð1; ζ ; εÞ þ μ3 3 ∂ζ ∂ζ #
¼ μ1 ϕ1;ð1;0Þ ðζ ; εÞ þ μ2 ϕ2;ð0;1Þ ðζ ; εÞ μ3 ð1 ρ1 ρ2 Þ þ ω
ð0Þ 0 þ μ1 Q ð1Þ 1 ð0; y; ζ Þ μ3 yð1 ρ1 ρ2 ÞL2 ðyÞP 0 ðζ Þ;
Q ð1Þ 1 ð1; 1; ζ Þ ¼
1 X
"
ð1Þ
Q ð1Þ 1 ðx; y; ζ Þ ¼
N 1 X x2 ¼ 0
ð1Þ yaðyÞQ 2ð1Þ ðy; ζ Þ ¼ λ2 yN ϕ3;ð0;x2 1Þ ðζ Þ μ1 Q ð1Þ 1 ð0; 0; ζ Þ μ2 Q 2 ð0; ζ Þ
ϕ3;ð0;x2 Þ ðζ ; εÞ μ3 ð1 ρ1 ρ2 Þε
" # ∂Q ð1; ζ ; εÞ μ3 ð1 ρ1 ρ2 Þ ∂2 Q 2 ð1; ζ ; εÞ 2 þ ω 2 þ ε þ Oðε3 Þ; 2 2 ∂ζ ∂ζ
λ1
ð23Þ
x2 ¼ 0
# ∂Q 1 ð1; 1; ζ ; εÞ μ3 ð1 ρ1 ρ2 Þ ∂2 Q 1 ð1; 1; ζ ; εÞ 2 þ ε þ Oðε3 Þ; 2 2 ∂ζ ∂ζ
x2 ¼ 1
ζÞ
N 1 X
ϕ2;ð0;x2 Þ ðζ ; εÞ þ μ2 ϕ2;ð0;1Þ ðζ ; εÞ
¼ μ1
Proof. Taking the PGF of the first-order terms of the resulting equations after the Taylor expansion of (6)–(8), we have ð1Þ 1 xyQ 2 ðy;
1 X x2 ¼ 1
μ2
xKðx; yÞQ 1ð1Þ ðx; y;
ω
þ
ϕ2;ð0;x2 Þ ðζ ; εÞ þ λ1
∂Q 3 ð1; ζ ; εÞ μ3 ð2 ρ1 ρ2 Þ ∂2 Q 3 ð1; ζ ; εÞ 2 þ ε þOðε3 Þ: 2 2 ∂ζ ∂ζ
ð32Þ
Summing over (30)–(32), we obtain ∂Q 1 ð1; 1; ζ ; εÞ ∂Q 2 ð1; ζ ; εÞ ∂Q 3 ð1; ζ ; εÞ þ þ 0 ¼ μ3 ð1 ρ1 ρ2 Þ ∂ζ ∂ζ ∂ζ " 2 ∂Q ð1; ζ ; εÞ μ ð1 ρ1 ρ2 Þ ∂ Q 1 ð1; 1; ζ ; εÞ εþ 3 þ μ3 3 2 2 ∂ζ ∂ζ ! ∂2 Q 2 ð1; ζ ; εÞ ∂2 Q 3 ð1; ζ ; εÞ ∂Q 1 ð1; 1; ζ ; εÞ þ þ þω 2 2 ∂ζ ∂ζ ∂ζ # 2 ∂Q ð1; ζ ; εÞ ∂Q 3 ð1; ζ ; εÞ μ ∂ Q 3 ð1; ζ ; εÞ 2 þ 2 þ ε þ Oðε3 Þ: ð33Þ þ 3 2 2 ∂ζ ∂ζ ∂ζ Now taking the Taylor expansion of (33) with respect to ε like (9) gives h 0 ¼ μ3 ð1 ρ1 ρ2 ÞP ″0 ðζ Þ þ ωP 00 ðζ Þ þ μ3 Q 30ð1Þ ð1; ζ Þ μ3 ð1 ρ1 ρ2 Þ
0 i 0 0 Q 1ð1Þ ð1; 1; ζ Þ þ Q 2ð1Þ ð1; ζ Þ þ Q 3ð1Þ ð1; ζ Þ ε2 þ Oðε3 Þ μ μ ¼ μ3 1 ρ1 ρ2 þ 3 ρ1 þ 3 ρ2 P ″0 ðζ Þ þ ωP 00 ðζ Þ ε2 þ Oðε3 Þ:
μ1
μ2
In the above equation, the first equation follows from (14) and the second equation follows from Proposition 1. From the above derivation procedure, we can conclude the following Proposition. Proposition 2. After taking the summation over all x1 and x2 of the Taylor series of all perturbed balance equations (6)–(8), the Oð1Þ and OðεÞ terms cancel and, moreover, equating the Oðε2 Þ terms yields the following differential equation for P 0 ðζ Þ: μ μ ωP 00 ðζ Þ ¼ ð1 ρ1 ρ2 Þ þ 3 ρ1 þ 3 ρ2 μ3 P ″0 ðζ Þ:
μ1
μ2
h i ð1Þ ð1Þ ð1Þ ð1 ρ1 ρ2 Þ Q ð1Þ 1 ð1; 1; ζ Þ þ Q 2 ð1; ζ Þ þ Q 3 ð1; ζ Þ Q 3 ð1; ζ Þ i μ ρ ð1 ρ ρ Þ2 1 h ð1Þ 3 1 1 2 ð1 ρ1 ρ2 ÞQ ð1Þ P 00 ðζ Þ 2 ð1; ζ Þ ρ2 Q 3 ð1; ζ Þ μ 1 ρ1 ð1 ρ1 Þ2 1 μ μ ¼ 3 ρ1 þ 3 ρ2 P 00 ðζ Þ:□
¼
4.4. The scaled number of customers in the critically loaded queue
4.3. Equating the second-order terms
Now we can finally present the density of the scaled number of customers in Q3, i.e. P 0 ðζ Þ. By combining the differential equation R1 in Proposition 2 with 0 P 0 ðζ Þdζ ¼ 1, we obtain
In this subsection, we consider the sum of all Oðε2 Þ terms in (6)–(8) to determine P 0 ðζ Þ.
