Conference PaperPDF Available

A Multiclass Retrial System with Coupled Orbits and Service Interruptions: Verification of Stability Conditions

Authors:

Abstract and Figures

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 accepting N classes 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.
Content may be subject to copyright.
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
_
_____________________________________________________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,i1,1,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
i}.
Note that in general case the capacity of the Miequals
2N1, and thus the number of possible conﬁgurations of the
system equals N2N1. 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.
_
_____________________________________________________PROCEEDING OF THE 24TH CONFERENCE OF FRUCT ASSOCIATION
---------------------------------------------------------------------------- 76 ----------------------------------------------------------------------------
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,n1}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,t0.(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=1Bi, we obtain
the following expression for this density:
c(i)(t)=eνitbi(t)+νieνit¯
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
period, the customer has to repeat from scratch his service.
Denote the LST of the generalized service time
c
i(s):=
0
estc(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)
1νi(1β
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 =1β
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)=eνitbi(t)+bi(t)
n=1
eνit(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 eνit(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.
_
_____________________________________________________PROCEEDING OF THE 24TH CONFERENCE OF FRUCT ASSOCIATION
---------------------------------------------------------------------------- 77 ----------------------------------------------------------------------------
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),t0, 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),t0}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
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
1iNˆμi
λiμi.(12)
In the next section we also verify the following sufﬁcient
stability condition obtained in [18]:
N
i=1
ρi+max
1iN
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
ρP(i)
0λi
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,
_
_____________________________________________________PROCEEDING OF THE 24TH CONFERENCE OF FRUCT ASSOCIATION
---------------------------------------------------------------------------- 78 ----------------------------------------------------------------------------
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,j1,(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
1
10 =30
1
01 =20
1
11 =25
,
M2=μ2
00 =20
2
10 =30
2
01 =20
2
11 =25
,
M3=μ3
00 =20
3
10 =30
3
01 =20
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
2=5
3=3,
γ1=10
2=20
3=13,
ν1=5
2=5
3=5,
α1=15
2=30
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
2=5
3=3,
γ1=5
2=10
3=15,
ν1=3
2=3
3=3,
α1=15
2=20
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=λii.
The following simulation results describe the dynamics of
orbits with RESUME interruptions. In this case we use the
following system parameters:
λ1=3
2=3
3=3,
γ1=10
2=20
3=13,
ν1=5
2=5
3=5,
α1=15
2=30
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
_
_____________________________________________________PROCEEDING OF THE 24TH CONFERENCE OF FRUCT ASSOCIATION
---------------------------------------------------------------------------- 79 ----------------------------------------------------------------------------
Fig. 2. RESUME mode: both conditions are violated, all orbits are unstable
Finally, for the RESUME interruptions system with the
parameters
λ1=3
2=3
3=3,
γ1=5
2=10
3=15,
ν1=5
2=5
3=5,
α1=15
2=20
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
λiμifor 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,
_
_____________________________________________________PROCEEDING OF THE 24TH CONFERENCE OF FRUCT ASSOCIATION
---------------------------------------------------------------------------- 80 ----------------------------------------------------------------------------
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.
REFERENCES
[1] Artalejo, J.R.: Accessible bibliography on retrial queues: Progress in
2000-2009. Mathematical and Computer Modelling 51(9-10), 1071-1081
(2010)
[2] Artalejo, J.R., Gomez-Corral, A.: Retrial Queueing Systems: A Compu-
tational Approach. Springer (2008)
[3] Avrachenkov, K., Morozov, E., Nekrasova, R., Steyaert, B.: Stability
analysis and simulation of N-class retrial system with constant retrial
rates and Poisson inputs. Asia Pac. J. Oper. Res. 31(2), 1440002 (18
pages) (2014)
[4] Avrachenkov, K., Morozov, E., Steyaert, B.: Sufﬁcient stability condi-
tions for multi-class constant retrial rate systems. Queueing Syst. 82,
149-171 (2016)
[5] Bonald, T., Borst, S., Hegde, N., Proutiere, A.: Wireless data perfor-
mance in multi-cell scenarios, in: Proc. ACM Sigmetrics/Performance
’04, pp. 378-388, ACM, New York (2004)
[6] Bonald, T., Massouli´
e, L., Prouti´
ere, A., Virtamo, J.: A queueing anal-
ysis of max-min fairness, proportional fairness and balanced fairness.
Queueing Syst. 53(1-2), 65-84 (2006)
[20] Morozov, E., Delgado, R.: Stability analysis of regenerative queues.
Automation and Remote control 70, 1977-1991 (2009)
[7] Dimitriou, I.: A two class retrial system with coupled orbit queues. Prob.
