An asymptotic property of branching-type overloaded polling networks

Yuejiao Wang; Zaiming Liu; Yuqing Chu; Yingqiu Li

doi:10.1515/math-2019-0116

Article Open Access

An asymptotic property of branching-type overloaded polling networks

Yuejiao Wang , Zaiming Liu , Yuqing Chu and Yingqiu Li

Published/Copyright: December 10, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] studied the fluid asymptotics of the joint queue length process for an overloaded cyclic polling system with multigated service discipline by exploiting the connection with multi-type branching processes. In contrast to the heavy traffic behaviors, the cycle time of the overloaded polling system increases by a deterministic times over times under passage to the fluid dynamics and the fluid limit preserves some randomness. The present paper aims to extend the overloaded asymptotics in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] to the corresponding polling system with general branching-type service disciplines and customer re-routing policy. A unifying overloaded asymptotic property is derived. Due to the exhaustiveness, the property is a natural extension of the classical polling model with multigated service discipline in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] and provides new exact results that have not been observed before for rerouting policy. Additionally, a stochastic simulation is undertaken for the validation of the fluid limit and the optimization of the gating indexes to minimize the total population is considered as an example to demonstrate the usefulness of the random fluid limit.

Keywords: polling networks; overloaded; general branching-type service policies; multi-type branching process; exhaustiveness

MSC 2010: 60J80; 90Bxx

1 Introduction

In this paper, we consider a cyclic N-queue (Q₁, ⋯, Q_N, N ≥ 2) polling system with general branching-type service discipline within each queue and customer re-routing policies: after completing service at Q_i, a customer is either routed to Q_j with probability p_i,j or leaves the system with probability p_i,0. The possibility for re-routing of customers further enhances the already-extensive modeling capabilities of polling models, since in many applications, customers require service at more than one facility of the system. Actually, the models of customer re-routing arise naturally in various models of computer, communication and robotic systems (see [1, 2, 3, 4] and references therein). One obvious example is a local area network in which terminals are interconnected in either a physical or logical structure (see [5]).

In the vast majority of papers that have appeared on polling models, it is almost invariably assumed that the system is stable and the stable performance measures are then concerned. With the advent of the era of Internet+, the study of critically or strictly super-critically loaded polling systems is vigorously pioneered due to the overloaded Internet channel or online shopping orders.

The heavy traffic (ρ → 1, ρ is the load of the system) behaviors have gained an ascending attention in the last two decades pioneered by Coffman et al. [6, 7]. By utilizing the connection with multi-type branching process, van der Mei[8] considered a unifying theory on branching-type polling models under heavy-traffic assumptions. In the similar way, Boon et al.[1] discussed the heavy-traffic asymptotic behaviors of a gated polling system with customer re-route policy. Furthermore, Liu et al.[9] extended the results in[1] to the analogous system with a general branching-type service policy in the same form. As an example for non-branching type polling systems, Liu et al.[10] investigated the heavy-traffic behavior of a priority polling system consisting of three M/M/1 queues with threshold policy and proved that the scaled queue-length of the critically loaded queue is exponentially distributed, independent of that of the stable queues.

The study of overloaded (ρ > 1) service system is important to control or predict how fast it blows up over time. However, hardly any attention has been given to the overloaded polling system. The few literature refers to [11, 12, 13, 14]. By using measure-valued state descriptor, Puha et al.[11] proved that the overloaded GI/GI/1 processor sharing queues converge in distribution to supercritical fluid models and a fluid limit result is proved as first order approximations to overloaded processor sharing queues. Using both fluid and diffusion limits, Jennings et al.[13] showed that the virtual waiting time process of an overloaded Multi-class FIFO(first-in-first-out) queue with abandonments converges to a limiting deterministic fluid process. Instead Remerova et al.[14] showed the fluid asymptotic process for the joint queue length process on an overloaded branching-type cyclic polling system by using the asymptotic properties of multi-type branching processes.

The motivation of the present paper is twofold. First, we dedicate to the investigation of a unified overloaded asymptotic fluid process for a very general class of branching-type polling models. Resing[15] observed that for a large class of polling models, including for example cyclic polling models with exhaustive and gated service, the evolution of the system at successive polling instants at a fixed queue can be described as a multi-type branching process with immigration. Models that satisfy this MTBP-structure allow for an exact analysis, whereas models that violate the MTBP-structure are often more intricate. Moreover, van der Mei[8] has given a unifying and insightful theory on branching-type polling models under heavy-traffic assumptions, which shows particularly attractive features. Second, by considering the polling system with re-routing policy, we aim to study how the strikingly simple overloaded fluid limit depends on the system parameters and in particular, on the routing probabilities p_i,j. Actually, the work carried out here is a natural progression from[9] and a natural extension of[14] due to the exhaustiveness. Our resulting expressions are very insightful, simple to implement and suitable for optimization purposes. Numerical results are presented to assess the accuracy of the results.

