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A Multiclass Retrial System With Coupled Orbits

And Service Interruptions: Veriﬁcation of Stability

Conditions

Ioannis Dimitriou

University of Patras, Greece

idimit@math.upatras.gr

Evsey Morozov, Taisia Morozova

Institute of Applied Mathematical Research, Karelian Research Centre of RAS

Petrozavodsk State University

Petrozavodsk, Russia

emorozov@karelia.ru, tiamorozova@mail.ru

Abstract—In this work, we investigate the stability conditions

of a multiclass retrial system with coupled orbit queues and

service interruptions. We consider a single server system accept-

ing Nclasses of customers according to independent Poisson

inputs and with class-dependent, arbitrarily distributed service

times. An arriving customer who ﬁnds the server unavailable

upon arrival, joins the corresponding orbit queue according to

its class. We assume that the ﬁrst (oldest) blocked customer in

an orbit queue attempts to connect with the server after an

exponentially distributed service time, which depends both on

its class, and on the current state (busy or idle) of the other

orbit queues. During service times, interruptions occur according

to class-dependent Poisson process, following by class-dependent

arbitrarily distributed setup times. We consider both preemptive-

repeat identical, and preemptive-resume interruptions. Potential

applications of such a system can be found in the modelling

of relay-assisted cooperative wireless networks. We focus on the

non-symmetrical orbits and perform simulation experiments for

the system with three classes of customers to verify stability

conditions for both types of the server interruptions.

I. INTRODUCTION

This research focuses on the numerical investigation of

the necessary stability conditions of a single-server multi-

class 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.

A. Related work

Queues with repeated attempts have been extensively stud-

ied so far. For more details on the theoretical development,

the interested reader is referred to the seminal books in [10],

[2], and the survey papers [1], [13] (not exhaustive list).

Clearly, the stationary analysis, and more importantly the

investigation of the stability conditions in a multiclass system

with repeated attempts, is much more challenging than that

of the single-class variant. For sake of clarity we mention the

works in [3], [4], in which necessary and sufﬁcient conditions

were investigated by using the regenerative approach [20]

for multiclass retrial systems under typical, i.e., non-queue

aware constant retrial policy. Simulation experiments were also

performed to validate the theoretical ﬁndings. Quite recently,

stability analysis of a multiclass retrial system under classical

retrial policy was also investigated in [17].

Recently, stability conditions for a novel class-dependent,

queue-aware constant retrial policy, i.e., coupled orbit queues

were investigated in [16], [18], using the regenerative ap-

proach, for the model with arbitrary number of orbit queues.

For more details on the use of regenerative approach on the

stability conditions of stochastic systems see e.g., [4], [3]. The

stability conditions for a two class retrial system with coupled

orbit queues have been obtained in [7], [8], [9] by using the

well-known Foster-Lyapunov drift criteria [11].

B. Our contribution

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. For such a model we

consider preemptive repeat different and preemptive resume

service policy. We show that the regenerative approach is an

adequate method to handle it.

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_____________________________________________________PROCEEDING OF THE 24TH CONFERENCE OF FRUCT ASSOCIATION

ISSN 2305-7254

Potential application of our system model with coupled

orbits can be found in the modelling of relay-assisted coop-

erative wireless systems [21] with interacting nodes. In such

systems the service rate of the nodes are severely affected by

the wireless interference, as a result of the number of active

(non-empty) nodes. This feature is adequately modelled in our

system by the concept of coupled orbit queues. Moreover, in

modern cognitive radio networks [15] wireless nodes are able

to dynamically adjust retransmission rates to improve spectrum

utilization [5], [7].

Furthermore, as it has been mentioned in [18], 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 [5]. A similar

effect is observed in the processor sharing models [6], [14].

This paper is organized as follows. In Section II we describe

the basic model with server interruptions and, in particular,

give a modiﬁed deﬁnition of the conﬁguration of the orbits.

In Section III, we describe in detail the server interruptions

mechanism and show how to calculate the mean generalized

service time for both types of interruptions, using Laplace-

Stieltjes Transform. In Section V, we provide theoretical

results describing the stationary regime of the system. Finally,

simulation experiments are presented in Section 5 which

demonstrate that the necessary stability condition is close to be

the stability criterion of the system with server interruptions.

