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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, tiamorozova@mail.ru

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

[7,14].

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

system.

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/λ ∈(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 (N−1)-dimensional vectors

J(i) = {j1, . . . , ji−1, 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) = t−I(t).

If the system is positive recurrent, then there exist the limits, with probability

(w.p.) 1,

lim

t→∞

I(t)

t=P0= 1 −Pb,

where P0is the stationary idle probability of the server, and

Pb= lim

t→∞

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-icustomers. It is proved

in [17] that the stationary probability that the server is occupied by class-i

customer is deﬁned as

lim

t→∞

Bi(t)

t=P(i)

b=ρi, i = 1, . . . , N.

Denote the traﬃc intensity for each class,

ρi=λi/γi, i = 1, . . . , N ,

and summary traﬃc intensity

ρ=Xρi.

Because Pb=PiP(i)

bthen it follows that Pb=ρ.

Now we introduce the maximal possible rate from orbit i:

ˆµi= max

J(i)∈G(i)µJ(i).

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

Pb=

N

X

i=1

ρi=ρ≤min

1≤i≤Nhˆµi

λi+ ˆµii<1.(1)

To formulate the next statement and suﬃcient stability conditions, we de-

note, for each class i, the minimal retrial rate

µ0

i= min

J(i)∈G(i)µJ(i).

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

λi

ˆµi

ρ≤P(i)

0≤λi

µ0

i

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

N

X

i=1

ρi+ max

1≤i≤N

λ

µ0

i+λ<1.

This condition implies a negative drift of the workload process, and as a result,

positive recurrence of the system, and can be written as

ρ=X

i

ρi<min

iµ0

i

λ+µ0

i.(3)

To compare (3) with the necessary stability condition (1), we deﬁne the gap

between the necessary and suﬃcient conditions,

∆=: min

i

ˆµi

λi+ ˆµi

−min

i

µ0

i

λ+µ0

i

>0.(4)

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

λi≡λ, γi≡γ, µi≡µ.

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

simulation.

Note that for the symmetric coupled orbits, the diﬀerence (4) becomes

∆:= ˆµ

λ/N + ˆµ−µ0

λ+µ0,

where ˆµ= ˆµi, µ0=µ0

i.Finally, for symmetric classical (non-coupled) orbits,

ˆµ=µ0and (4) becomes

∆=µ

λ/N +µ−µ

λ+µ.

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

λiPb=µiP(i)

0,

and we obtain the following explicit formula for the stationary probability that

orbit queue iis busy and server is idle, see [17]:

P(i)

0=λi

µi

N

X

k=1

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

i

ˆµi

λi+ ˆµi

−ρ, Γ2:= min

i

µ0

i

λ+µ0

i

−ρ.

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:

M1={µ1

00 = 20, µ1

10 = 30, µ1

01 = 15, µ1

11 = 25},

M2={µ2

00 = 20, µ2

10 = 30, µ2

01 = 15, µ2

11 = 25},

M3={µ3

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

01234567

t

N(t)

t

N(t)

t

N(t)

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

distribution

Fi(x)=1−(xi

0

x)α, x ≥xi

0(Fi(x)=0, x ≤xi

0),

and expectation

ES(i)=α xi

0

α−1, 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:

xi

0=1

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

t

N(t)

t

N(t)

t

N(t)

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

t

N(t)

t

N(t)

t

N(t)

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.

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.

References

1. Artalejo, J.R., G´omez-Corral, A.: Retrial Queueing Systems: A Computational

Approach. Springer-Verlag Berlin Heidelberg (2008), https://doi.org/10.1007/

978-3-540-78725- 9

500 1000 1500 2000 2500 3000 3500

0.00 0.05 0.10 0.15 0.20 0.25 0.30

t

probability of busy orbit 1 and idle server

Fig. 4. The symmetric system, Pareto service time. Estimation the probability P(1)

0=

P(busy orbit 1, idle server).

2. Artalejo, J.: Accessible bibliography on retrial queues: Progress in 2000-2009.

Mathematical and Computer Modelling pp. 9–10 (2010)

3. Asmussen, S.: Applied probability and queues. Springer, New York (2003)

4. Avrachenkov, K., Morozov, E., Nekrasova, R., Steyaert, B.: Stability analy-

sis and simulation of N-class retrial system with constant retrial rates and

poisson inputs. Asia-Paciﬁc Journal of Operational Research 31(2) (2014).

https://doi.org/10.1142/S0217595914400028

5. Avrachenkov, K., Morozov, E., Steyaert, B.: Suﬃcient stability conditions for

multi-class constant retrial rate systems. Queueing Systems 82(1-2), 149–171

(Feb 2016). https://doi.org/10.1007/s11134-015-9463-9,http://link.springer.

com/10.1007/s11134-015-9463-9

6. Bonald, T., Borst, S., Hegde, N., Proutiere, A.: Wireless data performance in multi-

cell scenarios. Proc. ACM Sigmetrics/Performance ’04 pp. 378–388 (2004)

7. Bonald, T., Massouli´e, L., Prouti´ere, A., Virtamo, J.: A queueing analysis of max-

min fairness, proportional fairness and balanced fairness. Queueing Syst. (2006)

8. Borst, S., Jonckheere, M., Leskela, L.: Stability of parallel queueing systems with

coupled service rates. Discrete Event Dyn. S. pp. 447–472 (2008)

9. Dimitriou, I.: Modeling and analysis of a relay-assisted cooperative cognitive net-

work. Springer (2017)

10. Dimitriou, I.: A queueing system for modeling cooperative wireless networks with

coupled relay nodes and synchronized packet arrivals. Perform. Eval. (2017).

https://doi.org/10.1016/j.peva.2017.04.002

11. Dimitriou, I.: A two class retrial system with coupled orbit queues. Prob. Engin.

Infor. Sc. pp. 139–179 (2017)

12. Falin, J., Templeton, J.G.C.: Retrial Queues. Chapman and Hall/CRC (1997)

13. Kim, J., Kim, B.: A survey of retrial queueing systems. Annals of Operations

Research pp. 3–36 (2016)

14. Liu, X., Chong, E., Shroﬀ, N.: A framework for opportunistic scheduling in wireless

networks. Comp. Netw. pp. 451–474 (2003)

15. Mitola, J., Maguire, G.: Cognitive radio: making software radios more personal.

IEEE Pers. Commun. 6(4) pp. 13–18 (1999)

16. Morozov, E., Dimitriou, I.: Stability analysis of a multiclass retrial system with cou-

pled orbit queues. Proceedings of 14th European Workshop, EPEW 2017, Berlin,

Germany, September 7-8, 2017 (2017). https://doi.org/10.1007/978-3-319-66583-

2-6

17. Morozov, E., Morozova, T.: Analysis of a generalized system with coupled orbits.

Proceedings of Fruct23, Bologna (2018)

18. Sadek, A., Liu, K., Ephremides, A.: Cognitive multiple access via cooperation:

Protocol design and performance analysis. IEEE Trans. Infor. Th. 53(10) pp. 3677–

3696 (2007)