The rest of the paper is organized as follows. In Section 2, we describe precisely the polling model. In Sections 3, we introduce the multi-type branching process-structure of polling system. Section 4 provides the main results. Section 5 provides the proofs of Theorems 4.1 and 4.2. Section 6 discusses some numerical issues including the stochastic simulation to test the validity of the overloaded asymptotic behaviors and the optimization of gating indexes to minimize the average growth rate of the total population as an example. Section 7 concludes and provides an outlook on potential further research of our paper.

2 Model description

Consider an asymmetric polling network that consists of N ≥ 2 queues, Q₁, ⋯, Q_N, and attended by a single server that visits the queues in a cyclic order. Customers arrive at Q_i according to a Poisson process E_i(⋅) with rate λ_i. The service time of each customer at Q_i is a random variable B_i with finite mean value 𝔼B_i = 1/μ_i. Indices throughout the paper are modulo N, so Q_N+1 actually refers to Q₁. The interarrival times and the service times (for different queues and for different visits) are assumed to be mutually independent.

In addition, we will consider the impact of customer rerouting policy. Upon completion of service at Q_i, i = 1, ⋯, N, a customer is either routed to Q_j, j = 1, ⋯, N with probability p_i,j or leaves the system with probability p_i,0, where

∑i=1Npi,0>0and∑j=0Npi,j=1.

We assume that all the switches of customers or servers between queues are instantaneous and when the system becomes empty, the server travels a full cycle and subsequently stops right before Q₁ until a new arrival occurs and then cycles along the queues to serve that customer.

Denote the total arrival rate at Q_i by γ_i, which is the unique solution of the following set of linear equation[2]:

γi=λi+∑j=1Nγjpj,ii=1,⋯,N.

For all i, we assume γ_i/μ_i < 1 and for overload we assume ∑i=1N ρ_i > 1.

In this paper, we will focus on branching-type service disciplines in a general parameter setting which satisfy the following property (see[15]).

Property 1

(Branching property ([15], Property 1)) If the server arrives at Q_i to find k_i customers there, then during the course of the server’s visit, each of these k_i customers will effectively be replaced in an i.i.d. manner by a random population having probability generating function (p.g.f) h_i(z₁, z₂, …, z_N), which can be any N-dimensional p.g.f..

According to this property, many classical service disciplines belong to the branching-type service discipline including exhaustive service discipline(per visit the server continues to serve all customers at a queue until it empties), gated service discipline (per visit the server serves only those customers at a queue which are found there upon his visit), binomial-gated[16] and binomial-exhaustive policies[15].

Furthermore, the multigated (X_i-gated) service discipline discussed in[14] is just a special case of Property 1. In Section 4, we can see that Theorem 4.2 in Section 4 is also an extention of[14] (for the special case of multigated (X_i-gated) service discipline and without customer rerouting policy (i.e. p_i,j = 0, i, j = 1, ⋯, N)). Moreover, we can know how the re-routing policy effect the queue length process from Figures 1 and 2 in Section 4.

$Figure 1 Fluid limit of Qi.$

Figure 1

Fluid limit of Q_i.

$Figure 2 Fluid limit of total population X̄1 + ⋯ + X̄N.$

Figure 2

Fluid limit of total population X̄₁ + ⋯ + X̄_N.

Define t⁽ⁿ⁾ as the time point that the server reaches right before Q₁ for the nth time and ti(n) as the time point that the server reaches Q_i for the nth time (n ∈ ℕ = {0, 1, 2, ⋯}, i = 1, 2, …, N). If the system is empty at t⁽ⁿ⁾, then the interval [t⁽ⁿ⁾, t1(n) ) is the period of waiting until the first arrival, otherwise t⁽ⁿ⁾ = t1(n) .

Let X(t⁽ⁿ⁾) = (X₁(t⁽ⁿ⁾), ⋯, X_N(t⁽ⁿ⁾)), n ∈ ℕ be the queue length process at time t⁽ⁿ⁾, where X_i(t⁽ⁿ⁾) is the number of customers at Q_i at time t⁽ⁿ⁾. By Resing[15], branching property implies that the queue length sequence {X(t⁽ⁿ⁾)}_n∈ℕ forms a multi-type branching process with immigration in state 0.

