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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 defined 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 sufficient stability conditions and focus on the analysis of symmetric model. Numerical experiments confirm theoretical analysis
Simulation of multiclass retrial system
with coupled orbits
Evsey Morozov1,2, Taisia Morozova2, and Ioannis Dimitriou3
1Institute of Applied Mathematical Research, Karelian Research Centre of the RAS,
Petrozavodsk, Russia
2Petrozavodsk State University, Petrozavodsk, Russia,
3University of Patras, Greece
Abstract. 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 Nclasses of customers, where an ar-
riving 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 defined 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 indepen-
dent Poisson inputs, while service times are class-dependent and have
general distributions. In particular, we verify necessary and sufficient
stability conditions and focus on the analysis of symmetric model. Nu-
merical experiments confirm theoretical analysis.
Keywords: Multiclass retrial queues ·Stability ·Constant retrial rates
·Coupled orbit queues ·Cognitive network.
1 Introduction
This research is devoted to verification by simulation a few performance re-
sults obtained in previous work [17,16]. Moreover, we focus on verification of
the obtained stability conditions of the system with coupled orbits. In general,
sufficient and necessary stability conditions are different, and the study of the
”gap” between these conditions is important in practice. Indeed, as we show, suf-
ficient condition is redundant, and possible extension of stability region seems
to be useful to increase the efficiency (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 anal-
ysis 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 con-
stant 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 mod-
elling 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 trans-
mission rate in a particular cell decreases as the number of users in the neighbor-
ing cells increase [6]. A similar effect is observed in the processor sharing models
This paper is organized as follows. In Section 2we describe the basic model.
In Section 3we 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 4we verify theoretical results simulating 3-
class symmetric model with exponential and Pareto service times. In particular,
we estimate the stationary probability that a fixed 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 sufficient stability conditions. Actually simulation
shows that the necessary stability condition is in fact stability criterion of the
2 Description of the model
We study a single-server with no buffer 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 Nof customer classes.
By this reason we will formulate below theoretical results for this general setting.
The class-icustomers form Poisson input with rate λi, i = 1, . . . , N . Because
all inputs are assumed to be independent, then the summary input is Poisson as
well, with rate λ:= Piλi,in which an arbitrary arrival belongs to class iwith
the probability pi=: λi/λ, i = 1, . . . , N . Then interarrival times of the input
are exponential with generic element τand expectation Eτ= 1(0,). We
also assume service times of class-icustomers, {S(i)
n, n 1}, to be independent
identically distributed (iid) with service rate
γi=: 1
ES(i)(0,), 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
imakes retrial attempts until he finds server idle to occupy it. The distance
between attempts are exponentially distributed with rate µJ(i), where J(i) is the
current configuration of the orbits: busy or empty. Thus each orbit acts as a FIFO
queueing system with state-dependent ”service” rate µJ(i). This dependence is
a key new property of the model.
To be more precise, for each i, we define (N1)-dimensional vectors
J(i) = {j1, . . . , ji1, ji+1, . . . , jN}
with binary components jk∈ {0,1}, 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 configuration. For each i, we introduce the set G(i) = {J(i)}of possible
configurations. It is assumed that there is given constant µJ(i), retransmission
rate from orbit i, if current configuration is J(i). We denote Mithe set of rates
for all configurations belonging to G(i).
This construction, proposed in [17], considerably generalizes the setting stud-
ied in previous works [16,11,10,9]. In these works, it is assumed that orbit ihas
rate µiif at least one (other) orbit is busy, otherwise, the rate is µ
i, i = 1, . . . , N .
Thus, in setting in [16], each set Mi={µ
i, µi}. In this work we continue to study
th new general setting from [17] with focus on verification some bounds and sta-
bility conditions proved in [17], for the system with three classes of customers.
Before to give main stationary performance measures to be verified by simu-
lation, we mention that the main stochastic processes describing the dynamics of
the system, such like accumulated work (workload) orbit size, ect., are regenera-
tive, with regeneration instants Tn. A regeneration occurs when a new customer
meets an idle system [3]. The distances Tn+1 Tnare iid regeneration periods,
and we denote Tthe generic period.
A queuing process is called positive recurrent if the mean generic period is
finite, that is ET < [3]. Under positive recurrence, there exists the stationary
regime of the system [3].
3 Preliminary results
Now we define the main stationary performance metrics and give necessary and
sufficient 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 defined
as B(t) = tI(t).
If the system is positive recurrent, then there exist the limits, with probability
(w.p.) 1,
t=P0= 1 Pb,
where P0is the stationary idle probability of the server, and
Pb= lim
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-icustomers. It is proved
in [17] that the stationary probability that the server is occupied by class-i
customer is defined as
b=ρi, i = 1, . . . , N.
