In this work, we verify by simulation some recent theoretical results describing the dynamics of the the retrial system with coupled orbits. In such a system, retransmission rate of customers blocked in a virtual orbit depends in general on the binary state, busy or idle, of other orbits. We consider a system with N classes of customers, where an arriving customer which meets server busy, joins the corresponding orbit depending on the class of customer. The top (oldest) blocked customer makes an attempt to enter server, with class-dependent exponential time between attempts. At that the retrial rate is de ned by the current states (busy or idle) of other orbits. To verify theoretical results, we simulate single-server retrial system with 3 classes of customers following independent Poisson inputs, while service times are class-dependent and have general distributions. In particular, we verify necessary and su cient stability conditions and focus on the analysis of symmetric model. Numerical experiments con rm theoretical analysis.
This research is devoted to veri cation by simulation a few performance results obtained in previous work [17,16]. Moreover, we focus on veri cation of the obtained stability conditions of the system with coupled orbits. In general, su cient and necessary stability conditions are di erent, and the study of the "gap" between these conditions is important in practice. Indeed, as we show, sufcient condition is redundant, and possible extension of stability region seems to be useful to increase the e ciency (throughput) of the system. In this work we pay the main attention to the so-called symmetric model, in which all classes of customers have the same parameters. This scenario allows to simplify analysis and detect some properties which turn out to be useful in a more general setting. In simulation, we focus on the symmetric system with three classes of customers which follow independent Poisson inputs. It is worth mentioning that this research complements and develops previous works [17,16].
To study classical retrial queues, we mention the books [12,1], and the survey papers [2,13]. Also the stability analysis of a multi-class retrial queue with constant retrial rates, which do not depend on the states of other orbits, has been developed in [4,5].
As to application of the system with coupled orbits, we mention the modelling of wireless multiple access systems, in particular, relay-assisted cognitive cooperative wireless systems [18]. Moreover, in the modern cognitive radio [15] there exists a possibility to dynamically adjusts retransmission rates to improve spectrum utilization [6,8,11]. Furthermore, as it has been mentioned in [16], this model is suitable to describe dynamics of cellular networks, in which the transmission rate in a particular cell decreases as the number of users in the neighboring cells increase [6]. A similar e ect is observed in the processor sharing models [7,14].
This paper is organized as follows. In Section 2 we describe the basic model. In Section 3 we summarize the main theoretical results obtained in [17] which we verify by simulation in Section 4. In particular, we introduce the symmetric model with coupled orbits. In section 4 we verify theoretical results simulating 3class symmetric model with exponential and Pareto service times. In particular, we estimate the stationary probability that a xed orbit is busy and server is idle, and demonstrate the correctness of the lower and upper bounds of this probability. Also the accuracy of stability conditions is studied, using the "gap" between the necessary and su cient stability conditions. Actually simulation shows that the necessary stability condition is in fact stability criterion of the system. 2
We study a single-server with no bu er for the waiting customers, and with three classes of customers. Nevertheless, it is worth mentioning that the theoretical results, which we will verify, hold for an arbitrary number N of customer classes. By this reason we will formulate below theoretical results for this general setting.
The class-i customers form Poisson input with rate i; i = 1; : : : ; N . Because all inputs are assumed to be independent, then the summary input is Poisson as well, with rate := Pi i; in which an arbitrary arrival belongs to class i with the probability pi =: i= ; i = 1; : : : ; N: Then interarrival times of the input are exponential with generic element and expectation E = 1= 2 (0; 1). We also assume service times of class-i customers, fSn(i); n 1g, to be independent identically distributed (iid) with service rate
1 i =: ES(i) 2 (0; 1); i = 1; : : : ; N: A customer, meeting server busy joins a class-dependent virtual orbit, where joins the end of the orbit queue. At that the head customer waiting in orbit i makes retrial attempts until he nds server idle to occupy it. The distance between attempts are exponentially distributed with rate J(i), where J (i) is the current con guration of the orbits: busy or empty. Thus each orbit acts as a FIFO queueing system with state-dependent "service" rate J(i). This dependence is a key new property of the model.
To be more precise, for each i, we de ne (N 1)-dimensional vectors
J (i) = fj1; : : : ; ji 1; ji+1; : : : ; jN g with binary components jk 2 f0; 1g, where the ith component is omitted. If the k-th orbit is currently busy, we put jk = 1, otherwise, jk = 0. Each vector J (i) is called con guration. For each i, we introduce the set G(i) = fJ (i)g of possible con gurations. It is assumed that there is given constant J(i), retransmission rate from orbit i, if current con guration is J (i). We denote Mi the set of rates for all con gurations belonging to G(i).
This construction, proposed in [17], considerably generalizes the setting studied in previous works [16,11,10,9]. In these works, it is assumed that orbit i has rate i if at least one (other) orbit is busy, otherwise, the rate is i ; i = 1; : : : ; N . Thus, in setting in [16], each set Mi = f i ; ig. In this work we continue to study th new general setting from [17] with focus on veri cation some bounds and stability conditions proved in [17], for the system with three classes of customers.
Before to give main stationary performance measures to be veri ed by simulation, we mention that the main stochastic processes describing the dynamics of the system, such like accumulated work (workload) orbit size, ect., are regenerative, with regeneration instants Tn. A regeneration occurs when a new customer meets an idle system [3]. The distances Tn+1 Tn are iid regeneration periods, and we denote T the generic period.