In this paper, we assume that all vectors are N-dimensional row vectors, all vectors are typeset in bold italic. The vector with all coordinates equal to 0 is denoted by 0 and the vector with coordinate i equals to 1 and the other coordinates equal to 0 by e_i.

3 The MTBP-structure of polling system

In the above section, we have known the queue length process {X(t⁽ⁿ⁾)}_n∈ℕ forms a multi-type branching process with immigration in state 0. Let M = {mi,j}i,j=1N be the mean offspring matrix. Also let the vectors u = (u₁, ⋯, u_N) and v = (v₁, ⋯, v_N) be the right and left eigenvectors corresponding to the maximal real-valued, positive eigenvalue θ of M, commonly referred to as the maximum eigenvalue ([17]), normalized such that vu^⊤ = 1. In this section, we will give the mean offspring matrix associated with the branching process. Moreover, Theorem 5.6.1(supercritical limit theorem) in[17] leads to our main results.

To start with, we give some notations associated with the branching-type polling system.

Define Ľ_i = (Ľ_i,1, ⋯, Ľ_i,N) as the visit offspring of a customer at Q_i, which equals in distribution to X( ti+1(n) ) given that X( ti(n) ) = e_i (its distribution does not depend on n) with m̌_i = (m̌_i,1, ⋯, m̌_i,N) = 𝔼Ľ_i.
Define L_i := (L_i,1, ⋯, L_i,N) as the session offspring of a customer at Q_i, which equals in distribution to X(t⁽ⁿ⁺¹⁾) given that X(t⁽ⁿ⁾) = e_i (its distribution does not depend on n) with m_i = (m_i,1, ⋯, m_i,N) = 𝔼L_i. In order to ensure the non-degenerate, we assume that 𝔼L_i,j log L_i,j < ∞ for all 1 ≤ i, j ≤ N.

To proceed, we need further to define the exhaustiveness f_i of the service descipline at Q_i by (see[18], (55), (56))

fi=1−∂∂zihi(z1,z2,⋯,zN)|z=1=1−ELˇi,i.

It has an appealing interpretation: during the course of the server’s visit at Q_i, each customer present at the start of the visit to Q_i will be replaced by a number of customers with mean 1 – f_i at the end of the visit to Q_i.

Remark 3.1

In particular, the exhaustiveness of the multigated (X_i-gated) service discipline at Q_i in [14] (where X_i = 1 and ∞ of gating index corresponding to conventional gated and exhaustive, respectively) equals fi=1−E(λiμi+pi,i)Xi.

Let BiE be the total service time of a customer in Q_i before he is either routed to Q_j, j ≠ i or leaves the system. Then biE:=EBiE=1/(μi(1−pi,i)).

Define T_i as the busy period in Q_i. This busy period consists of the service of its first customer at Q_i, the services of the customers arriving at Q_i during the service of the first customer (i.e., the children), the services of the customers arriving at Q_i during the service of the children (i.e., the grandchildren), and so forth. Then, we have

ti:=ETi=fibiE1−λibiE. (3.1)

By Lemma 1 in [9], the mean offspring matrix M is given in the following proposition.

Proposition 3.1

For the cyclic branching-type polling system, the mean matrix M is given by

M=M1…MN,

where Mk=mi,j(k) and

mi,j(k)=δ{i=j}, i≠k,1−fi, i=k=j,ti(μipi,j+λj), i=k≠j, (3.2)

where δ_F denotes the indicator function on F.

Actually, M_k is the mean session offspring during the visit time on Q_k. Hence, for all i,

mi,j(i)=mˇi,j=1−fi, i=j,ti(μipi,j+λj), i≠j.

Proof

In the Multi-type branching process, by the definition of M = {mi,j}i,j=1N , we have that mi,j=∂f(i)(z)∂zj|z=1, i, j = 1, ⋯, N, where f⁽ⁱ⁾(z) is the ith generating function of the distribution of the number of offspring of various types to be produced by a type i particle, and

f(i)(z)=∑j1,⋯,jN≥0p(i)(j1,⋯,jN)z1j1⋯znjN,|zk|≤1,i,k=1,⋯,N,

where z = (z₁, ⋯, z_N) and p⁽ⁱ⁾(j₁, ⋯, j_N) = the probability that a type i parent produces j₁ particles of type 1, j₂ of type 2, ⋯, j_N of type N.