P 0 ðζ Þ ¼ ηe ηζ ; h i with ωη ¼ 1 ρ1 ρ2 þ μμ3 ρ1 þ μμ3 ρ2 μ3 .
μ1
μ2
1
2
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
As a special case, we may take ω ¼ μ3 , which gives
ζ ¼ ð1 ρÞX 3 , then 1 μ μ ¼ 1 ρ1 ρ2 þ 3 ρ1 þ 3 ρ2 : η μ1 μ2
By applying the multiclass distributional law of Bertsimas and Mourtzinou [16], we get that the scaled waiting time at Q3 follows an exponential distribution with parameter μ3 η. 4.5. Main result Theorem 1. For λ3 ¼ μ3 ð1 ρ1 ρ2 Þ εω, we have limPfX 1 r x1 ; X 2 r x2 ; εX 3 r ζ g ¼ Lðx1 ; x2 Þð1 e ηζ Þ; ε↓0
where Lð; Þ is the joint cumulative distribution function (cdf) of the queue lengths of the degraded model described in Remark 1 with PGFs of (20)–(22). The main result stated in Theorem 1 can be interpreted as follows: in the heavy-traffic regime, R1. The queue lengths in the stable queues have the same distribution as a two-class priority queue with N-policy vacation. R2. The scaled number of customers in the critically loaded queue is exponentially distributed with parameter η. R3. The queue lengths in the stable queues and the (scaled) number of customers in the critically loaded queue are independent. We now elaborate on these interpretations. For R1, since Q3 is critically loaded, Q3 would be visited during each cycle. From the perspective of type 1 or type 2 customers, Q3 behaves like an Npolicy vacation in heavy traffic which starts when Q1 and Q2 become empty and terminates once a type 1 customer arrives or there are N type 2 customers queueing. For R2, we note that the total workload in the system equals the amount of workload in an M/G/1 queue with arrival rate λ1 þ λ2 þ λ3 and hyperexponentially distributed service times, i.e. the service time is exponentially distributed with parameter μi with probability λ þ λλi þ λ , i ¼ 1; 2; 3. Based on the heavy-traffic 1 2 3 results for the M/G/1 queue (see [16]), the distribution of the scaled total workload converges to an exponential distribution with mean ρE½R, where R is a residual service time and 1 E½R ¼
μ1
ρ1 þ
1
1
ρ þ ρ μ2 2 μ3 3 : ρ
In the heavy traffic, since almost all customers are located in Q3, the total number of customers at this queue is also exponentially distributed with mean μ3 μ1 ρ1 þ μ1 ρ2 þ μ1 ρ3 . Since λ3 ↑μ3 1 2 3 ð1 ρ1 ρ2 Þ, the scaled number of customers in Q3 is exponentially distributed with parameter η. Finally, R3 follows from the time-scale separation in the heavy traffic which implies that the dynamics of the stable queues evolve at a much faster time scale than the dynamics of the critically loaded queue. Since the amount of “memory” of the stable queues asymptotically vanishes compared to that of the critically loaded queue, the queue lengths in the stable queues are independent of the (scaled) number of customers in the critically loaded queue in the limit. Remark 2. For the degraded model, we provide a stochastic decomposition of the queue length distribution in Q2, which ðlÞ explains N appears only via Lð0Þ 3 ðyÞ. Let L ðÞ be the PGF of the number of type 2 customers.