II. THE SYSTEM MODEL

We consider a single server multiclass retrial system accept-

ing Nclasses of customers according to Poisson independent

inputs. The service station can handle at most one customer,

and class-icustomer arrives according to Poisson process with

rate λi,i =1,...,N. Denote λ=iλi,the total input

rate and let pi=λi/λ, be the probability that an arbitrary

arrival belongs to class i=1,...,N. Then, interarrival times

of the input are exponential with rate λ. We also assume

service times of class-icustomers, {S(i)

n,n ≥1},tobe

independent identically distributed (iid) with with cumulative

distribution function (cdf) Bi, probability density function

(pdf) bi, Laplace-Stieltjes Transform (LST) β∗

iand service

rate γi,i=1,...,N.

This model is characterized by the following features: the

server is subject to class-dependent interruptions, following by

also class-dependent setup periods. In particular, we assume

that during the service time of a type-icustomer, interruptions

occur according to a Poisson process with rate νi,i=1,...N.

Upon service interruption, the server becomes unavailable

for an arbitrarily distributed time R(i)(called setup time) with

pdf ri, LST r∗

iand the mean ER(i)=¯ri,i=1,...,N. During

setup time customers continue to arrive at the system and join

the corresponding orbits. The interrupted customer remains in

the service area to return for service after the setup period.

We consider both preemptive repeat different, and preemp-

tive resume service discipline. In the former case, the service

time of the preempted customer begins again from scratch, but

each time another interruption is cleared, i.e., upon setup time

completion, a new independent (potential) service time begins.

Thus, a customer departs from the system when, for the ﬁrst

time, such a service time elapses without interruption. In the

latter case, upon the setup completion, the customer resumes

his service from the interruption point. In both cases we call

ageneralized service time the total time a customer spends in

the service area [12].

As in [16], [18], [19], we also adopt a novel class-dependent

queue-aware constant retrial policy which is described as

follows. A customer, meeting server busy upon arrival, joins

a class-dependent FIFO (First In First Out) orbit queue. The

customer at the head of orbit irepeats his attempt until he ﬁnds

server idle to occupy it. The time between successive attempts

from each orbit iis exponentially distributed with a rate which

depends on the current conﬁguration of other orbits: busy or

empty. Thus, each orbit acts as a FIFO queueing system with

a state-dependent ”service” rate, and this dependence is a key

property of the model.

To be more precise, for each i, we deﬁne the set G(i)of

N-dimensional vectors

J(i)

n={j(i)

n,1,...,j(i)

n,i−1,1,j

(i)

n,i+1,...,j(i)

n,N }

with binary components j(i)

n,k ∈{0,1},ifk=i, while the

ith component always equals 1, j(i)

n,i =1. Each vector J(i)

n

is called conﬁguration and has the following interpretation: if

the kth orbit is non-idle (busy), we put j(i)

n,k =1, otherwise,

j(i)

n,k =0. We assume that G(i)is the ordered set of possible

conﬁgurations, then index ndenotes the nth element of this

set. We stress that each conﬁguration from the set G(i)relates

to the case when orbit iis non-idle. For a given conﬁguration

J(i)

n, we denote μ(i)

nthe retransmission rate of orbit i. Finally,

we denote Mithe set of rates for all conﬁgurations belonging

to the set G(i),i=1, ..., N .

We recall that this construction has been proposed in [16],

and it generalizes the setting studied in previous works [18],

[7], [8], [9], where it is assumed that orbit ihas rate μi

if at least one (other) orbit is busy, otherwise, the rate is

μ∗

i,i=1,...,N. In this case each the set G(i)contains only

two conﬁgurations, J(i)

1,J

(i)

2, such that maxk=ij(i)

1,k =0, and

maxk=ij(i)

2,k =1. Moreover, in this case Mi={μ∗

i,μ

i}.

Note that in general case the capacity of the Miequals

2N−1, and thus the number of possible conﬁgurations of the

system equals N2N−1. Also we note that, in general, not

all different conﬁgurations have different retrial rates, and it

means that the input rate, from a ﬁxed orbit, is insensitive

to switching between these conﬁgurations. It reﬂects such

situation in practice when the effect of interference caused

by the change of the state (idle/non-idle) of some orbits turns

out to be negligible for the analysis of the system.