Let X(t⁽ⁿ⁾) = (X₁(t⁽ⁿ⁾), ⋯, X_N(t⁽ⁿ⁾)), n ∈ ℕ be the queue length process at time t⁽ⁿ⁾, where X_i(t⁽ⁿ⁾) is the number of customers at Q_i at time t⁽ⁿ⁾. By Resing [15], branching property implies that the queue length sequence {X(t⁽ⁿ⁾)}_n∈ℕ forms a multi-type branching process with immigration in state 0. Therefore, we have

f(i)(z)=hi(z1,z2,⋯,zi,f(i+1)(z),⋯,f(N)(z)),|zk|≤1,i=1,⋯,N,

where

hi(z)=ψi(∑j≠iλj(1−zj),zi,pi,01−pi,i+∑j≠ipi,j1−pi,izj),i=1,⋯,N,

where ψi(u,z1,z2)=E(e−uTi,z1Li,Z2Mi), L_i is the so-called busy period residue, i.e., the number of type-i children of the original customer that generates this busy period and M_i is the number of customers leave Q_i in the busy period T_i. Then, by the definition of M, we obtain,

mi,j=hi,jI{j≤i}+∑k=i+1Nhi,kmk,j,i,j=1,⋯,N, (3.3)

hi,j=∂hi(z)∂zj|z=1=1−fi,i=j,tk(λj+pi,jμi),i≠j, (3.4)

Therefore, by (3.3) and (3.4), we obtain (3.2).□

It follows that the auxiliary process {X(t⁽ⁿ⁾)}_n∈ℕ has the following asymptotics (see [17], Theorem 5.6.1), which will be important for proving the main results in the next section.

Proposition 3.2

If the first arriving customer arrives at Q_i after t = 0, then

X(t(n))θn→ζivalmost surely (a.s.)asn→∞,

where the distribution of the random variable ζ_i has a jump of magnitude q_i = ℙ(X(t⁽ⁿ⁾) = 0 for some n|X(t⁽⁰⁾) = e_i) < 1 at 0 and a continuous density function on (0, ∞) and 𝔼ζ_i = u_i.

4 Main results

To give the main results, three more notations are needed.

Let B̄_i be the total service time of a customer arriving at Q_i from outside, c̄_i = 𝔼B̄_i, we have c̄_i = 1/μ_i + ∑j=1N p_i,j c̄_j. It is also easy to deduce that ρ = ∑i=1N λ_i c̄_i.
For n ∈ ℕ, let ηn:=min{k:t(k)≥θn}, if n≥0;0,if n<0.
For n ∈ ℕ, define the scaled queue length process X̄⁽ⁿ⁾(t) := X(θnt)θn , t ∈ [0, ∞).

Theorem 4.1

There exist constants b̄_i ∈ (0, ∞) and ā_i = (ā_i,1, ⋯, ā_i,N) ∈ [0, ∞)^N, i = 1, ⋯, N + 1, and a random variable ξ with values in [1, θ) such that, for all k ∈ ℕ and i = 1, 2, ⋯, N,

ti(ηn+k)θn→θkb¯iξandX(ti(ηn+k))θn→ξθka¯ia.s.asn→∞.

The b̄_i and ā_i are given by

b¯1=1,b¯i+1=b¯i+[viα+λi(b¯i−b¯1)+∑j=1i−1pj,iμj(b¯j+1−b¯j)]ti

and for j = 1, ⋯, N,

a¯1=vα,a¯i+1,j=a¯i,j+[λj+μipi,j](b¯i+1−b¯i)j≠i,a¯i,i+[λi−μi(1−pi,i)](b¯i+1−b¯i),j=i.

where

α=∑i=1Nvic¯iρ−1.

The distribution of ξ see [14].

Proof

See Section 5.□

Remark 4.1

Specially, for a polling model with multigated (X_i-gated) service discipline and without customer rerouting policy (i.e. p_i,j = 0, i, j = 1, ⋯, N), the asymptotics in Theorem 4.1 remains true while the b̄_i and ā_i turn to be

b¯1=1,b¯i+1=b¯i+[viα+λi(b¯i−b¯1)]ti

and for j = 1, ⋯, N,

a¯1=vα,a¯i+1,j=a¯i,j+λj(b¯i+1−b¯i)j≠i,a¯i,i+[λi−μi](b¯i+1−b¯i)j=i.

where ti=ETi=1−E(λiμi)Xiμi−λi and α=∑i=1Nvi/μiρ−1, which is in accord with Theorem 1 in [14].