23
Kella and Yechiali introduced the concept of the delay-cycle in [14] and succeeded in deriving the waiting times in an M/G/1 queue with priorities and vacations. Here we do some modifications and apply it to our degraded model. Since FIFO or LIFO (Last In First Out) discipline lead to the same distribution of the queue length in the system, without loss of generality, we assume a nonpreemptive LIFO queueing discipline (among the same class customers) here. Define vacation customers to be customers who arrive during the vacation time. According to the argument in [14], we divide the whole time into two delaycycles: T1-cycle and TU-cycle (see Fig. 1). T1-cycle starts with the service time B1 of the type 1 vacation customer and the customers served thereafter are all his descendants only. TU-cycle is the vacation time U, plus the service time of all the type 2 vacation customers and their descendants. For type 2 customers, each delay-cycle behaves like an M=G=1 queue with a vacation. We consider a type 2 customer and his type 1 descendants as a generalized type 2 customer, denoted by a type 20 customer. Hence, type 20 customers arrive with intensity λ2 and have service time B20 with LST and mean B20 ðsÞ ¼
μ2 ; μ2 þ s þ λ1 ð1 θ1 ðsÞÞ
EB20 ¼
1
μ2 ð1 ρ1 Þ
:
ð34Þ
Since B20 is the time between the service initiations of two consecutive type 2 customers, the number of type 2 customers at an arbitrary time is equal to that of type 20 customers. It is noted that any point in time is either within T1-cycle or TU1 cycle. The length of a T1-cycle equals 1 EB ρ1 ρ2 . Hence, the fraction 1 of the time that the system is in a T1-cycle equals λ1 ρ3 1 EB ρ1 ρ2 ¼ ρ1 and the fraction of the time in a TU-cycle equals 1 ρ1 . By (16), the PGF of the number of type 2 customers present at a random point during the vacation time equals βðÞ. Due to the LIFO discipline, if there exist type 2 vacation customers when T1-cycle begins, they will wait until the T1-cycle ends and then get service. By applying the Fuhrmann–Cooper Decomposition Theorem (see [15]), the number of type 2 customers at a random point in a T1cycle equals the number of type 2 customers present when B1 begins, plus the number of type 2 customers at a random point during B1 and the number of type 2 customers in an M/G/1 queue. That is B 0 ðλ2 ð1 yÞÞð1 ρ20 Þð1 yÞ 1 θ1 ðλ2 ð1 yÞÞ LðlÞ ðyÞ ¼ 2 βðyÞ ρ1 þ ð1 ρ1 Þ ; B20 ðλ2 ð1 yÞÞ y λ2 ð1 zÞEθ1 ð35Þ ρ
where ρ20 ¼ ð1 2ρ Þ is the traffic load of type 20 customers. Sub1 stituting (34) into the above equation leads to 1 αðyÞ 1y Lð0Þ ðyÞ: L ðyÞ ¼ 1 αðyÞ 3 ρ2 μ2 λ2 y λ1 y 1y ðlÞ
1
λ2 þ λ1
Since the vacation time depends on the arriving process in the degraded model, N appears only via β ðÞ in (35). Remark 3. Since the singular-perturbation technique only depends on the balance equations, by using the supplementary variable method, it also applies to the system with general services. However there would be one more parameter. Thus the derivation will be much more tedious. In order to make the
Fig. 1. Diagram of T1-cycle and TU-cycle.
24
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
methodology easier to handle and get a better understanding of the conclusions, we only consider the polling system with exponential services. Remark 4. In a K-queue polling system with threshold priority, because of the priority policy, there would be a single critically loaded queue (QK for example) in heavy traffic. Along the same derivation procedure, the stable queues are transferred into a multi-class priority polling system with N-policy vacations and the scaled number of customers in the critically loaded queue is exponentially distributed with parameter η such that PK 1 P K 1 μK 1 i ¼ 1 ρi þ i ¼ 1 μ ρi . η ¼ 1
Proof. Rewriting (20), we get Lð0Þ 2 ðyÞ ¼
6 1 ¼ ð1 ρ1 ρ2 Þ6 4 y
5. Exact tail asymptotics in the degraded model