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III. CALCULATION OF THE MEAN GENERALIZED SERVICE

TIME

Clearly, due to the presence of interruptions, the ”service”

time of a customer will deﬁnitely be extended. To cope with

this issue, we introduce the concept of the generalized service

time, as the time elapsed from the instant a customer starts his

service until the instant the server is ready to start the service

of another customer. Denote C(i)

nthe generalized service time

of the nth arriving customer, who belongs to class i, and let

C(i)be the generic generalized class-iservice time. Then,

{C(i)

n,n≥1}are the iid random variables, with rate

γi:= 1

EC(i)<∞,i=1,...,N.

Similarly as in [16], denote Ai(t)the number of class-i

customers arriving in the interval [0,t]. Then, the work which

class-icustomers bring in the system in the interval [0,t]

equals,

Vi(t):=

Ai(t)

n=1

C(i)

n,(1)

and the summary amount of the work that arrive in the system

during [0,t]equals,

V(t):=

N

i=1

Vi(t)=

N

i=1

Ai(t)

n=1

C(i)

n,t≥0.(2)

Before proceeding further, we provide some expressions for

the generalized service times that are necessary in the follow-

ing.

A. Preemptive repeat different service discipline

We ﬁrst consider the preemptive repeat different setting.

Denote the density c(i)(t)of the generalized service time of a

class-icustomer, that is,

PC(i)∈(t, t +dt)=: c(i)(t)dt.

Then, denoting the tail distribution ¯

Bi=1−Bi, we obtain

the following expression for this density:

c(i)(t)=e−νitbi(t)+νie−νit¯

Bi(t)∗ri(t)∗c(i)(t),(3)

where ∗means convolution.

The 1st term in the right-hand side of (3) corresponds to

the case when no interruption occurs during the service time

of a class-icustomer, and thus, the generalized service times

of a type icustomer equals his service time. The 2nd term

in the right-hand side of (3) describes the case where an

interruption occurs at some instant u<tduring the service

time of a class-icustomer. Then, the server is immediately

stop working, and becomes unavailable for a setup period

of type i. The interrupted class-icustomer remains in the

service area, awaiting the server to return after the setup. Then,

the interrupted customer starts his service from scratch, and

thus, a new generalized service time begins. This situation is

continued until the instant a service time is completed without

interruption.

Note that in this case the clear service time (i.e., the time

that a customer spends receiving service) of a customer is

extended as many times an interruption occurs. In particular,

each time an interruption occurs, it cancels the service time a

customer already received until this instant, and after a setup

period, the customer has to repeat from scratch his service.

Denote the LST of the generalized service time

c∗

i(s):=∞

0

e−stc(i)(t)dt, s > 0.

Then, using (3) and the property of the LST of the convolution,

we obtain, after some algebra,

c∗

i(s)= β∗

i(νi+s)

1−νi(1−β∗

i(νi+s))

νi+sr∗

i(s).(4)

Using (4), we derive both the mean generalized service time

EC(i)=−d

dsc∗

i(s)|s=0 =1−β∗

i(νi)

β∗

i(νi)1

νi

+¯ri,(5)

and the trafﬁc intensity of class-icustomers:

ρi=λiEC(i),i=1,...,N. (6)

B. Preemptive-resume service discipline

In this case, the clear service time that a customer receives,

is equal to his original service time. Contrary to the case

discussed previously, the original service time of a customer is

extended to a time period which is equal to the summary setup

period (i.e., the sum of the setup periods that occur during a

service time). Then, we obtain the following expression for the

density of the generalized service time, for each i=1,...,N:

c(i)(t)=e−νitbi(t)+bi(t)

∞

n=1

e−νit(νit)n

n!∗r(n)

i(t),(7)

where r(n)

i(t)denotes the nth convolution of the density

ri(t)with itself. The 2nd term in the right-hand side of (7)

corresponds to the case when, during the clear (original)

service time of a class-icustomer, exactly ninterruptions

occur (with the probability e−νit(νit)n

n!within time interval

of the length t), and then we take into account these nsetup

times by means of the convolution r(n)

i(t).