Theorem 4.2

There exists a deterministic function X̄(⋅) = (X̄₁, ⋯, X̄_N)(⋅) ∈ [0, ∞)^N such that,

X¯(n)(⋅)→ξX¯(⋅ξ)a.s.asn→∞, (4.1)

uniformly on compact sets. For all i = 1, ⋯, N, the function X̄(⋅) is continuous and piecewise linear as depicted in Figure 1 and specified by

X¯(t)=0,if t=0;θka¯i+(t−θkb¯i)λ+(t−θkb¯i)μipi , if t∈[θkb¯i,θkb¯i+1), (4.2)

where λ = [λ₁, ⋯, λ_N] and p_i = [p_i,1, ⋯, p_i,i–1, p_i,i – 1, p_i,i+1, ⋯, p_i,N].

Proof

See Section 5.□

Remark 4.2

Theorem 4.2 is also an extention of [14] (for the special case of multigated (X_i-gated) service discipline and without customer rerouting policy (i.e. p_i,j = 0, i, j = 1, ⋯, N)).

Corollary 4.1

Under passage to the fluid dynamics, the fluid total population (X̄₁ + X̄₂ + ⋯ + X̄_N)(⋅) grows at the rate

(λ1+⋯+λN)−pi,0μi

when t ∈ [θ^k b̄_i, θ^k b̄_i+1) for all k ∈ ℕ and i = 1, ⋯, N. (see Figure 2)

Remark 4.3

In [9], the fluid asymptotics of the queue length process in the heavy traffic for the same polling model have been discussed. In the heavy traffic, the total scaled workload is effectively constant while the individual queue workload is emptied and refilled at a rate during the course of a cycle. In contrast to the heavy traffic asymptotics, the total overloaded asymptotic workload is always increasing as shown in Corrollary 4.1 during the course of a cycle. In addition, the overloaded fluid limit always contains a random variable ξ. However, the individual queue workload is emptied and refilled at the same rate as in the heavy traffic like a fluid model. Hence, our result is a further progress of[9].

Remark 4.4

For different branching-type service discipline, our main results have shown that the overloaded fluid asymptotics just depend on the exhaustiveness of each service discipline, which also applies to the heavy traffic asymptotics in [9] and the asymptotics with the large-switchover times in [18]. This can be easily interpreted by the fluid approximation. It also proves that the branching-type polling system deserves much more attention.

Remark 4.5

The rerouting policy only affects the flow rate both in the heavy traffic asymptotics (see [9]) and in the overloaded asymptotics. In Theorem 4.2, we can see that the fluid limit depends on the re-routing probability p_i,j. Upon completion of service at Q_i, i = 1, ⋯, N, a customer is either routed to Q_i with probability p_i,i, which leads the decreasing rate of the length of Q_i to be λ_i – μ_i(1 – p_ii), or routed to Q_j, j ≠ i with probability p_i,j, which leads the increasing rate of the length of Q_j to be λ_j + μ_i p_i,j at time [θ^kb̄_i, θ^kb̄_i+1).

According to Theorem 4.2 and Corrollary 4.1, the fluid limit processes both demonstrate an oscillation waveform with increasing amplitude and cycle time over time. To be more specific, the amplitude and cycle time both increase by θ – 1 times each cycle. Hence, the average growth rate of the scaled total population, denoted by β, equals to the average growth rate in each cycle, as shown in Figure 2. Therefore, we have

∑i=1N∑j=1Na¯i,j+∑j=1Na¯i+1,j2(b¯i+1−b¯i)=∫b¯1θb¯1βtdt,

which yields

β=1θ2−1∑i=1N∑j=1Na¯i,j+∑j=1Na¯i+1,j(b¯i+1−b¯i). (4.3)

By the definition of the scaled queue length process, the fluid limit could approximate the original queue length process in steady state. Furthermore, the average growth rate in (4.3) allows us to study the optimization problem of how to choose the gating indexes of each queue to minimize the total queue length. Since each of the queues adheres to a branching-type service discipline, we study how to choose the exhaustiveness f_i with the same objective in mind. We would provide an optimization example by utilizing the genetic algorithm in Section 6.