5.1. Preliminary First we introduce some necessary notations. The marginal distributions for the high- and low-priority customers are denoted by π iðhÞ and π ðlÞ j , respectively. The joint stationary distribution is ð0Þ ð0Þ ð0Þ ð0Þ denoted by π ð0Þ ði;jÞ such that π ð0;jÞ ¼ π 2;ð0;jÞ and π ði;jÞ ¼ π 1;ði;jÞ (i Z1). The distribution of the total number of customers is denoted by π ðTÞ n . Let λ ¼ λ1 þ λ2 and ρ 1 ¼ λ=μ1 . Without loss of generality, throughout this section we assume that λ1 þ λ2 þ μ1 þ μ2 ¼ 1. To completely derive the exact tail asymptotics, we also need the following notations:
λ2 λ2 b1 ¼ pffiffiffiffiffiffi pffiffiffiffiffi ; b2 ¼ pffiffiffiffiffiffi pffiffiffiffiffi ; λ2 þ ð μ1 λ1 Þ2 λ2 þ ð μ1 þ λ1 Þ2 ΔðyÞ ¼ ðλ þ μ1 λ2 yÞ2 4λ1 μ1 ¼ λ22 ð1 b1 yÞð1 b2 yÞ=b1 b2 ; pffiffiffiffiffiffiffiffiffiffi ðλ þ μ1 Þ Δð0Þ λ2 c 0 c0 ¼ ; c1 ¼ pffiffiffiffiffiffiffiffiffiffi; 2μ 1 Δð0Þ x1 ðyÞ ¼
ðλ þ μ1 λ2 yÞ 2λ1
pffiffiffiffiffiffiffiffiffiffi ΔðyÞ
¼ αðyÞ;
x1 ¼ x1 ð0Þ ¼
c0
; pffiffiffiffiffiffiffiffiffiffi ðλ þ μ1 λ2 yÞ þ ΔðyÞ 1 ; x2 ¼ x2 ð0Þ ¼ ; x2 ðyÞ ¼ c0 2λ1 xKðx; yÞ ¼ λ1 x2 þ ðλ þ μ1 λ2 yÞx μ1 ¼ λ1 ðx x1 ðyÞÞðx x2 ðyÞÞ;
ρ1
FðyÞ ¼ λ2 y2 ð1 2μ1 þ μ2 Þyþ 2μ2 ; pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi T n ðyÞ ¼ FðyÞ þy ΔðyÞ; TðyÞ ¼ FðyÞ y ΔðyÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2μ1 Þ þ ð1 2μ1 Þ2 þ4ðμ1 μ2 Þλ2 η1 ¼ ; 2μ2 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2μ1 Þ ð1 2μ1 Þ2 þ4ðμ1 μ2 Þλ2 η2 ¼ ; 2μ2 TðyÞT n ðyÞ ¼ 4μ22 ð1 yÞð1 η1 yÞð1 η2 yÞ; 1 ρ1 ρ2 η1 1 ρ1 ρ2 η2 ; b¼ ; a¼ 2μ 2 η1 η2 2μ 2 η η1 qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 2 D ¼ ðλ þ μ1 2 λ1 μ1 Þðμ1 μ2 λ1 μ1 Þ þ λ2 μ2 : Using the above notations, we rewrite (20) as the following Proposition. Proposition 3. n 1 aT n ðyÞ bT ðyÞ ð0Þ L2 ðyÞ ¼ þ ð1 ρ1 ρ2 Þ βðyÞ: y 1 η1 y 1 η2 y
3
μ2
7 17 5β ðyÞ 1 αðyÞ μ2 λ2 y λ1 y 1y n 1 2T ðyÞμ2 ð1 yÞ ¼ ð1 ρ1 ρ2 Þ 1 β ðyÞ n y TðyÞT ðyÞ 1 1 ρ1 ρ2 T n ðyÞ ð1 ρ1 ρ2 Þ β ðyÞ ¼ y ð1 η1 yÞð1 η2 yÞ 2μ2 n n 1 aT ðyÞ bT ðyÞ þ ð1 ρ1 ρ2 Þ β ðyÞ:□ ¼ y 1 η1 y 1 η2 y
i
In Section 4, we have derived the PGFs of the queue-length distributions in the degraded model expressed by (20)–(22) to approximate the stable queue lengths in the primary model in heavy traffic. In this section, to give an approximation of the distributions of the stable queues, we carry out a detailed analysis on the exact tail asymptotics for the stationary distributions of the degraded model.
1 αðyÞ 1y Lð0Þ ðyÞ 1 αðyÞ 3 μ2 λ2 y λ1 y 1y 2
λ2 þ λ1
5.2. Analysis of singularities and asymptotic expansions Along the same idea used for the classical priority model in [11], asymptotics of the coefficients are obtained by using the following Tauberian-like theorem, which is Corollary 2 given in [17]. For a function f(y) that is analytic at y ¼0, we denote the coefficient of yk in the Taylor expression of f(y) by C k ½f ðyÞ. Define
Δ~ ðϕ; ε; aÞ ¼ fz : j azj r 1 þ ε; j Argðaz 1Þj Z ϕ for 0 o a o 1; ε 4 0 and 0 o ϕ o π =2g f1=ag: Lemma 1 (Flajolet and Odlyzko). Assume that f(z) is analytic in Δðϕ; ε; 1Þ and f ðzÞ Kð1 zÞs
as z-1
in Δðϕ; εÞ:
Then as n-1: 1. If s2 = f0; 1; 2; …g, K n s 1: C n ½f ðzÞ Γ ð sÞ 2. If s is a nonnegative integer, then C n ½f ðzÞ ¼ oðn s 1 Þ:
From Lemma 1, the key goal is to locate the dominant singularity and to characterize the nature of the dominant singularity, which determines the decay, the prefactor and the singularity coefficient. For this goal, we also need the following Lemmas and Propositions. The proofs can be referred to [11]. Lemma 2. For the non-unit zeros 1=η1 and 1=η2 , we have Both 1=η1 and 1=η2 are real. η1 4 0. η1 4 η2 , and η1 o j η2 j implies η2 o0. η2 o 0, η2 ¼ 0 or η2 4 0 if and only if μ2 o μ1 , μ2 ¼ μ1 or μ2 4 μ1 , respectively. 5. η2 a b1 , and either T nð1=η2 Þ ¼ 0 or j η2 j ob1 .
1. 2. 3. 4.
Lemma 3 (Key Lemma). There are three cases for the dominant singularity of L2ð0Þ ðyÞ: 1. If D 4 0, then 1 o 1=η1 o 1=b1 and 1=η1 is a zero of T(y) (but not T n ðyÞ), and therefore 1=η1 is the dominant singularity of Lð0Þ 2 ðyÞ, which is a simple pole.
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
2. If D¼0, then 1 o 1=η1 ¼ 1=b1 and 1=η1 is a zero of T(y) and T n ðyÞ, and therefore 1=b1 is the dominant singularity of Lð0Þ 2 ðyÞ, which is both a branch point and a simple pole. 3. If D o 0, then 1 o 1=η1 o 1=b1 and 1=η1 is a zero of T n ðyÞ (but not T(y)), and therefore 1=b1 is the dominant singularity of L2ð0Þ ðyÞ, which is a branch point.