Engin. Infor. Sc. 31(2), 139-179 (2017)
[8] Dimitriou, I.: A queueing system for modeling cooperative wireless
networks with coupled relay nodes and synchronized packet arrivals.
Perform. Eval. 114: 16-31 (2017)
[9] Dimitriou, I.: Modeling and analysis of a relay-assisted cooperative
cognitive network. In: N. Thomas, M. Farshaw (eds.), ASMTA 2017,
LNCS, Springer, Newcastle, UK (2017)
[10] Falin G.I., Templeton, J.G.D.: Retrial Queues. Chapman & Hall, London
(1997)
[11] Fayolle G., Malyshev V.A., Menshikov M.V.: Topics in the constructive
theory of countable Markov chains. Cambridge University Press (1995)
[12] Gaver, D.P.: A Waiting Line with Interrupted Service, Including Priori-
ties. Journal of the Royal Statistical Society. Series B (Methodological),
24 (1), 73-90 (1962).
[13] Kim, J., Kim, B.: A survey of retrial queueing systems. Annals of
Operations Research 247(1), 3-36 (2016)
[14] Liu, X., Chong, E., Shroff, N.: A framework for opportunistic scheduling
in wireless networks. Comp. Netw. 41, 451-474 (2003)
personal. IEEE Pers. Commun. 6(4), 13-18 (1999)
[16] E Morozov, T.Morozova. Analysis of a generalized system with coupled
orbits. Submitted for FRUCT23, 2018.
[17] Morozov, E., Phung-Duc, T.: Stability analysis of a multiclass retrial
system with classical retrial policy. Perform. Eval. 112: 15-26 (2017)
[18] Morozov, E., Dimitriou, I. Stability Analysis of a Multiclass Re-
trial System with Coupled Orbit Queues, Proceedings of 14th Euro-
pean Workshop, EPEW 2017, Berlin, Germany, September 7-8, 2017,
DOI10.1007/978 3319 66583 26
[19] Evsey Morozov, Taisia Morozova, and Ioannis Dimitriou. Simula-
tion of multiclass retrial system with coupled orbits. Proceedings of
SMARTY’18: First International Conference Stochastic Modeling and
Applied Research of Tech- nology Petrozavodsk, Russia, September 21-
25, 2018,pp. 6-16.
[21] Sadek, A., Liu, K., Ephremides, A.: Cognitive multiple access via
cooperation: Protocol design and performance analysis. IEEE Trans.
Infor. Th. 53(10), 3677-3696 (2007)
_
_____________________________________________________PROCEEDING OF THE 24TH CONFERENCE OF FRUCT ASSOCIATION
---------------------------------------------------------------------------- 81 ----------------------------------------------------------------------------
... Finally, we would like to mention a series of recent works devoted to regenerative stability analysis of the multiclass retrial systems with coupled orbits (or state-dependent retrial rates), being a far-reaching generalization of the constant retrial rate systems, in which the retrial rate of each orbit depends on the binary state (busy or idle) of all other orbits, see [24,25,29,26,27,28]. In particular, this analysis is based on PASTA and a coupling procedure connecting the real processes of the retrial with the independent Poisson processes corresponding to various 'configurations' of the (binary) states of the orbits. ...
Preprint
Full-text available
We establish stability criterion for a two-class retrial system with Poisson inputs, general class-dependent service times and class-dependent constant retrial rates. We also characterise an interesting phenomenon of partial stability when one orbit is tight but the other orbit goes to infinity in probability. All theoretical results are illustrated by numerical experiments.
Chapter
In this paper, we consider a retrial queuing system with unreliable servers and analyze the distribution of the stationary generalized service time which includes also the unavailable periods (setup times) occurring during service of the customer. We consider three service interruption disciplines: preemptive resume (PRS), preemptive repeat different (PRD), and preemptive repeat identical (PRI); and provide the stationary distribution of the generalized service time and the remaining generalized service time for these disciplines in Laplace transform (LT) domain.
Chapter
In this chapter, we study the stability of some important extensions of the classical GI/G/m system : a multiserver system with non-identical servers (that was already in part discussed in Chap. 3), a finite-buffer system, a system with an infinite number of servers, and a system with a regenerative input flow. The multiserver system with non-identical servers is difficult to analyze because it does not possess the monotonicity property and belongs to a special class of state-dependent systems where the service time of a customer depends on the assigned server.
Chapter
In this chapter, we extend the analysis developed in Chap. 7 to more complicated retrial systems, in which the retrial rate of each orbit depends on a binary state of the other orbits. This setting is well-motivated and suited for modelling wireless multiple-access systems, see Sect. 8.4.