5 Proof of Theorems 4.1 and 4.2

5.1 Proof of Theorem 4.1

Proof

By the tool of Lemma 8 in [14], if we can prove

ti(n)θn→biξandX(ti(n))θn→ξaia.s.as n→∞, (5.1)

where b_i = αb̄_i, a_i = αā_i, then Theorem 4.1 is concluded. Hence, we focus on the proof of (5.1).

Limit of t1(n) /θⁿ. Define index ν by

ν=maxn∈N+,such thatX(t(n))=0andX(t(m))≠0for allm>n.

By the total workload process, we have, for n > ν,

t1(n)=t(n)=W+t(n)A1(n)−θnA2(n), (5.2)

where B¯i(k) are i.i.d. copies of B̄_i. By definition of ν, we know that it is a.s. finite, so that we obtain W = ∑l=0ν(t1(l)−t(l))<∞ a.s..

A1(n)=∑i=1N∑k=1Ei(t(n))B¯i(k)Ei(t(n))Ei(t(n))t(n),A2(n)=∑i=1N∑k=1Xi(t(n))B¯i(k)Xi(t(n))Xi(t(n))θn,

Since, we know that B¯i(k) are i.i.d. copies of B̄_i, and 𝔼B̄_i = c̄_i, then, by the SLLN and Proposition 3.2, we obtain, as n → ∞,

A1(n)→∑i=1NλiEB¯i(k)=∑i=1Nλic¯i=ρ,A2(n)→∑i=1NviξEB¯i(k)=∑i=1Nvic¯iξa.s..

Therefore, by (5.2), we have, as n → ∞,

t(n)θn→b1ξandt1(n)θn→b1ξ,

where b1=∑i=1Nvic¯iρ−1.
Limit of ti(n) /θⁿ. In (1), by utilizing the index ν and equation t1(n) = t⁽ⁿ⁾, we proved lim_n→∞ t1(n) /θⁿ = b₁ξ. By the symmetry, there also exist positive numbers b_i such that

limn→∞ti(n)θn=biξ,i=1,⋯,N.

It remains to prove the iteration of b_i, which refers to (4) below.
Limit of X_j( ti(n) )/θⁿ. Define the renewal processes

Yi(t)=maxn∈N+,such that∑i=1nBi(k)≤t,

where Bi(k) are i.i.d. copies of B_i. Also let I_i(t) be the whole time that the server has spent at Q_i before time t, i.e.,

Ii(t)=∫0tI( queueiis in service in time s)dst∈(0,∞).

Let A_i be the position of a customer after completion of service at Q_i, for i = 1, ⋯, N, i.e.,

Ai=j,after receiving service at Qi, a customer is routed to Qj;0,after receiving service at Qi, a customer leaves the system.

Then ℙ(A_i = j) = p_i,j, j = 0, 1, ⋯, N. Hence, we have

Xj(ti+1(n))=Xj(ti(n))+Ej(ti+1(n))−Ej(ti(n))+∑k=1Yi(Ii(tn+1))−Yi(Ii(tn))δ{Ai(k)=j}, j≠i;Xi(ti(n))+Ei(ti+1n)−Ei(ti(n))−∑k=1Yi(Ii(tn+1))−Yi(Ii(tn))δ{Ai(k)≠i}, j=i, (5.3)

where Ai(k) are i.i.d. copies of A_i. Since Eδ{Ai(k)=j}=P(Ai(k)=j)=pi,j is finite, so that, by SLLN, we have

Xj(ti+1(n))θn→ai+1,jξ,

where

ai+1,j=ai,j+λj(bi+1−bi)+pi,jμj(bi+1−bi), j≠i;ai,i+λi(bi+1−bi)−μi(1−pi,i)(bi+1−bi), j=i.
The iteration of b_i. Recall that

ti+1(n)=ti(n)+∑k=1Xi(ti(n))Ti(k), (5.4)

Xi(ti(n))=Xi(t1(n))+Ei(ti(n))−Ei(t1(n))+∑j=1i−1∑k=1Yj(Ij(t(n)))−Yj(Ij(t(n−1)))δ{Aj(k)=i}, (5.5)

where Ti(k) are i.i.d. copies of T_i. As n → ∞, if X_i( t1(n) ) = c, where c is a positive constant, then ∑k=1Xi(ti(n))Ti(k)/θn→0, so that b_i+1 = b_i, this case is obviously to us, so we study the case that X_i( t1(n) ) → ∞. θⁿ corresponds to T_n in the Proposition 2 of [14], Ti(k) corresponds to Yn(k) in Proposition 2 of [14], a_i,i corresponds to τ in the Proposition 2 of [14], t_i corresponds to 𝔼Y in Proposition 2 of [14]. Therefore, by Proposition 2 in [14] and (5.4), we get

bi+1−bi=ai,iti. (5.6)