In this subsection, we provide a complete exact tail asymptotic of the stationary distributions (the joint and marginal queue lengths and the total number of customers) by using the Lemma 1 to the related generating functions. Theorem 2. The exact tail distribution in the marginal stationary distribution of the high-priority queue is given by
π nðhÞ ¼ ð1 ρ1 Þρn1 :
which readily yields the conclusion.□
Proof. Taking the Taylor expansion of (21) with respect to x gives 0 1
μ2
C n1 Lð0Þ ðyÞC Ax 1 αðyÞ 3 μ2 λ2 y λ1 y 1y 2 3 1 1 1 X X X ð0Þ n1 j5 n 1 4 ¼ f n ðyÞx ¼ π 1;ðn;jÞ y x ;
1 ½x2 ðyÞn
μ2
Lð0Þ 3 ðyÞ;
n ¼ 1; 2; …
ð37Þ
ðjÞ
f n ð0Þ ; j!
n ¼ 0; 1; 2; …:
ð38Þ
By elementary calculation, we have 2 j 6 d 1 ðjÞ lim f ð0Þ ¼ lim j 6 4 n-1 n n-1dy ½x2 ðyÞn y¼0
3. (Geometric decay with prefactor n 3/2) In the region of D o0, ffi ð0Þ ~ pffiffiffiffiffi π ði;nÞ C l;3 ð1 þ iBÞð ρ1 Þi n 3=2 bn1 :
1
λ1 μ1
2
1 μ2 λ2 η uðηÞ ¼ 2μ1
pffiffiffiffiffiffiffiffiffiffi ΔðηÞ :
Proof. Case 1: We consider i¼0. In this case, we have P1 ð0Þ n yLð0Þ n ¼ 1 π ð0;nÞ y . 2 ðyÞ ¼ i. if D 40, then Tðη1 Þ ¼ 0. Hence, 1 " # yLð0Þ 2 ðyÞ ¼ a lim T n ðyÞβ ðyÞ lim η1 y-1 ð1 η yÞ 1 η1 y-1 1 ¼ aT n ð1=η1 Þβ ð1=η1 Þ ¼ 2aFð1=η1 Þη1 β ð1=η1 Þ: ~ ðϕ; ε; η Þ. By Lemma 1, we obtain Clearly, yLð0Þ ðyÞ is analytic in Δ 1
ii. if D ¼0, then Tðb11 Þ ¼ Tðη1 Þ ¼ 0, hence, 1 pffiffiffiffiffiffiffiffiffiffi T n ðyÞ FðyÞ Fð1=b1 Þ y ΔðyÞ βðyÞ ¼ βðyÞ þ βðyÞ 1 η1 y 1 b1 y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b1 y 0 F ð1=b1 Þβð1=b1 Þ λ2 1 b2 =b1 βð1=b1 Þ þ pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 b1 b2 1 b1 y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 1 b2 =b1 βð1=b1 Þ pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b1 b2 1 b1 y and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n T ðyÞ 1 b1 y βðyÞ ¼ 0: 1 η2 y b1 y-1 lim
3 7 μ 7 5 1 αðyÞ μ2 λ2 y λ1 y 1 y y¼0
Applying Lemma 1 again, we get
ð0Þ 2 L3 ðyÞ
¼ lim ð 1Þj nj x2 ð0Þ n j x02 ð0Þj Lð0Þ 3 ð0Þ: n-1
2. (Geometric decay with prefactor n 1/2) In the region of D¼0, pffiffiffiffiffiffi n ð0Þ π ði;nÞ C l;2 ð ρ1 Þi n 1=2 b1 :
ð36Þ
By applying Taylor expansion to y, we get
π ð0Þ 1;ðn;jÞ ¼
1. (Exact geometric decay) In the region of D 4 0, ð0Þ π ði;nÞ C l;1 ½uðη1 Þi ηn1 :
2
j¼0
1 αðyÞ μ2 λ2 y λ1 y 1y
ð41Þ
ð0Þ π 2;ð0;nÞ C 2;l;1 ηn1 :
where f n ðyÞ ¼
where the last equation follows from the definition of c1 and x2 ð0Þ ¼ c10 . Taking (40) into (39), we get
l;3
Theorem 3. The exact tail asymptotics in the joint stationary distribution along the high-priority queue is characterized by: for a fixed number jZ 0 of low-priority customers, ! cj1 j n j ð0Þ π ðn;jÞ βð0Þð1 ρ1 ρ2 Þ n c0 : j!