Chapter
In this paper, we consider the LoRa technology to expand sensor network coverage in smart sustainable cities. A model of a LoRa mesh network is proposed using the AODV protocol in packet routing. With a simulation model developed based on OMNET++, a series of computer experiments was carried out with changing various parameters. In the experiments results, the end-to-end delay and packet loss ratio were analyzed in the dependence on the number of nodes and packet size in the network. The simulation results show that the latency is relatively high in the LoRa mesh network, but it might be accepted for some applications.
Chapter
In this paper, we consider a multiserver retrial queuing system with unreliable servers class-dependent retrial rates and N classes of customers following Poisson input processes. We analyze the distribution of the stationary generalized remaining service time which includes all unavailable periods (setup times) occurring during service of the customer. During service of a class-i customer, the interruptions occur according to the i-dependent Poisson process and the following i-dependent random setup time of the server. We consider two following disciplines caused by the service interruptions: preemptive repeat different and preemptive resume. Using coupling method and regenerative approach, we derive the stationary distribution of the generalized remaining service time in an arbitrary server. For each class i, this distribution is expressed as a convolution of the corresponding original service times and setup times, and in general is available in the terms of the Laplace-Stieltjes transform allowing to calculate the moments of the target distribution. Some numerical examples are included as well.
Conference Paper
Full-text available
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
Article
Full-text available
We study multi-class retrial queueing systems with Poisson inputs, general service times, and an arbitrary numbers of servers and waiting places. A class-i blocked customer joins orbit i and waits in the orbit for retrial. Orbit i works like a single-server (Formula presented.) queueing system with exponential retrial time regardless of the orbit size. Such retrial systems are referred to as retrial systems with constant retrial rate. Our model is not only motivated by several telecommunication applications, such as wireless multi-access systems, optical networks, and transmission control protocols, but also represents independent theoretical interest. Using a regenerative approach, we provide sufficient stability conditions which have a clear probabilistic interpretation. We show that the provided sufficient conditions are in fact also necessary, in the case of a single-server system without waiting space and in the case of symmetric classes. We also discuss a very interesting case, when one orbit is unstable, whereas the rest of the system is stable.
Article
Full-text available
Retrial queueing systems have been extensively studied because of their applications in telephone systems, call centers, telecommunication networks, computer systems, and in daily life. This survey deals with various retrial queueing models. The main focus of this survey is to show analytic results for queue length distributions, waiting time distributions, and tail asymptotics for the queue length and waiting time distributions. This survey also considers the stability analysis of retrial queueing models.
Conference Paper
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.
Conference Paper
In this work we consider a single-server system accepting $N$ types of retrial customers, which arrive according to independent Poisson streams. In case of blocking, type $i$ customer, $i=1,2,...,N$ is routed to a separate type $i$ orbit queue of infinite capacity. Customers from the orbit queues try to access the server according to the constant retrial policy. We consider coupled orbit queues. More precisely, the orbit queue $i$ retransmits a blocked customer of type $i$ to the main service station after an exponentially distributed time with rate $\mu_{i}$, when at least one other orbit queue is non-empty. Otherwise, if all other orbit queues are empty, the orbit queue $i$ changes its retransmission rate from $\mu_{i}$ to $\mu_{i}^{*}$. Such an operation arises in the modeling of cooperative cognitive wireless networks, in which a node is aware of the status of other nodes, and accordingly, adjusts its retransmission parameters in order to exploit the idle periods of the other nodes. Using the regenerative approach we obtain the necessary conditions of the ergodicity of our system, and show that these conditions have a clear probabilistic interpretation. We also suggest a sufficient stability condition. Simulation experiments show that the obtained conditions delimit the stability domain with remarkable accuracy.
Article
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.
Article
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.
Article
We consider a multiserver retrial queueing system with a renewal input, $K$ classes of customers, and a finite buffer. Service times are class-dependent, however, for each class, are independent, identically distributed (iid). A new class-$i$ customer joins the primary system (servers and buffer), otherwise, if all servers and buffer are full, he joins the class-$i$ (virtual) orbit, and attempts to enter the system after an exponentially distributed time with rate $\gamma_i,\,i=1,\ldots,K$. The retrial discipline is classical because the attempts of different orbital (blocked) customers are independent. We exploit the regenerative structure of the (non-Markovian) queue-size process (total number of customers in the primary system and in orbits) to develop the stability analysis. First we establish the necessary stability conditions, and then show that these conditions are sufficient for stability as well. These conditions coincide with the known stability conditions of a conventional multiclass multiserver system with {\it infinite buffer}. Our analysis covers also the model in which, when a server is free, it makes an outgoing call which occupies the server for a server-dependent random time. Also we extend the stability analysis to the retrial system with a feedback when a served customer returns to the corresponding orbit with a positive probability.
Article
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.