By (5.5) and the SLLN, we obtain

ai,i=vi+λi(bi−b1)+∑j=1i−1μj(bj+1−bj)pj,i. (5.7)

Then the iteration of b_i can be proved by substituting (5.7) into (5.6).
The equivalence of b_N+1 and θb₁. By tN+1(n)=t1(n+1), we obtain b_N+1 ξ = limn→∞tN+1(n)/θn = limn→∞θt1(n+1)/θn+1 = θb₁ξ, i.e. b_N+1 = θb₁. This can be proved as follows. By the definition of M_i in Lemma 3.1, it is easy to give

(Mi−I)c¯T=(ρ−1)tiei,

where I is the identity matrix, ⋅^⊤ denotes the operation of transposition and c̄ = (c̄₁, …, c̄_N). Substituting the above equation into (5.6) yields

bi+1−bi=ai,iti=aieiti=1ρ−1ai(Mi−I)c¯T=1ρ−1ai+1−aic¯T.

where a_i = (a_i,1, ⋯, a_i,N), which gives immediately

bN+1=∑i=1N(bi+1−bi)+b1=1ρ−1aN+1−a1c¯T+b1=1ρ−1vM−vc¯T+b1=(θ−1)vc¯Tρ−1+b1=(θ−1)b1+b1=θb1.

□

5.2 Proof of Theorem 4.2

Proof

For each i, by (4.2), we know that the function X̄_i(⋅) might have discontinuities only at t = 0 and t = θ^k b̄_i for each k ∈ ℕ. Since the function X̄_i(⋅) is càdlàg, the continuity of X̄(⋅) is evident in combination with the definition of a_i. Additionally, the uniform convergence on compact sets can be proved in the same way as in the proof of Theorem 2 in [14]. Hence, it suffices to prove the point-wise convergence (4.2) for each i = 1, 2, …, N.

For t = 0, the convergence of (4.2) holds since the system starts empty. For each i = 1, …, N, if t ∈ [θ^k b̄_i, θ^k b̄_i+1), it remains to prove

X¯j(t)=θka¯i,i+[λi−μi(1−pi,i)](t−θkb¯i), j=i;θka¯i,j+[λj+μipi,j](t−θkb¯i), j≠i.

For all n big enough, ti(ηn+k)θn<t<ti+1(ηn+k)θn implying that Q_i is in service during [ ti(ηn+k) , θⁿt). Hence,

Xj(θnt)=Xi(ti(ηn+k))+Ei(θnt)−Ei(ti(ηn+k))−∑k=1Yi(Ii(θnt))−Yi(Ii(t(ηn+k)))δ{Ai(k)≠i}, j=i;Xj(ti(ηn+k))+Ej(θnt)−Ej(ti(ηn+k))+∑l=1Yi(Ii(θnt))−Yi(Ii(t(ηn+k)))δ{Ai(l)=j}, j≠i.

Combining the above equation with Theorem 4.1, we have

X¯j(n)(t)=Xj(θnt)θn→ξθka¯i,i+[λi−μi(1−pi,i)](t−θkb¯iξ), j=i;ξθka¯i,j+(λj+μipi,j)(t−θkb¯iξ), j≠i,

where the right hand-side actually equals ξX¯j(tξ). Therefore, we proved (4.2).□

6 Numerical validation and optimization of gating indexes

6.1 Numerical validation

This subsection is devoted to test the validity of the fluid limits of the scaled queue length process. For simplicity, we consider a 3-queue polling system described in Table 1 with exponentially distributed service time. For this model, it is readily to obtain ρ₁ = 0.4749, ρ₂ = 0.5194, ρ₃ = 0.8625 and ρ = 1.8568, which belongs to the overloaded traffic case studied in this paper.

Table 1

Parameter values in 3-queue polling network.

Parameter	Considered parameter values
Arriving rate	λ₁ = λ₂ = λ₃ = 1
Service rate	μ₁ = 8, μ₂ = 5, μ₃ = 2
Transition probability	P=(pi,j)3×3=0.10.250.20.20.10.20.20.10.25

We utilize the SimEvents toolbox of Matlab to undertake the simulations of the polling networks. The exhaustive and gated service policies are taken for example and some vital variables are given in Table 2. In order to illustrate the convergence of the scaled queue length process, we take n = 1, 5, 8, 10 in polling network with exhaustive service policies and n = 1, 10, 18, 20 in the gated counterpart. The corresponding scaled queue length process of Q₂ and the scaled total queue length process are depicted in Figure 3 and Figure 4, respectively. Apparently, the scaled queue length sample paths get closer and closer as n increases.