n¼1
ð40Þ
Here C l;1 , C l;2 , C l;3 , uðηÞ and B~ are given below: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aλ2 1 b2 =b1 C l;1 ¼ 2aFð1=η1 Þβð1=η1 Þ; C l;2 ¼ pffiffiffiffi pffiffiffiffiffiffiffiffiffiffi βð1=b1 Þ; π b1 b1 b2 pffiffiffiffiffiffiffiffiffiffiffi μ μ1 μ2 b1 þ λ1 μ1 pffiffiffiffiffiffiffiffiffiffiffi ; C ¼ ½aσ ðη Þ þ bσ ðη Þβ ð1=b1 Þ; B~ ¼ 2
Proof. Letting y-1 in (19) leads to
1 X ρ 1 ρ1 ; π ðnhÞ xn 1 ¼ 1 L1ð0Þ ðx; 1Þ ¼ 1 ρ1 x n¼1
n¼1
λ2 x2 ð0Þ λ2 x2 ð0Þ c1 ¼ pffiffiffiffiffiffiffiffiffiffi ¼ 2 ; 2λ1 x2 ð0Þ þ λ þ μ1 Δð0Þ c0
Theorem 4. The exact tail asymptotics in the joint stationary distribution along the low-priority queue is characterized by: for a fixed number i Z 0 of high-priority customers,
b1 b2 π ðη b1 Þ
1 B X B 1 @½x ðyÞn 2 n¼1
x02 ð0Þ ¼
Combining (41) with (38) yields the conclusion.□
5.3. Main results of exact tail asymptotics
L1ð0Þ ðx; yÞ ¼
Differentiating xKðx; yÞ ¼ 0 yields
ðjÞ ð0Þnj cn0 j cj1 : lim f ð0Þ ¼ lim Lð0Þ n-1 n n-1 3
Proposition 4. If D o0, then for η ¼ ηi , i¼1,2, n T ðyÞ n b1 σ ðηÞn 3=2 b1 ; Cn 1 ηy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ 1 b =b with σ ðηÞ ¼ p2ffiffiffiffiffiffiffipffiffiffi 2 1 . 2
25
n ð0Þ π 2;ð0;nÞ C 2;l;2 n 1=2 b1 :
ð39Þ
iii. if D o 0, the conclusion is a direct consequence of Proposition 4.
26
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
Case 2: We consider i 40. Rewriting Eq. (37), we get 1 μ2 Lð0Þ 3 ðyÞ; ½x2 ðyÞi μ λ y λ y1 αðyÞ 2 1 2 1y n n T ðyÞ T ðyÞ þb ½uðyÞi βðyÞ; i ¼ 1; 2; … ¼ a 1 η1 y 1 η2 y
Table 1 The exact tail asymptotics of total number of customers when μ1 a μ2 .
f i ðyÞ ¼
Tail asymptotic
D40
(i) ρ 1 Z 1; or (ii) ρ 1 o 1 and ρ 1 o η1
π Tn C t;1a ηn1
ρ 1 o 1 and ρ 1 4 η1
π Tn C t;1b ðρ 1 Þn
ρ 1 o 1 and ρ 1 ¼ η1 D ¼0
π Tn C t;1c nηn1
ρ1 Z 1
n
π Tn C t;2a n 1=2 b1 π Tn C t;2b ðρ 1 Þn
ρ1 o 1
1 x1 ðyÞ ¼ ¼ ρ1 x1 ðyÞ ¼ uðyÞ: x2 ðyÞ x1 ðyÞx2 ðyÞ
Do0
i. if D 4 0, applying the same methodology in Case 1-i leads to the conclusion. pffiffiffiffiffiffi ii if D ¼0, we have uðb11 Þ ¼ ρ1 . The proof is similar to Case 1-ii. iii if D o 0, then for η ¼ ηi , i¼1,2, we have 6 lim 6 b1 y-14
Case
ð42Þ
where the last equation follows from:
2
Region
3 d T n ðyÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1 λ2 1 b2 =b1 dy 1 ηy 7 7¼ u 1 p ffiffiffiffiffiffiffiffiffiffi ; β 5 b1 b1 2 b1 b2 ð1 η=b1 Þ ð1 b1 yÞ 1=2
½uðyÞi β ðyÞ
ð43Þ
ρ1 Z 1
n
pffiffiffiffiffi ρ 1 o 1 and ρ 1 a ρ1 pffiffiffiffiffi ρ 1 o 1 and ρ 1 ¼ ρ1 , (ρ 1 ¼ b1 a η1 )
π Tn C t;3a n 3=2 b1 π Tn C t;3b ðρ 1 Þn π Tn C t;3c ðρ 1 Þn
C t;1b ¼ C t;2b ¼ C t;3b ¼ C t;3c are given below: ðμ μ2 Þη1 ðμ μ2 Þ C ; C t;1c ¼ 1 C 2;l;1 ; C t;1a ¼ 1 μ1 ðη1 ρ 1 Þ 2;l;1 μ1
μ2 μ1 b1 C ; μ1 ρ1 b1 2;l;2
C t;3a ¼
C t;2a ¼ C t;1b ¼
μ2 μ1 b1 C ; μ1 ρ1 b1 2;l;3
μ2 ð1 ρ1 ρ2 Þ μ1 μ2 T n ð1=ρ 1 Þ T n ð1=ρ 1 Þ þ a þb 1 η 1 =ρ 1 1 η 2 =ρ 1 μ1 μ1
β
1
ρ1
:
and 3 1 T n ðyÞ d
i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i b1 F 61 ηy dy ½uðyÞ βðyÞ 7 1 1 λ 1 b =b b1 7¼i u p2ffiffiffiffiffiffiffiffiffiffi 2 1 : lim 6 β 5 4 1=2 1 b b b1 y-1 ð1 b1 yÞ 1 1 2 b1 b2 ð1 η=b1 Þ 2 μ1 u b1 2
ð44Þ In addition, it can be deduced that 1 b1 F pffiffiffiffiffiffi b 1 ; uðb1 Þ ¼ ρ1 : B~ ¼ 1 2μ1 u b1
ð45Þ
