Stability analysis of two-class retrial systems with constant retrial rates and general service times

10/19/2021
by   Konstantin Avrachenkov, et al.
Inria
Mail.Ru Group
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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.

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1 Introduction

In this work, we consider a two-class retrial system with a single server and no waiting space associated with the server. If an incoming job finds the server busy, the job goes to the orbit associated with its class. The jobs blocked on a class-dependent orbit attempt to access the server after class-dependent exponential retrial times in FIFO manner. The jobs initially arrive to the system according to Poisson processes. The service times are generally distributed. The arrival processes as well as service times are class-dependent.

An interested reader can find the description of various types of retrial systems and their applications in the books and surveys [Falin1997, FalinSurvey, Artalejo, Artalejo1, Artalejo2008book]. Specifically, the multi-class retrial systems with constant retrial rate can be applied to computer networks [PEISarticle, Nain], wireless networks [dim2, dimi, dim3, dimitriou2019power, Dimitriou2020Valuetools, Tuan] and call centers [PhungDuc2016, PEVA].

Let us outline our contributions and the structure of the paper. After providing a formal description of the system in Section 2, we first establish equivalence in terms of stability between the original continuous-time system and a discrete-time system embedded in the departure instants, see Section 3.1. Then, in Section 3.2 we prove the stability criterion for our retrial system. In Section 3.3 we give an extension of the stability criterion to the modified system with balking, which is useful for modelling two-way communication systems. In Section 4 we characterize a very interesting regime of partial stability when one orbit is tight and the other orbit goes to infinity in probability. In particular, we show that in this regime, as time progresses, the original two-orbit system becomes equivalent to a single orbit system. Curiously enough, this new single orbit system gains in stability region due to the jobs lost at infinity. Namely, the stability of one orbit is attained in part due to a ‘displacement’ of the customers going from other (unstable) orbit, and it gives a new insight to the transience phenomena.

We mention the most related works in the next paragraph, leaving the detailed description of related works and the comparison of various stability conditions to Section 5. Finally, in Section 6 we demonstrate all theoretical results by simulations with exponential and Pareto distributions of the service times.

Stability conditions for the single-class retrial systems with constant retrial rates have been investigated in [Fayolle1986, Lillo, PEISarticle, KostiaUri, MMOR]. In [AMNS] the authors have established the necessary stability conditions for the present system that coincide with the sufficient conditions obtained here. In fact, the necessary conditions have been obtained for -orbit systems, with . We would like to note that the proof of the necessary conditions turns out to be much less challenging than the proof of sufficiency of the same conditions. In [Nain] the necessary and sufficient conditions have been established by algebraic methods for the case of two classes in a completely Markovian setting with the same service rates. In [dim2] the author, also in the Markovian setting, has generalized the model of [Nain] to the case of coupled orbits and different service rates. Then, the author of [Dimitriou2018EJOR] conjectured sufficient conditions for the two-class retrial system in the case of general service times. In [Questa2015], using an auxiliary majorizing system, the authors have obtained sufficient stability conditions for a very general multi-class retrial system with classes. Their sufficient conditions coincide with the necessary conditions of the present model in the case of homogeneous classes. Recently, the authors of [PEVA] have also obtained sufficient (but not generally necessary) conditions for the multi-class retrial system with balking. To the best of our knowledge, in this paper we for the first time establish stability criterion for the two-class retrial system with constant retrial rates and general service times. We credit this to the combination of the regenerative approach [MSbook2021] with the Foster-Lyapunov approach for stability analysis of random walks [FMM]. Finally, the concept of partial (local) stability has been studied in [Questa2015] in the context of retrial systems and in [FOSS]

in a more general context of Markov chains. In the present work, we use both approaches from

[Questa2015] and [FOSS] to obtain a refined characterisation of the phenomenon of partial stability in multi-class retrial systems.

2 System description

Consider a single-server two-class retrial queueing system with constant retrial rates. The system has two Poisson inputs with class-dependent rates and generic service times with distribution functions , . There is no waiting space but two orbits. Define the basic three-dimensional process

(1)

where if the server is busy at instant ( otherwise) and is the state (size) of orbit at instant . If an incoming customer of class- comes to the system and sees that the server is busy, he/she goes to the -th orbit. The class- customers retries from orbit- in FIFO manner with exponential retrial times with rate .

In general, the continuous-time process is not Markovian. Now let us construct a discrete-time process, embedded in the process at the departure instants, which turns out to constitute a Markov chain. Denote by the sequence of the departure instants, and let be the number of customers in orbit just after the -th departure, . Construct the following two-dimensional discrete-time process

(2)

It is easy to check that the process is a homogeneous irreducible aperiodic Markov chain (MC). Let us define the increments

(3)

and introduce the sequence of vectors

Then, the dynamics of MC dynamics is described by

(4)

where the distribution of depends on the value of only.

2.1 Transition probabilities of the embedded MC

Denote by and , the idle and busy periods of the server between the -th and the -st departures, respectively, . Thus, the -st departure instant can be recursively presented as

Then, let and be the corresponding generic times. Next, define by the event, when the -st customer in the server belongs to class-, . Note, that on the event , is distributed as service time . On the other hand, the distribution of depends on the state of the orbits: idle/busy. Now we consider all possible cases separately.


1. Both orbits are empty. In this case and the server stays idle until the next arrival. Thus, the idle period

is exponentially distributed with rate

and the mean .

Denote by the probability that customers join the -st orbit in the interval , provided that both orbits are empty at instant and the -st customer arriving to the server is from class , and let be the similar probability for the 2-nd orbit. Thus, for we can write

(5)
(6)

In fact, the following two subcases are possible:

  1. With probability

    (7)

    a -st class customer occupies the server and customers join the -st orbit, resulting in . Moreover, with probability

    (8)

    this customer, during the service, generates new class-2 orbital customers, resulting in, .

