An asymptotic property of branching-type overloaded polling networks

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.


Introduction
In this paper, we consider a cyclic N-queue (Q , · · · , Q N , N ≥ ) polling system with general branchingtype 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, . 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 Indices throughout the paper are modulo N, so Q N+ actually refers to Q . The interarrival times and the service times (for di erent queues and for di erent 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 = , · · · , N, a customer is either routed to Q j , j = , · · · , N with probability p i,j or leaves the system with probability p i, , where N i= p i, > and N j= p i,j = .
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 Q 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]: For all i, we assume γ i /µ i < and for overload we assume N i= ρ i > . In this paper, we will focus on branching-type service disciplines in a general parameter setting which satisfy the following property (see [15]). ([15], Property 1)) If the server arrives at Q i to nd k i customers there, then during the course of the server's visit, each of these k i customers will e ectively be replaced in an i.i.d. manner by a random population having probability generating function (p.g.f) h i (z , z , . . . , z N ), which can be any Ndimensional p.g.f..

Property 1. (Branching property
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 = , i, j = , · · · , N)). Moreover, we can know how the re-routing policy e ect the queue length process from Figures 1 and 2 in Section 4.
De ne t (n) as the time point that the server reaches right before Q for the nth time and t (n) i as the time point that the server reaches Q i for the nth time (n ∈ N = { , , , · · · }, i = , , . . . , N). If the system is empty at t (n) , then the interval [t (n) , t (n) ) is the period of waiting until the rst arrival, otherwise t (n) = t (n) .
Let X(t (n) ) = (X (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 .
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 and the vector with coordinate i equals to and the other coordinates equal to 0 by e i .

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 . Let M = {m i,j } N i,j= be the mean o spring matrix. Also let the vectors u = (u , · · · , u N ) and v = (v , · · · , v N ) be the right and left eigenvectors corresponding to the maximal realvalued, positive eigenvalue θ of M, commonly referred to as the maximum eigenvalue ( [17]), normalized such that vu = . In this section, we will give the mean o spring 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.
-De neĽ i = (Ľ i, , · · · ,Ľ i,N ) as the visit o spring of a customer at Q i , which equals in distribution to X(t (n) i+ ) given that X(t (n) i ) = e i (its distribution does not depend on n) withm i = (m i, , · · · ,m i,N ) = EĽ i . -De ne L i := (L i, , · · · , L i,N ) as the session o spring of a customer at Q i , which equals in distribution to X(t (n+ ) ) given that X(t (n) ) = e i (its distribution does not depend on n) with m i = (m i, , · · · , m i,N ) = EL i . In order to ensure the non-degenerate, we assume that EL i,j log L i,j < ∞ for all ≤ i, j ≤ N.
To proceed, we need further to de ne the exhaustiveness f i of the service descipline at Q i by (see [18], (55), (56)) 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 − 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 = and ∞ of gating index corresponding to conventional gated and exhaustive, respectively) equals Let B E i 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.
. De ne T i as the busy period in Q i . This busy period consists of the service of its rst customer at Q i , the services of the customers arriving at Q i during the service of the rst 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 By Lemma 1 in [9], the mean o spring 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 where δ F denotes the indicator function on F.
Actually, M k is the mean session o spring during the visit time on Q k . Hence, for all i, Proof. In the Multi-type branching process, by the de nition of is the ith generating function of the distribution of the number of o spring of various types to be produced by a type i particle, and where z = (z , · · · , z N ) and p (i) (j , · · · , j N ) = the probability that a type i parent produces j particles of type 1, j of type 2, · · · , j N of type N. Let X(t (n) ) = (X (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 . Therefore, we have , 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 de nition of M, we obtain,

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 rst arriving customer arrives at Q i after t = , then where the distribution of the random variable ζ i has a jump of magnitude q i = P(X(t (n) ) = for some n|X(t ( ) ) = e i ) < at 0 and a continuous density function on ( , ∞) and Eζ i = u i .

Main results
To give the main results, three more notations are needed.
-LetB i be the total service time of a customer arriving at Q i from outside, -For n ∈ N, de ne the scaled queue length processX (n) (t) := X(θ n t) θ n , t ∈ [ , ∞).

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 = , i, j = , · · · , N), the asymptotics in Theorem 4.1 remains true while theb i andā i turn to beb , which is in accord with Theorem 1 in [14].

Corollary 4.1.
Under passage to the uid dynamics, the uid total population (X +X + · · · +X N )(·) grows at the rate (λ + · · · + λ N ) − p i, µ i when t ∈ [θ kb i , θ kb i+ ) for all k ∈ N and i = , · · · , N. (see Figure 2) Remark 4.3. In [9], the uid asymptotics of the queue length process in the heavy tra c for the same polling model have been discussed. In the heavy tra c, the total scaled workload is e ectively constant while the individual queue workload is emptied and re lled at a rate during the course of a cycle. In contrast to the heavy tra c 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 uid limit always contains a random variable ξ . However, the individual queue workload is emptied and re lled at the same rate as in the heavy tra c like a uid model. Hence, our result is a further progress of [9].