Combining (43)–(45) yields 3 2 n d T ðyÞ ½uðyÞi β ðyÞ 6dy 1 ηy 7 pffiffiffiffi 1 ffii ~ pffiffiffiffiffi 7 ¼ ð1 þ iBÞð Þ lim 6 b1 π σ ðηÞ; ρ β 1 4 5 1=2 b b1 y-1 ð1 b yÞ 1 1
where σ ðηÞ is defined in 4. Applying Corollary 4.1 in [12], we get n T ðyÞ 1 ffi ~ pffiffiffiffiffi ½uðyÞi βðyÞ ð1 þ iBÞð ρ1 Þi β σ ðηÞn ð3=2Þ bn1 ; Cn 1 ηy b1 ð46Þ Taking (46) into (42) leads to the conclusion.□ Theorem 5. The exact tail asymptotics in the marginal stationary distribution of the low-priority queue is given by
π ðlÞ n ¼
μ2 π : λ2 2;ð0;n þ 1Þ
Proof. It is ¼ μλ 2 Lð0Þ 2 ðyÞ.□
clear
since
LðlÞ ðyÞ ¼ Lð0Þ 1 ð1; yÞ þ
yL2ð0Þ ðyÞ þLð0Þ 3 ðyÞ
2
Theorem 6. For the degraded model, 1. If μ1 ¼μ2 , then 1 π Tn ¼ β ð1 ρ 1 Þðρ 1 Þn ; 1 ρ1
n ¼ 0; 1; 2; …:
2. If μ1 a μ2 , then the exact tail asymptotics in the stationary distribution of total number of customers in the system is characterized in Table 1, where C t;1a , C t;1c , C t;2a , C t;3a and
Proof. Let LðTÞ ðyÞ ¼
P1
n¼0
n π ðTÞ n y . Then we have
ð0Þ ð0Þ LðTÞ ðyÞ ¼ yLð0Þ 1 ðy; yÞ þyL2 ðyÞ þ L3 ðyÞ 1 αðyÞ μ2 μ1 y λ2 þ λ1 1y Lð0Þ ðyÞ ¼ 1 αðyÞ 3 μ1 ð1 ρ 1 yÞ μ2 λ2 y λ1 y 1y 2
6 1 6μ þ ðμ μ Þ ¼ 1 2 μ1 ð1 ρ 1 yÞ4 2 ¼
3
μ2
7 ð0Þ 7L ðyÞ 1 αðyÞ 5 3 μ2 λ 2 y λ 1 y 1y
1 T n ðyÞ T n ðyÞ μ2 ð1 ρ1 ρ2 Þ þ ðμ1 μ2 Þ a βðyÞ; þb μ1 ð1 ρ 1 yÞ 1 η1 y 1 η2 y
where the last equation follows the same idea used in Proposition 3. 1. If μ1 ¼ μ2 , then LðTÞ ðyÞ ¼ 1 1ρ yL3ð0Þ ðyÞ and ρ 1 ¼ ρ1 þ ρ2 . Hence, the 1 conclusion is true. 2. If μ1 a μ2 , the proof is similar to Theorem 4. It is noted that two cases exist inpffiffiffiffi the region of D ¼0: ρ 1 Z 1 and ρ 1 o 1. Since 2 pffiffiffiffiffiffi ρ 1 b1 ¼ ρð1ρþ1 1 2ρp1 Þffiffiffiffi Z 0, we have ρ 1 ¼ b1 iff ρ 1 ¼ ρ1 and ρp 1 ffiffiffiffiffiffi ρ 1 4 b1 if ρ 1 a ρ1 . It is assumed that ρ 1 ¼ b1 . Then pffiffiffiffiffiffi D ¼ μ1 ð1 ρ1 Þ2 ðμ1 μ2 Þ a 0, which contradicts with D¼ 0. Hence, ρ 1 4 b1 .□ Remark 5. When N ¼1, the conclusions in tail asymptotics are consistent with [11]. Theorems 3 and 4 when i4 0 can also be proved using the mathematical induction based on the balance equations (see Theorems 3.1 and 3.2 in [11]). However, our proofs based on the PGFs could illustrate the value of parameters c1, B~ and u(y) much better. Remark 6. From Theorems 3–6, the tail asymptotics of the joint and marginal queue lengths and the total number of customers in the degraded model have the same decay rates and prefactors as the classical two-class preemptive priority queue in [11] and the coefficients in the degraded model are β ðÞ times of that in the second model. According to the stochastic decomposition in [15], the PGF of the number of customers at an arbitrary time in a vacation system is the product of two parts: one is the PGF of the number of customers in a classical M/G/1 queue and the other is the PGF of the number of customers present in the system at a random point
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
in the vacation. It is noted that the second part is analytic. Hence, the decay rates and the prefactors just depend on the M/G/1 queue. From this point of view, for all queueing systems with vacations, the vacation policy would not change the decay rates and prefactors in the tail asymptotics. They only appear via adding a factor in the coefficients.
