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The Necessary Stability Conditions of a Tandem System
With Feedback
xEvsey Morozov, 2Gurami Tsitsiashvili •
1 Institute for Applied Mathematical Researches, Karelian Center of Russian Academy Sciences, Petrozavodsk University, Petrozavodsk, Russia, 2 Institute for Applied Mathematics, Far Eastern Branch of Russian Academy Sciences, Far Eastern Federal University, Vladivostok, Russia, 1 emorozov@karelia.ru, 2guram@iam.dvo.ru 1emorozov@karelia.ru, 2 guram@iam.dvo.ru
Abstract
In this paper, we consider Markovian model of a two-station tandem network with the following feedback admission control policy: the first station rejects new arrivals when the queue size in the second station exceeds a certain threshold N. We provide necessary stability conditions of this model. Each station operates as a multiserver queuieng system, and thus work in part generalizes the results from the paper [1] in which single-server stations have been considered. The analysis is based on the Burke's theorem and stochastic monotonicity of the Birth-Death process describing the number of customers in the second station.
Keywords: queuing system, ergodicity, input flow, feedback
I Introduction
We consider the following two-station queueing system with a feedback admission control policy. The input flow in this system is Poisson with the parameter A. Station i has Ni servers, and the service time of each server in station i is exponentially distributed with parameter i = 1,2.
We consider a feedback admission control when the 1st station closes the admission gate provided the queue size (number of customers) in the 2nd station exceeds a fixed threshold N > 1. When the queue length of the 2nd station falls below the threshold, admission gate opens again. With this non-idling control policy, the system losses arrivals during the period when the gate is closed. We assume the FIFO service discipline at both stations. (In general, under the same conditions, stability of the system holds true for any work-conserving service discipline.) The detailed motivation of this model can be found in [1].
Our analysis is based on the dependencies between the rates of the flows, in particular, input rate and output rate from the first station, in stationary regime. Also the analysis is heavily based on the Burke's theorem stating the equality of the input and output rates in the stationary (non-overloaded) multiserver first station. Finally, we apply stochastic monotonicity of the Birth-Death (BD) process, describing the multiserver queuing system.
II Stability Conditions
In this section, we establish the necessary stability conditions of the basic model described shortly above.
First of all, we give more detailed description of the model. We consider the described above two-station tandem system with Piosson input with rate A and feedback admission control, assuming that the first station operates as a queueing system MIMIN1 with N1 identical servers and infinite buffer. The second station is the system MIMIN2, also with infinity capacity buffer. The service rate is ^ at each server of station i = 1,2. Because all governing distributions are exponential, this feedback system is completely defined by the parameters A, Nh N.
The dynamics of this model can be described by a continuous-time discrete-valued Markov process Z(t) =: (z1(t),z2(t)), t > 0, where component zt(t) is the number of customers at station i at instant t, i = 1,2. Denote y(t) the number of arrivals in the interval (0, t], y(0) = 0, in the Poisson input flow (with the intensity A), and define x(t), the actual number of arrivals to the 1st station in interval (0, t], x(0) = 0.
The following statement generalizes the necessary stability conditions found in [1] for the single-server stations.
Theorem 1. Assume the Markov process Z is ergodic. If i) N1^1 < N2^2, then A < Fn(N1^1);
ii) otherwise, if N1^1 > N2^2, that there are no other restrictions except A < ro.
Proof. Assume that the Markov process Z is in steady state, and denote PN = P(z2(t) > N) the stationary probability that there are at least N customers in the 2nd station. The Poisson arrivals with the intensity A enter the 1st station. Then, at an arrival instant a transition y(t) ^ y(t) + 1 happens , and moreover, transition x(t) ^ x(t) + 1 happens if and only if z2(t) < N. Thus, the transition rate x(t) ^ x(t) + 1 equals v: = APN.
Therefore, for each t and constant T, the number of customers entering the 1st station in interval [t, t + T) does not depend on the number of customers arriving in interval (0, t], t > 0. Then it follows from [2], [3] that the rate of the arrivals entering the 1st station equals v = APN as well. Since the flow of arrivals entering the 1st station is Poisson with rate v and the process Z is ergodic, then the process z1 (t), t > 0, turns out to be ergodic also. As a result , the process z1 (t) is distributed as a BD process with the birth rate v and the death rates nk = min(k,N1)^1 [§ 1.2][4]. It then follows from Karlin - McGregor criterion [6], we obtain the inequality v < N1^1. Because the stationary output from the 1st station is also Poisson process with the rate v = APN, then we may notation PN = PN (v) which is heavily used below.
Apply now a similar analysis to the 2nd station. Since the input to the 2nd station (output from the 1st station) is Poisson with rate v, and the process Z is ergodic then the process z2(t), t > 0, is ergodic also.