Table 2

Essential variable values in 3-queue polling network.

Variable	Values

	Exhaustive	Gate
Gating index	∞	1
Exhaustiveness	1	1−(λiμi+pi,i)
Maximum eigenvalue	θ = 3.7497	θ = 1.6394
Left eigenvector	v = [0.9731, 0.683, 0]	v = [0.7454, 0.5301, 0.4774]

Moreover, as shown in Figure 3 and Figure 4, the fluid limit processes both demonstrate an oscillation waveform with increasing amplitude and cycle time forward and oscillate at an infinite rate when approaching zero. According to Theorem 4.2, the amplitude and cycle time increase by θ – 1 times within each cycle, which has been easily illustrated by the sample paths.

$Figure 3 The scaled queue length process of Q2 for different n (left:t ∈ [0, 100], right:t ∈ [0, 8]) with exhaustive service policy.$

Figure 3

The scaled queue length process of Q₂ for different n (left:t ∈ [0, 100], right:t ∈ [0, 8]) with exhaustive service policy.

$Figure 4 Left: the scaled total queue length process for different n with exhaustive service policy. Right: the scaled queue length process of Q2 for different n with gated service policy.$

Figure 4

Left: the scaled total queue length process for different n with exhaustive service policy. Right: the scaled queue length process of Q₂ for different n with gated service policy.

6.2 Optimization of gating indexes

Subsequently, we consider the optimization of the gating indexes by numerical method. We assume that the gating indexes are integers. Virtually, the fluid limits only depend on the exhaustiveness of the service discipline at each queue (moment of gating index), which allows us to minimize the total queue length through the accommodation of the integer gating indexes.

By (4.3), the average growth rate of the total queue length process β with exhaustive and gated service policies equals 1.5025 and 1.2416 respectively (see Figure 5). This can be intuitively interpreted from the growth rate during each visiting period on different queues. The visiting period at Q₃ with the maximal growth rate (minimal service rate) takes 4 times as much time as others in exhaustive service policy. Instead, it takes less than 2 times as much time as others in gated service policy. Therefore, to minimize the average growth rate, we need to increase the visiting time at Q₁ and Q₂ and decrease the visiting time at Q₃.

To optimize the gating indexes turns to be an integer programming with three variables. The GA toolbox of Matlab is undertaken here to search for the optimal gating indexes. For our model, it just takes 51 iterations to find the optimal solution: Q₁ and Q₂ both take exhaustive service policy while Q₃ takes gated service policy. The minimal average growth rate equals to 1.19262 and the corresponding exhaustiveness is f₁ = f₂ = 1, f₃ = 0.25. Figure 5 depicts the process of the optimal average growth rate in each generation. Apparently, the convergence process turns to be very effective. Hence, the average growth rate of the fluid limit provides a simple and transparent method to optimize the gating index.

$Figure 5 Left: the fluid limit of the scaled total queue length process (exhaustive and gated). Right: the convergence process of the optimal average growth rate by GA solver.$

Figure 5

Left: the fluid limit of the scaled total queue length process (exhaustive and gated). Right: the convergence process of the optimal average growth rate by GA solver.

7 Conclusions and further Research

Inspired by [14], we present the fluid limit of an overloaded polling system with general branching-type service discipline and customer re-routing policies. These results provide new fundamental insight in the impact of exhaustiveness. As a by-product, we propose an optimization problem of gating indexes to minimize the total queue length.

This work gives rise to a variety of directions for further research. A logical follow-up step would be to study the case with non-zero switch-over time. In addition, the asysmptotic behaviors of discrete-time polling systems are also direct extensions to this study. Furthermore, the fluid limit allows us to propose control strategies of the growth depression, which requires substantially more effort.

Acknowledgement

This research is partially supported by the National Natural Science Foundation of China (11901186, 11671404) and by the Fundamental Research Funds for the Central Universities (WUT: 2017 IVA 069, WUT: 2018IB018). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.

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Received: 2019-01-02

Accepted: 2019-09-06

Published Online: 2019-12-10

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0116

Keywords for this article

polling networks; overloaded; general branching-type service policies; multi-type branching process; exhaustiveness

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