Denote the traffic intensity for each class,
ρi=λii, i = 1, . . . , N ,
and summary traffic intensity
Because Pb=PiP(i)
bthen it follows that Pb=ρ.
Now we introduce the maximal possible rate from orbit i:
ˆµi= max
The following statement, which has been proved in [17], contains the necessary
stability (positive recurrence) condition of our system.
Theorem 1. If the N-class retrial system with coupled orbits is positive recur-
rent, then
λi+ ˆµii<1.(1)
To formulate the next statement and sufficient stability conditions, we de-
note, for each class i, the minimal retrial rate
i= min
Also let P(i)
0be the stationary probability that server is idle and orbit iis busy.
The following statement is proved in [17].
Theorem 2. The following inequalities hold
ρ, i = 1, . . . , N. (2)
Now we formulate the sufficient stability condition [17].
Theorem 3. The sufficient stability condition of N-class retrial system with
coupled orbits is
ρi+ max
This condition implies a negative drift of the workload process, and as a result,
positive recurrence of the system, and can be written as
To compare (3) with the necessary stability condition (1), we define the gap
between the necessary and sufficient conditions,
=: min
λi+ ˆµi
An important special class constitute the symmetric systems. To explain sym-
metry for the system with coupled orbits, we first note that this system becomes
standard retrial system with constant rate provided µJ(i)µi,implying equal-
ity ˆµi=µ0
i=µi. The latter standard retrial system with constant rate, becomes
symmetrical, if the corresponding parameters are equal, that is
λiλ, γiγ, µiµ.
However, the notion symmetrical system with coupled orbits is more flexible.
To explain it in more detail, we define, for each i, the set of vectors describing
retrial rates for each configuration J(i) = {j1, . . . , jN}. Because in our case
N= 3, then the capacity |J(i)|= 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 configurations J(i) are possible:
Mi=: {(ij= 0, ik= 0),(ij= 1, ik= 0),(ij= 0, ik= 1),(ij= 1, ik= 1)}.(5)
(Recall that ik= 1 means that orbit kis busy, while ik= 0 means that it
is empty.) We denote µi
00, µi
10, µi
01, µi
11,the retrial rates corresponding to each
configuration in (5). Then we obtain that, in the symmetrical system, all sets
G(i) = {µi
00, µi
10, µi
01, µi
11}, i = 1,2,3,
are identical, although rates within G(i) may differ. One can expect that this
structure leads to similar behavior of the orbits, and it is confirmed below by
Note that for the symmetric coupled orbits, the difference (4) becomes
:= ˆµ
λ/N + ˆµµ0
where ˆµ= ˆµi, µ0=µ0
i.Finally, for symmetric classical (non-coupled) orbits,
ˆµ=µ0and (4) becomes
λ/N +µµ
In the next section, containing simulation results, we analyze the symmetric
model and leave studying a more general model for a further work.
Remark. For non-coupled orbits, µJ(i)=µifor all configurations J(i), and each
busy orbit ihas a fixed retrial rate µi. Then relation (2) becomes
and we obtain the following explicit formula for the stationary probability that
orbit queue iis busy and server is idle, see [17]:
ρk, i = 1, . . . , N.
4 Simulation results
In this section we verify by simulation some obtained above theoretical results for
three classes of customers considering symmetric model. To this end, we define
the following variables,
Γ1:= min
λi+ ˆµi
ρ, Γ2:= min
which delimit the boundary of stability region. More exactly, if Γi>0, i =
1,2, then both stability conditions (1) and (3) are satisfied. If Γi<0, i =
1,2, then instability of the orbits is expected. However, if the intermediate case
Γ1>0, Γ2<0 holds, then, as simulation shows, the orbits are stable in all
experiments. It indicates that the necessary stability condition is in fact stability
criterion, while sufficient stability condition is redundant. (However we can not
prove it strictly for the system with general service times.) These observations
are illustrated by simulation below. We emphasize again that in this work we
pay attention simulation symmetric model with coupled orbits.
Now we present numerical results obtained by simulation, to verify theoreti-
cal analysis of stationary regime and necessary and sufficient conditions (1), (3).
Everywhere we use the black, grey and dotted curve to demonstrate the dynam-
ics of the 1st, 2nd and 3rd orbit, respectively. (The axis tcounts the number
of discrete events: arrivals, departures, attempts, in the applied discrete-event
simulation algorithm.)
First we perform some experiments for the completely symmetric model and
demonstrate stability/instability of all orbits. This analysis also shows the re-
dundancy of sufficient condition (3) for stability because simulation stays all
orbits stable even for Γ1<0.