As above then the process z2 (t) is distributed as a BD process with the birth rate v and the death rates = min(k,N2)^2. Then, as above it follows from Karlin - McGregor criterion, that the inequality v < N2^2 holds. Thus, we obtain the following relations:
v = APn, v < N1^1, v < N2^2. (1)
Consider another BD process z'2(t), t >0, with the same death rates and a birth rate v' > v. Moreover, we assume the same initial state in both processes, that is z2(0) = z2(0). Then it follows from Theorem 4.2.1 in [8], that the following inequality holds:
limP(z2(t) > N) = PN(v) > limP(z't) > N) =: PN(v'). (2)
Because = min(/, N2)^2, j > 1, then it follows from [5] (Chapter 2, Section 3), that for each fixed N > 0 and for all v, 0 < v < N2^2, the function PN(v) has the following explicit expression
Pn(V) = 1 + z"=1 vk/nu tj1 + zm=1 vk/nu xpj,
and moreover, is monotonically decreasing (2) and continuous in v. Because, under condition v > N2^2, the process z2(t) is not ergodic, then we obtain PN(v) = 0 for v > N2n2. Therefore, for the fixed N > 0, the function
Fn(V) (3)
is continuous and monotonically increases in v, as long as 0 < v < N2^2, while we put FN(v) = ro if v > N2h2. Then the equality
v = APN(v) = Fn(v)Pn(v) in (1) can be rewritten as v = F-1(A), where F-1 is the inverse function to function F. Hence, by the monotonicity, we obtain from (1) that, for N1^1 < N2n2,
A < Fn(N1^1). (4)
Assume that N1^1 > N2^2. Take an arbitrary e £ (0,N2^2). Then, by the ergodicity of the Markov process Z(t), t > 0, the inequality v < N2^2 — e < N1^1 follows, which in turn, is equivalent to the inequality v < N2^2 — e. The latter inequality implies A < Fn(N2^2 — e) by the monotonicity of function Fn . Because £ is arbitrary and
then (4) becomes A < ro, and the proof is completed.
III A Generalization
In the paper [1], also the following more general m-station system, m>2, is considered: the external input (with rate A) is rejected at the first station, if the number of customers zk (t) in each remaining station k exceeds a given threshold N(k\ Moreover, the output from station k is the input to station k + 1,k = 1,... ,m — 1. Denote zk(t) the number of customers at station k at instant t. In more detail, keeping other notation, consider an m - station exponential queueing system, in which station k has Nk (stochastically equivalent) servers with exponential service time with rate k = 1,...,m. It is assumed that a customer of the external Poisson input is rejected if the following inequalities hold true:
Z2(t)>N(2).....zm(t)>N(m).
The dynamics of this system is described by the following m-dimensional Markov process
Z = (z1(t).....zm(t)), t>0.
Theorem 2. Assume the process Z is ergodic. If
N1^1 < m™ Nk^k,
2<k<m
then A < Fn(N1^1). Otherwise, if
N1^1 > min NkVk,
2<k<m
that only requirement is A < ro.
Proof. Denote v the output rate of the (Poisson) flow of each station 1,...,m. (This rate is the same for all stations by the ergodicity.) By the product-form theorem for stationary regime [9], the joint stationary distribution of the basic process satisfies
P(z2(t) > N(2).....zm(t) > N(m)) = Vtf=2 P(zk(t) > N(k)) =: Pn(2).....^(v). (5)
The component processes z2(t), ...,zm(t) are the BD processes. Moreover, the process zk(t) has the birth rate v and, if zk(t) = i, the death rate = min(i,Nk)^k, k = 2, ...,m. It follows by Theorem 4.2.1 [8] and from analysis of the proof of Theorem 1 above, that the kth multiplier P(zk(t) > N(k)) in (5) (as function of v) is continuous and decreases for all v, 0 < v < Nk^k, k = 2, ...,m. Thus, function Pn(2) N(m) (v) is monotonically decreasing (and continuous) in v as long as
0 < v < min Nk^k.
2<k<m
Because the process Z is ergodic, then the rate of the (Poisson) process entering the 1st station is v = APn(2) N(m). Furthermore, the output flows of all stations in the system are Poisson with the same rate v. Now, repeating the arguments used in the proof of Theorem 1, we obtain the following relations
V = ¿Pn(2).....N(m) (V), V < N1H1, ...,V < NmVm. (6)
At that, the equality
V = APn(2).....N(m)(v) =: Fn(2).....N(m)(v)
in (6) can be rewritten as
V = FN(2).....N(m)W,
where F-2) N(m) is the inverse function to function FN(2) N(m). Now, by the monotonicity, we
obtain from (6), for N1^1 < min2sksmNk^k, the following inequality
¿<Fn(2).....N(m)(N1^1). (7)
If N1^1 > min2<k<mNky.k, then again repeating arguments used in the proof of Theorem 1, we obtain finally the inequality A < ro, which completes the proof.
IV Conclusion
The necessary stability conditions of the Markovian model of a two-station tandem queueing network with a special type of feedback are found. Under this feedback, the input to the first station is rejected as long as the queue size in the second station exceeds a predefined fixed level. The analysis is based on the introduction of a function expressing the dependence between the rates of input and output at the first station. We apply stochastic monotonicity of the Birth-Death process describing the dynamics of the system, to obtain the necessary conditions in an explicit form. Analysis of the two-station system is then generalized to multi-station system.
V Acknowledgements
The research of Evsey Morozov was carried out under state order to the Karelian Research Centre of the Russian Academy of Sciences (Institute of Applied Mathematical Research KRC RAS) and is partly supported by Russian Foundation for Basic Research, projects 18-07-00147, 18-07-00156. The research of Gurami Tsitsiashvili is partially supported by Russian Foundation for Basic Research, project 17-07-00177.
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