6. Stochastic simulation In this section, some simulations are undertaken to test the validity of our main results in Theorem 1 and the tail asymptotic results in Theorem 5. We consider the model with fixed parameters λ1 ¼ 1, λ2 ¼ 3, μ1 ¼ 5, μ2 ¼ 10, μ3 ¼ 15 and N ¼ 7. We take ρ ¼ 0:7; 0:8; 0:9; 0:95 to describe the procedure of ρ-1 and λ3 can be determined by λ3 ¼ μ3 ðρ ρ1 ρ2 Þ. With the help of Matlab, we undertake
1 0.9 0.8
0.6 0.5 0.4 0.3
heavy−traffic ρ=0.7 ρ=0.8 ρ=0.9 ρ=0.95
0.2 0.1 0
2
4
6
8
simulations under different traffic loads. Due to the discrete property of the queue lengths which accumulates too many datum to one isolated point and definitely increases the errors, the relative error of the pth percentile is not a good criterion here. Hence we use the cdf (cumulative distribution function) of the queue lengths to observe the validity of the asymptotics. Each simulation runs until at least 5 105 customers are served and the first 1000 customers are discarded due to the warm-up phase. For this model, the scaled queue-length in the critically loaded queue is exponentially distributed with parameter η in the heavytraffic scenario. In Fig. 2, the cdf of ð1 ρÞX 3 is plotted. It is shown that the approximation performs well either for ρ↑1 or for moderate ρ. Meanwhile, we take down the waiting times of an arbitrary type 1 customer and type 2 customer under different traffic load and plot the cdfs in Fig. 3. In the heavy-traffic regime, the queue lengths in the stable queues have the same distributions as a twoclass priority queue with vacations (see the cdf with the legend of heavy-traffic in Fig. 3). Because of the preemptive priority, the waiting time of an arbitrary type 1 customer is independent of the traffic load ρ. When ρ is very close to 1 (when ρ ¼ 0:9 and 0.95 in Fig. 3), the empirical cdf of the waiting time of an arbitrary type 2 customer is very close to that in the degraded model. Since the degraded model is independent of ρ3, the asymptotic illustrates that the queue lengths in the stable queues are independent of the queue length in the critically loaded queue in heavy traffic. In order to illustrate the accuracy of the tail asymptotics in Section 5, another three simulations are taken here. We take ρ ¼ 0:99 to denote the heavy traffic regime and take the tail asymptotic of the marginal queue length of Q2 (see conclusions Table 2 Parameter values in three cases of tail asymptotics.
10
x
Cases
λ2
D
Tail asymptotic
Case 1 Case 2 Case 3
λ2 ¼ 4 λ2 ¼ 3 λ2 ¼ 2
D¼ 0.0042 D¼ 0 D ¼ 0:0051
Exact geometric decay Geometric decay with prefactor n ð1=2Þ Geometric decay with prefactor n ð3=2Þ
Fig. 2. Empirical cdf of ð1 ρÞX 3 for different loads.
1
1
0.98
0.9 0.8
0.96
0.7
0.94 P{W2 ≤ x}
0
P{W1 ≤ x}
P{(1−ρ)X3 ≤ x}
0.7
27
0.92 0.9
0.86 0.84
0
0.5
x
1
0.5 0.4
heavy−traffic ρ=0.7 ρ=0.8 ρ=0.9 ρ=0.95
0.88
0.6
heavy−traffic ρ=0.7 ρ=0.8 ρ=0.9 ρ=0.95
0.3 0.2 1.5
0.1
0
2
4 x
Fig. 3. Empirical cdf of waiting times in Q1 (left) and Q2 (right) for different loads.
6
8
28
Z. Liu et al. / Computers & Operations Research 65 (2016) 19–28
0.05 P{X2=n}
simulation tail asymptotic 0
5
10
15
20 25 30 n (case1: D=0.0042)
35
40
45
0.05 P{X2=n}
simulation tail asymptotic 0
4
6
8
10
12 14 16 n (case2: D=0)
18
20
22
24
0.05 P{X2=n}
simulation tail asymptotic 0
4
6
8
10 12 n (case3: D=−0.0051)
14
16
18
Fig. 4. Tail asymptotic of the marginal queue length of Q2 in heavy traffic.
in Theorem 5) as an example. Since the tail asymptotic depends on the parameter D, we fix λ1 ¼ 2, μ1 ¼ 5, μ2 ¼ 10, μ3 ¼ 15 and N ¼7 while taking three different λ2 such that the following three cases in Table 2 show up. The corresponding marginal queue lengths of Q2 in heavy traffic are shown in Fig. 4. It is shown that the methodology that uses the result obtained by singular perturbation technique to get the tail asymptotics in the stable queues is effective.
7. Conclusions In this paper, using the singular-perturbation technique, we first derived the exact heavy-traffic limits of a three-queue priority polling system with threshold service policy. Then based on the result obtained, we provided an approximation of the tail asymptotics of the stable queues, which describes the heavy-traffic behaviors more distinctly. The singular-perturbation technique is based on the balance equations and hence can be easily extended to polling systems with multiple queues and with general services. It can also be used to analyze the heavy-traffic limits of polling systems without multitype branching properties [18]. However, if we apply the singular technique to the models with more than one critically loaded queues, then, initially, we need to know which queue becomes unstable first. This can be referred to the concepts of relative stability and, further, the degree of stability in [19]. In addition, when ρ is moderate, the approximation seems not so accurate. Hence, it is necessary to seek for more efficient approximation techniques. Acknowledgments This research is partially supported by the National Natural Science Foundation of China (11271373, 11201489, 11371374) and Independent Innovation Project of Graduate School of Central South University (2014zzts009). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.
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