In the 1st experiment, we use the following input and service rates,
λ1=λ2=λ3= 3, γ1=γ2=γ3= 15,
and the following retrial rates:
00 = 20, µ1
10 = 30, µ1
01 = 15, µ1
11 = 25},
00 = 20, µ2
10 = 30, µ2
01 = 15, µ2
11 = 25},
00 = 20, µ3
10 = 30, µ3
01 = 15, µ3
11 = 25}.(6)
Thus, with those parameters we receive ρ= 0.6 and Γ1= 0.3, Γ2= 0.
Therefore, both stability conditions are satisfied and all orbits are stable, as
expected, and we can see that at Fig. 1
0 500 1000 1500 2000 2500 3000
Fig. 1. The symmetric system, exponential service time. Condition (1) and (3) hold:
Γ1>0, Γ2>0; all orbits are stable.
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. 4shows the dynamics of the orbits in Pareto model with service time
x)α, x xi
0(Fi(x)=0, x xi
and expectation
ES(i)=α xi
α1, α > 1, xi
0>0, i = 1,2,3.
We select α= 2 and the following values of the shape parameter xi
0for orbit
i= 1,2,3, respectively:
24, i = 1,2,3.
This choice gives the following service rates
γ1=γ2=γ3= 12,
ceteris paribus. Here ρ= 0.76, implying Γ1= 0.14 and Γ2=0.16. Thus
condition (1) holds while condition (3) is violated. As we see, all three orbits
remain stable, however the stability is reached at a higher level.
In the 3rd experiment, shown on the Fig.3, we further increase service rate
0 5000 10000 15000
0 5 10 15 20
Fig. 2. The symmetric system, Pareto service time. Condition (1) holds, condition (3)
is violated: Γ1>0, Γ2<0; all orbits are stable at a higher level.
(ceteris paribus)
γ1=γ2=γ3= 10.
Here we obtain ρ= 0.9 and it gives Γ1= 0, Γ2=0.3. Thus both conditions
(1) and (3) are violated. (Γ1= 0 is called boundary case.) As we see on Fig. 3,
all orbits become now unstable.
Thus in the these experiments, a gradual decreasing service rates (which im-
plies reduction Γ1and Γ2) makes orbits unstable only if the necessary condition
(1) is violated. So we suggest that condition (3) is redundant, and moreover that
the necessary condition (1) is indeed stability criterion.
In the following experiment we demonstrate the estimation the stationary
probability P(1)
0for the symmetric model with exponential service time. In that
experiment we use the following input and service rates,
λ1=λ2=λ3= 3γ1=γ2=γ3= 15,
while the retrial rates remain (6). Thus in this case µ0
iand ˆµiare different,
and exact value of the target probability is unknown. However, by the positive
recurrence, the sample mean estimate still converges to a limit. Fig. 4shows that
the sample mean estimator of P(1)
0satisfies the corresponding inequality in (2).
(The dynamics of the estimators of P(2)
0and P(3)
0is similar and is omitted.)
0 1000 2000 3000 4000 5000 6000 7000
0 10 20 30 40
Fig. 3. The symmetric system, Pareto service time. Conditions (1) and (3) are violated:
Γ1<0, Γ2<0; all orbits are unstable.
5 Conclusion
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 correspond-
ing infinite capacity orbit. The retrial rate from orbit idepends on the current
configuration of other orbits: busy or idle. We verify by simulation some sta-
tionary performance measures and the accuracy of the found earlier stability
conditions of this model.
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.
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... This research focuses on the numerical investigation of the necessary stability conditions of a single-server multiclass retrial system with coupled orbit queues and service interruptions. Thus we generalize the model considered in previous works [16], [18], [19] where a reliable server has been assumed. This system is characterized by the fact that the retransmission rate of an orbit queue depends on the state of the other orbit queues, which in turn provides a remarkable flexibility to the orbits to adapt their characteristics with ultimate goal the optimal system's performance. ...
... In general, sufficient and necessary stability conditions are different, and in our previous our works we extensively studied the "gap" between these conditions. Indeed, as it has been shown by simulation in [16], [19] the sufficient condition is redundant, and necessary stability conditions solely allow to delimit stability region with a high accuracy. As we show in this note, this effect holds for the system with server interruptions as well. ...
... This work is devoted to the investigation of the stability conditions of a retrial system with an arbitrary number of orbit queues under a class-dependent queue-aware constant retrial policy and service interruptions. This research is a continuation and extension of our previous works [16], [18], [19]. Using the regenerative approach, we extend the works [16], [18], [19] with an important goal to investigate the impact of service interruptions on the stability conditions. ...