By applying similar arguments as above, we can easily

obtain the following expression for the LST of the general-

ized service time density for the preemptive-resume service

discipline:

c∗

i(s)=β∗

i(νi(1 −r∗

i(s)) + s).(8)

Now, by a standard way, we obtain the mean generalized

service time:

EC(i)=−d

dsc∗

i(s)|s=0 =ES(i)(1 + νi¯ri),(9)

ρi=λiEC(i),i=1,...,N.

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Remark 1. One can give an intuitive explanation of the

expression (9) rewritten in the following form:

EC(i)=ES(i)+νiES(i)¯ri.

The 1st term is evident, while the 2nd term shows that the

average number of the interruptions during original service

time equals νiES(i)(by the Wald’s identity), and this quantity

then is multiplied by the average duration of the setup time

¯ri. Thus the term νiES(i)¯riequals an extra time caused by

the interruptions. Moreover, we note that expression (9) is

consistent with the case when the server is reliable, that is

when the interruption rate νi=0for each i, because in this

case (9) becomes EC(i)=ES(i).

IV. PERFORMANCE ANALYSIS

As in [16], we apply the regenerative approach to obtain the

stationary performance measures or the corresponding bounds.

A remarkable feature, demonstrating the power and ﬂexibility

of the regenerative approach, is that, in this new setting, we

can easily establish the performance results presented below

by the same arguments that have been applied in works [18],

[16] for the systems with reliable server. By this reason we

omit the corresponding details of the proof.

Let N(t)=(N1(t),...,N

N(t)) be the N-dimensional

vector describing the number of customers in the

corresponding orbit queues at instant t. Denote Wi(t)

the remaining work in orbit i, at instant t−, and consider the

process X(t):=N(t)+Q(t),t≥0, where Q(t)∈{0,1}

denotes the number of customers in the server, and

N(t)=iNi(t)is the summary size of all orbit queues, at

instant t−. The process X=: {X(t),t≥0}is regenerative,

and its regenerations are deﬁned as the arrival instants when

customers see an idle system. As in [16], denote generic

regeneration period Tand recall that the system is stable

if the process Xis positive recurrent that is the mean

regeneration period if ﬁnite, ET<∞.

Remark 2. It is worth mentioning that the instances {Tn}

are also regenerations of the multidimensional queue-size

process {N(t)}. However, instead of this vector process, we

study much more simple regenerative process X, and this

dimension reduction is a key advantage of the regenerative

approach.

Recall that due to service interruptions the server is occu-

pied by a customer for a generalized service period. Denote

ˆ

Bi(t)the summary time, in the interval [0,t], when the

server is occupied by class-icustomers, and let Wi(t)be

the remaining workload of class-icustomers in the system

at instant t−. As in [16], we can deduce from the balance

equation Vi(t)=Wi(t)+Bi(t)for the summary work Vi(t)

generated by class-icustomers in the interval [0,t], that the

stationary busy probability P(i)

bthat the server is occupied by

class-icustomers is

lim

t→∞

ˆ

Bi(t)

t=P(i)

b=ρi,i=1,...,N. (10)

Analogously, we obtain the stationary busy probability of the

server as the limit w.p.1,

lim

t→∞

V(t)

t=

N

i=1

ρi=Pb=: ρ. (11)

We emphasize that the stationary busy probability of the

server Pbin this setting includes the time when the server is

blocked because of the interruptions.

Now we introduce the maximal and the minimal possible

retrial rate from orbit i:

ˆμi=max

J(i)

n∈G(i)

μ(i)

n,μ

0

i=min

J(i)

n∈G(i)

μ(i)

n,i=1,...,N.

Then, analysis developed in [16] can be extended to our model

to establish the following statement.

Theorem 1. The necessary stability condition of the system

under consideration is

i

ρi<min

1≤i≤Nˆμi

λi+ˆμi.(12)

In the next section we also verify the following sufﬁcient

stability condition obtained in [18]:

N

i=1

ρi+max

1≤i≤N

λ

μ0

i+λ<1.(13)

These conditions have been proved for a less general model

with two-state conﬁgurations [18], but they can be readily

extended to the current model [19].