A queuing process is called positive recurrent if the mean generic period is nite, that is ET < 1 [3]. Under positive recurrence, there exists the stationary regime of the system [3]. 3
Now we de ne the main stationary performance metrics and give necessary and su cient stability conditions proved in [17] by regenerative method. Let I(t) be the summary idle time of the server in interval [0; t], then busy time is de ned as B(t) = t I(t).
If the system is positive recurrent, then there exist the limits, with probability (w.p.) 1,
I(t) lim = P0 = 1 Pb; t!1 t where P0 is the stationary idle probability of the server, and
Pb = lim t!1
B(t) t is the stationary busy probability of the server. Denote Bi(t) the summary time, in interval [0; t], when the server is occupied by class-i customers. It is proved and summary tra c intensity Because Pb = Pi P(i) then it follows that Pb = :
b Now we introduce the maximal possible rate from orbit i:
N Pb = X i = i=1 min h 1 i N
^i
in [17] that the stationary probability that the server is occupied by class-i
customer is de ned as
0
i
+ 0
i
:
To compare (
^i An important special class constitute the symmetric systems. To explain symmetry for the system with coupled orbits, we rst note that this system becomes standard retrial system with constant rate provided J(i) i; implying equality ^i = i0 = i. The latter standard retrial system with constant rate, becomes symmetrical, if the corresponding parameters are equal, that is i ; i ; i : However, the notion symmetrical system with coupled orbits is more exible. To explain it in more detail, we de ne, for each i, the set of vectors describing retrial rates for each con guration J (i) = fj1; : : : ; jN g. Because in our case N = 3, then the capacity jJ (i)j = 4 for each i = 1; 2; 3. To compose J (i), we use lexicographical order, that is, for each orbit i, and two remaining orbits j < k, with k; j 6= i, the following four con gurations J (i) are possible:
Mi =: f(ij = 0; ik = 0); (ij = 1; ik = 0); (ij = 0; ik = 1); (ij = 1; ik = 1)g: (
G(i) = f i00; i10; i01; i11g; i = 1; 2; 3; are identical, although rates within G(i) may di er. One can expect that this structure leads to similar behavior of the orbits, and it is con rmed below by simulation.
Note that for the symmetric coupled orbits, the di erence (
^ =N + ^ = =N +
0 + 0
; +
:
iPb = iP(0i);
where ^ = ^i; 0 = i0: Finally, for symmetric classical (non-coupled) orbits,
^ = 0 and (
P(i) = 0
N i X i k=1
k; i = 1; : : : ; N: 4
In this section we verify by simulation some obtained above theoretical results for three classes of customers considering symmetric model. To this end, we de ne the following variables, 1 := min i
^i
which delimit the boundary of stability region. More exactly, if i > 0; i =
1; 2, then both stability conditions (
Now we present numerical results obtained by simulation, to verify
theoretical analysis of stationary regime and necessary and su cient conditions (
First we perform some experiments for the completely symmetric model and
demonstrate stability/instability of all orbits. This analysis also shows the
redundancy of su cient condition (
In the 1st experiment, we use the following input and service rates,
Thus, with those parameters we receive = 0:6 and 1 = 0:3, 2 = 0. Therefore, both stability conditions are satis ed and all orbits are stable, as expected, and we can see that at Fig. 1 t() N
( xi0 ) ; x x xi0 ;
1
In the following experiments we show the results for Pareto service times (Pareto model), and not equal service rates, ceteris paribus. That is we still keep a symmetry in the input rates.
Fig. 4 shows the dynamics of the orbits in Pareto model with service time distribution
Fi(x) = 1 xi0 (Fi(x) = 0; x xi0); and expectation
ES(i) = > 1; xi0 > 0; i = 1; 2; 3: We select = 2 and the following values of the shape parameter xi0 for orbit i = 1; 2; 3, respectively:
1 = 2 = 3 = 12;
ceteris paribus. Here = 0:76, implying 1 = 0:14 and 2 = 0:16. Thus
condition (
In the 3rd experiment, shown on the Fig.3, we further increase service rate
t()N 01
Here we obtain = 0:9 and it gives 1 = 0, 2 = 0:3. Thus both conditions
(
Thus in the these experiments, a gradual decreasing service rates (which
implies reduction 1 and 2) makes orbits unstable only if the necessary condition
(
In the following experiment we demonstrate the estimation the stationary
probability P(
3 = 3 1 = 2 = 3 = 15;
while the retrial rates remain (
0 (The dynamics of the estimators of P(02) and P(03) is similar and is omitted.) t()N 02 In this work, we simulate a 3 -class symmetric retrial system with independent Poisson inputs and the coupled orbits to verify some theoretical results found earlier. In this system, a new customer meeting server busy joins the corresponding in nite capacity orbit. The retrial rate from orbit i depends on the current con guration of other orbits: busy or idle. We verify by simulation some stationary performance measures and the accuracy of the found earlier stability conditions of this model.
ACKNOWLEDGEMENTS
The study was carried out under state order to the Karelian Research Centre of
the Russian Academy of Sciences (Institute of Applied Mathematical Research
KRC RAS). The research of EM is partly supported by Russian Foundation
for Basic Research, projects 18-07-00147, 18-07-00156. The research of TM is
supported by Petrozavodsk State University and Russian Foundation for Basic
Research, project 18-07-00147.
t
Fig. 4. The symmetric system, Pareto service time. Estimation the probability P(
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