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The performance of wireless data systems has been extensively studied in the context of a single base station. In the present paper we investigate the flow-level performance in networks with multiple base stations. We specifically examine the complex, dynamic interaction of the number of active flows in the various cells introduced by the strong impact of interference between neighboring base stations. For the downlink data transmissions that we consider, lower service rates caused by increased interference from neighboring base stations result in longer delays and thus a higher number of active flows. This in turn results in a longer duration of interference on surrounding base stations, causing a strong correlation between the activity states of the base stations. Such a system can be modelled as a network of multi-class processor-sharing queues, where the service rates for the various classes at each queue vary over time as governed by the activity state of the other queues. The complex interaction between the various queues renders an exact analysis intractable in general. A simplified network with only one class per queue reduces to a coupled-processors model, for which there are few results, even in the case of two queues. We thus derive bounds and approximations for key performance metrics like the number of active flows, transfer delays, and flow throughputs in the various cells. Importantly, these bounds and approximations are insensitive, yielding simple expressions, that render the detailed statistical characteristics of the system largely irrelevant.
We investigate a novel queueing system that can be used to model relay-assisted cooperative cognitive networks with coupled relay nodes. Consider a network of two saturated source users that transmit packets towards a common destination node under the cooperation of two relay nodes. The destination node forwards packets outside the network, and each source user forwards its blocked packets to a dedicated relay node. Moreover, when the transmission of a packet outside the network fails, either due to path-loss, fading or due to a hardware/software fault in the transmitter of the destination node, the failed packet is forwarded to a relay node according to a probabilistic policy. In the latter case a recovery period is necessary for the destination node in order to return in an operating mode. Relay nodes have infinite capacity buffers, and are responsible for the retransmission of the blocked/failed packets. Relay nodes have cognitive radio capabilities, and there are fully aware about the state of the other. Taking also into account the wireless interference, a relay node adjusts its retransmission parameters based on the knowledge of the state of the other. We consider a three-dimensional Markov process, investigate its stability, and study its steady-state performance using the theory of boundary value problems. Closed form expressions for the expected delay are also obtained in the symmetrical model.
In this work we analyze a novel queueing system for modeling cooperative wire less networks. We consider a network of three saturated source users, say a central and two background users, two relay nodes and a common destination node. When the central user fails to transmit a packet directly to the destina tion, it forwards a copy of the blocked packet at both relay nodes in order to exploit the spatial diversity they provide. Moreover, each relay node receives also blocked packets from a dedicated background user. Relay nodes assist source users by retransmitting their blocked packets to the destination. Due to the complex interdependence among relays, and the wireless interference, the retransmission rate of a relay is affected by the state of the other. We con sider a three-dimensional Markov process, investigate its stability, and study its steady-state performance using the theory of boundary value problems. Explicit expressions for the expected delay in the symmetrical model, and a generalization to N > 2 relay nodes are also given. Numerical examples are obtained and show insights into the system performance.
We consider a single server system accepting two types of retrial customers, which arrive according to two independent Poisson streams. The service station can handle at most one customer, and in case of blocking, type i customer, i =1, 2, is routed to a separate type i orbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and thus, when both orbit queues are non-empty, the orbit queue i tries to re-dispatch a blocked customer of type i to the main service station after an exponentially distributed time with rate μ i . If an orbit queue empties, the other orbit queue changes its re-dispatch rate from μ i to $\mu_{i}^{\ast}$ . We consider both exponential and arbitrary distributed service requirements, and show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann (–Hilbert) boundary value problems. For exponential service requirements, we also investigate the exact tail asymptotic behavior of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method. Performance metrics are obtained, computational issues are discussed and a simple numerical example is presented.
The application of auto-repeat facilities in telephone systems, as well as the use of random access protocols in computer networks, have led to growing interest in retrial queueing models. Since much of the theory of retrial queues is complex from an analytical viewpoint, with this book the authors give a comprehensive and updated text focusing on approximate techniques and algorithmic methods for solving the analytically intractable models. Retrial Queueing Systems: A Computational Approach also •Presents motivating examples in telephone and computer networks. •Establishes a comparative analysis of the retrial queues versus standard queues with waiting lines and queues with losses. •Integrates a wide range of techniques applied to the main M/G/1 and M/M/c retrial queues, and variants with general retrial times, finite population and the discrete-time case. •Surveys basic results of the matrix-analytic formalism and emphasizes the related tools employed in retrial queues. •Discusses a few selected retrial queues with QBD, GI/M/1 and M/G/1 structures. •Features an abundance of numerical examples, and updates the existing literature. The book is intended for an audience ranging from advanced undergraduates to researchers interested not only in queueing theory, but also in applied probability, stochastic models of the operations research, and engineering. The prerequisite is a graduate course in stochastic processes, and a positive attitude to the algorithmic probability. © 2008 Springer-Verlag Berlin Heidelberg. All rights are reserved.