We also note that, exactly as in [16], one can establish the

following bounds of the stationary probability P(i)

0that the

server is idle and orbit iis busy:

λi

ˆμi

ρ≤P(i)

0≤λi

μ0

i

ρ, i =1,...,N. (14)

In the next section, we verify by simulation stability conditions

(12) for a set of parameters for the system with 3 classes of

customers.

V. S IMULATION RESULTS

In this section we verify stability conditions contained in

Theorem 1 for a particular case of a 3-class system.

We simulate this type of system using discrete-event mod-

eling. In more detail, we consider the system only at such

time instants (called key instants), when one of the following

events occurs: arrival to the system, departure from the

system, retrial or interruption. We denote Zjthe instant when

the jth key instant occurs. There exist another events, say,

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service beginning instant or the setup time-over, but they match

with key instants. For instance, a service beginning instant

may happen only if a new customer meets free server, or a

customer makes a successful retrial. These instants are deﬁned

recursively as follows:

Zj+1 =Zj+mint(j)

A,t

(j)

D,t

(j)

R,t

(j)

I,j≥1,(15)

where t(j)

A,t

(j)

D,t

(j

R)and t(j)

Iare the time intervals since in-

stant Zjuntil the next arrival, departure, retrial or interruption,

respectively. (Z1is the ﬁrst arrival instant.)

Recall that there are two possible types of the server

interruptions in this system. In the ﬁrst scenario, called pre-

emptive repeat different service (REPEAT), the interrupted

class-icustomer starts the service over again, with the new

independent service time sampled from the given service time

distribution Bi. In the second scenario, preemptive-resume

service (RESUME), the class-iinterrupted customer continues

being served after each setup instant until he accumulates his

predeﬁned service time.

In all experiments, we study the dynamics of orbit sizes

Ni(t)in an exponential model vs. number of key instances

(x-axis), and verify the stability conditions for both types of

the interruptions. Moreover, in each experiment we consider

600 arrivals, and in summary sample 300 such independent

experiments. Then we average these observations (calculate

the sample mean estimate) to obtain smooth output plots.

The following retrial rates are used in all experiments:

M1=μ1

00 =20,μ

1

10 =30,μ

1

01 =20,μ

1

11 =25

,

M2=μ2

00 =20,μ

2

10 =30,μ

2

01 =20,μ

2

11 =25

,

M3=μ3

00 =20,μ

3

10 =30,μ

3

01 =20,μ

3

11 =25 .

(16)

Note that these parameters reﬂect a partial symmetry of the

system: the identical response of each orbit on the identical

conﬁgurations. However, in general it is not a symmetric

system, because not all corresponding parameters are identical

[19].

Denote 1/¯ri=αithe rate of the setup time when a class-i

customer is being served. Fig. 3 shows dynamics of the orbits

for the REPEAT interruptions model and the following input

parameters:

λ1=2,λ

2=5,λ

3=3,

γ1=10,γ

2=20,γ

3=13,

ν1=5,ν

2=5,ν

3=5,

α1=15,α

2=30,α

3=20.

To calculate the trafﬁc intensity ρ, we need to ﬁnd LST β∗

iin

formula (5). Note that for the exponential service time with

parameter γi, we obtain

β∗

i(νi)= γi

γi+νi

.

As a result, these parameters give ρ= 0.85, see (11). On

the other hand, it is easy to calculate, that the r.h.s. of

conditions (12) and (13), equals 0.9 and 0.8, respectively, so

only condition (12) hold while condition (13) is violated. As

we see, despite breaking the sufﬁcient condition, all orbits are

stable as shown at Fig. 3.

For the next experiment with the REPEAT interruptions

model, we take the following set of system parameters:

λ1=2,λ

2=5,λ

3=3,

γ1=5,γ

2=10,γ

3=15,

ν1=3,ν

2=3,ν

3=3,

α1=15,α

2=20,α

3=15.