Remark 4.4.
For di erent branching-type service discipline, our main results have shown that the overloaded uid asymptotics just depend on the exhaustiveness of each service discipline, which also applies to the heavy tra c asymptotics in [9] and the asymptotics with the large-switchover times in [18]. This can be easily interpreted by the uid approximation. It also proves that the branching-type polling system deserves much more attention.
Remark 4.5. The rerouting policy only a ects the ow rate both in the heavy tra c asymptotics (see [9]) and in the overloaded asymptotics. In Theorem 4.2, we can see that the uid limit depends on the re-routing probability p i,j . Upon completion of service at Q i , i = , · · · , 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 ( − 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 kb i , θ kb i+ ).
According to Theorem 4.2 and Corrollary 4.1, the uid limit processes both demonstrate an oscillation waveform with increasing amplitude and cycle time over time. To be more speci c, the amplitude and cycle time both increase by θ − 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 By the de nition of the scaled queue length process, the uid 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.

Proof of Theorems 4.1 and 4.2 . Proof of Theorem 4.1
Proof. By the tool of Lemma 8 in [14], if we can prove t (n) i θ n → b i ξ and

X(t (n)
i ) θ n → ξ a i a.s. as n → ∞, (5.1) where b i = αb i , a i = αā i , then Theorem 4.1 is concluded. Hence, we focus on the proof of (5.1).
By the total workload process, we have, for n > ν, whereB (k) i are i.i.d. copies ofB i . By de nition of ν, we know that it is a.s. nite, so that we obtain W = ν l= (t (l) − t (l) ) < ∞ a.s..
Since, we know thatB (k) i are i.i.d. copies ofB i , and EB i =c i , then, by the SLLN and Proposition 3.2, we obtain, as n → ∞, Therefore, by (5.2), we have, as n → ∞, (2) Limit of t (n) i /θ n . In (1), by utilizing the index ν and equation t (n) = t (n) , we proved limn→∞ t (n) /θ n = b ξ . By the symmetry, there also exist positive numbers b i such that It remains to prove the iteration of b i , which refers to (4) below.
(3)Limit of X j (t (n) i )/θ n . De ne the renewal processes where B (k) i 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., Let A i be the position of a customer after completion of service at Q i , for i = , · · · , N, i.e., A i = j, after receiving service at Q i , a customer is routed to Q j ; , after receiving service at Q i , a customer leaves the system.
Then P(A i = j) = p i,j , j = , , · · · , N. Hence, we have where T (k) i are i.i.d. copies of T i . As n → ∞, if X i (t (n) ) = c, where c is a positive constant, then i /θ n → , so that b i+ = b i , this case is obviously to us, so we study the case that X i (t (n) ) → ∞. θ n corresponds to Tn in the Proposition 2 of [14], T (k) i corresponds to Y (k) n 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 By (5.5) and the SLLN, we obtain Then the iteration of b i can be proved by substituting (5.7) into (5.6).
where a i = (a i, , · · · , a i,N ), which gives immediately

. Proof of Theorem 4.2
Proof. For each i, by (4.2), we know that the functionX i (·) might have discontinuities only at t = and t = θ kb i for each k ∈ N. Since the functionX i (·) is càdlàg, the continuity ofX(·) is evident in combination with the de nition 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 su ces to prove the point-wise convergence (4.2) for each i = , , . . . , N.
For t = , the convergence of (4.2) holds since the system starts empty.
For all n big enough, Combining the above equation with Theorem 4.1, we havē where the right hand-side actually equals ξX j ( t ξ ). Therefore, we proved (4.2).

Numerical validation and optimization of gating indexes . Numerical validation
This subsection is devoted to test the validity of the uid 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 ρ = .
, which belongs to the overloaded tra c case studied in this paper.
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 = , , , in polling network with exhaustive service policies and n = , , , in the gated counterpart. The corresponding scaled queue length process of Q 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.

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

. 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 uid 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 di erent queues. The visiting period at Q 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 Q and Q and decrease the visiting time at Q . 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 nd the optimal solution: Q and Q both take exhaustive service policy while Q takes gated service policy. The minimal average growth rate equals to 1.19262 and the corresponding exhaustiveness is f = f = , f = . . Figure 5 depicts the process of the optimal average growth rate in each generation. Apparently, the convergence process turns to be very e ective. Hence, the average growth rate of the uid limit provides a simple and transparent method to optimize the gating index.

Conclusions and further Research
Inspired by [14], we present the uid 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 uid limit allows us to propose control strategies of the growth depression, which requires substantially more e ort.