This choice gives ρ= 1.4, which evidently violates both

necessary and sufﬁcient stability conditions. Note that this

result is mainly caused by the choice of the setup rates, which

are taken smaller than that in the 1st experiment, making setup

time, and hence generalized service time, longer. As a result,

Fig. 4 demonstrates that all orbits indeed become unstable

and grow approximately linearly. Moreover, this linear grow

is speciﬁc for each orbit, and it is mainly because of different

values of the trafﬁc intensities ρi=λi/γi.

The following simulation results describe the dynamics of

orbits with RESUME interruptions. In this case we use the

following system parameters:

λ1=3,λ

2=3,λ

3=3,

γ1=10,γ

2=20,γ

3=13,

ν1=5,ν

2=5,ν

3=5,

α1=15,α

2=30,α

3=20.

In this case we obtain ρ= 0.9, while the r.h.s of (12) equals

0.83. Thus, condition (13) is violated while condition (12) is

satisﬁed, and Fig. 1 indicates that dynamics of the orbits is

indeed similar to a stable dynamics observed on Fig. 3.

0 200 400 600 800 1000

0 5 10 15 20 25 30

t

N(t)

t

N(t)

t

N(t)

Fig. 1. RESUME mode: condition (13) is violated, condition (12) holds, all

orbits are stable

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Fig. 2. RESUME mode: both conditions are violated, all orbits are unstable

Finally, for the RESUME interruptions system with the

parameters

λ1=3,λ

2=3,λ

3=3,

γ1=5,γ

2=10,γ

3=15,

ν1=5,ν

2=5,ν

3=5,

α1=15,α

2=20,α

3=15,

we obtain ρ= 1.8. Thus, both stability conditions are violated,

and Fig. 2 reﬂects this property. It is also important to note

that, despite the orbits are non-symmetric, their dynamic

is quite similar. We suppose that it is because the trafﬁc

intensities are equal, ρi=0.6(while other parameters remain

different).

Fig. 3. REPEAT mode: condition (13) is violated, condition (12) holds, all

orbits are stable

It is worth mentioning that the results demonstrated by Fig.

3 and Fig. 1 seem to be unexpected, because in both cases

the sufﬁcient stability condition is violated, while all orbits

remain stable. This result has been ﬁrst detected in the work

[16] and it supports our conjecture that condition (13) is indeed

redundant and the necessary stability condition (12) is close

Fig. 4. REPEAT mode: both conditions are violated, all orbits are unstable

to be the stability criterion (or even is stability criterion) of

the system.

VI. CONCLUSION

In this work, we introduce a retrial model with coupled

orbit queues and two type of interruptions: preemptive-repeat

different service, and preemptive-resume service. In this sys-

tem, a new customer meeting server unavailable joins the

corresponding inﬁnite capacity orbit. The retrial rate from orbit

idepends on the current conﬁguration of other orbits: busy

or idle, which gives rise to a novel class-dependent, queue-

aware constant retrial policy. Service interruptions occurs

according to a class-dependent Poisson process following a

class-dependent setup periods. For both types of the models,

we formulate and verify by simulation the stability conditions.

These conditions have been proved earlier in our previous

works [18], [16] for the system with reliable server. But they

are readily extended, again by the regenerative approach, to the

system with interruptions. This research veriﬁes by simulation

that the necessary stability conditions indeed are stability cri-

terion for the model with coupled orbits and unreliable server

when setup times have class-dependent general distributions.

Moreover, this work again shows that regenerative approach

is a powerful method to analysis complicated models of the

modern communication systems.

For a future research it would be important to simulate

non-exponential models, however, in the model with REPEAT

service interruptions, it can be a hard problem to calculate the

LST of a non-exponential service time distribution present in

(5). Another goal of a future research is to verify stability

condition related to each orbit separately, that is, instead

of (12), verify condition ρ< ˆμi

λi+ˆμifor each orbit i.In

addition, we are planning to extend the observed model by

making it possible to switch between the interruption modes

depending on the customer’s class. It is very motivated setting,

for instance, for Windows and Unix-like operations system

because both of these modes are supported for multitasking,

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and the interruption type depends on which task has been sent

to the server.

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, 19-07-00303.

The research of TM is supported by Petrozavodsk State Uni-

versity and Russian Foundation for Basic Research, projects

18-07-00147, 19-07-00303.

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