  2. With probabilities

    (9)

    a -nd class external arrival captures the server and we obtain and , respectively, with the above probabilities.


2. Only the -st orbit is empty. Note that in this case . Then consider the following cases.

  1. With probability

    (10)

    a -st class customer occupies the server and customers join the -st orbit, resulting in . Moreover, with probability

    (11)

    this customer, during the service, generates new class-2 orbital customers, that is, .

  2. With probabilities

    (12)

    a -nd class external arrival captures the server and we obtain and , respectively, .

  3. With probabilities

    (13)

    an orbital class-2 customer occupies the server and we obtain

    respectively, .


3. Only the -nd orbit is empty. In this case , and we have . Next we consider the following three possible cases.

  1. A class- external arrival occupies the server, class- customers join the 1-st orbit with probability

    (14)

    implying , and, simultaneously, class- customers join the orbit with probability

    (15)

    implying

  2. A retrial attempt from the -st class orbit was successful (a secondary customer occupies the server before the next external arrival) and and customers join class- and class- orbits with probabilities

    (16)

    respectively. As a result, we obtain

  3. The server becomes busy with the -nd class external arrival and with probabilities

    (17)

    we have

    respectively.


4. Both orbits are busy. In this case , and similarly to the above, we obtain , with probability

(18)

In the case of a successful class- retrial attempt, we have with probability

(19)

Similarly, with probability

(20)

and with probability

(21)

3 Stability criterion

3.1 Stability of the embedded MC and underlying continuous-time process

In this section we establish a connection between the notion of stability (ergodicity) of the embedded MC introduced in the previous section and the concept of positive recurrence, which is an analogous notion of stability for regenerative processes in continuous time. Although it seems quite intuitive that stability of the embedded MC implies the positive recurrence of the underlying continuous-time process and vice versa, it is instructive to give a formal proof of this fact.

Recall the definition of the basic three-dimensional process

where is the indicator function of the server occupancy at time instant and is the size of orbit . Denote by the arrival instants of the (superposed) Poisson process and let We stress that the new hat-notation reflects the fact that the discrete-time process in general is not a Markov chain and evidently differs from the original Markov chain obtained by embedding at the departure instants.

We recall that the process is called regenerative with regeneration instants defined recursively as

(22)

Note that the equality is component-wise. We note that represents the arrival instant of such a customer which meets the system totally idle in the th time. We assume that the 1st customer arrives in an empty system at instant . Such a setting is called zero initial state [MorozovDelgado], and the corresponding regenerative process is called zero-delayed [Asmus]. Denote by generic regeneration period (which is distributed as any difference ). Then the regenerative process is called positive recurrent if . Denote by the generic interarrival (exponential) time in the superposed Poisson input process, which has rate . Because the input is Poisson, the regeneration period is non-lattice. Then, the positive recurrence implies the existence of the stationary distribution of the process as and hence the stability of the system [Asmus]. To study stability, it is much more convenient to work with a one-dimensional process

counting the total number of customers in the system, which is regenerative with the same regeneration instants (22). In the following lemma we establish the equivalence between the stability of the embedded MC and the stability of the original continuous-time process for the case of zero initial state.

Lemma 1. The zero-initial state Markov chain is positive recurrent if and only if the process is positive recurrent, that is if .

Proof.

If the process is positive recurrent, then it follows by a regenerative argument [Asmus] that the stationary probability , the probability of the system being totally free, exists and is equal to

(23)

where denotes the generic inter-arrival time in the superposed Poisson input process. On the other hand, is the embedded discrete-time regenerative process with the regeneration instants

(24)

and denoting generic regeneration period of this discrete-time process. Namely, the generic regeneration cycle is given by . It is well known that the discrete-time length of the regeneration cycle is connected with the continuous-time length by the following stochastic equality [Asmus] (Chapter X, Propositions 3.1 and 3.2):

where is the -th inter-arrival interval and the summation index is a (randomized) stopping time. It then immediately follows from the Wald’s identity that

Note that, because , then implies , and vice versa. Thus we obtain that , and hence the positive recurrence of the basic process implies the positive recurrence of the process embedded (in the basic process) at the arrival instants. Conversely, implies as well.

It remains to connect the process with the embedded MC we studied above. Note that, as in (22), regenerations , defined in (24), are generated by the arrivals meeting empty system. On the other hand, represents both the number of arrivals and the number of departures from the system within a continuous-time regeneration period . Thus, is also generic regeneration period of the embedded MC . It is worth mentioning that, at each instant of time, the index of a customer which see empty system (and generating new regeneration period of the processes and ), differs from the index of a customer leaving empty system by no more than one. Thus we obtain that

Hence , and because the MC we consider is aperiodic and irreducible, then it is also ergodic. Thus we see that both concepts of stability (in continuous and discrete time) agree and the lemma hereby is proven. ∎

We note that, at the first sight, the equality seems rather surprising because relates to the MC while relates to the regenerative process which in general is not Markov.

3.2 Stability of the embedded Markov chain

The results of this section are based on the general stability conditions for two-dimensional MCs, obtained in [FMM].

Let us first introduce some additional notations for the embedded Markov chain (2) needed for the application of the results from [FMM]. Specifically, let us derive the drifts of the embedded MC in various regions of the state space.

Denote by the mean number of customers joining the class- orbit in the time interval , provided . Recall that and denote

Then it follows from (10)–(13) after a simple algebra that

(25)
(26)

Similarly, denote by the mean number of customers joining the class- orbit in the time interval , provided . Then by analogy with (25) and (26) from (14)–(17) we obtain

(27)
(28)

Continuing in the same way, we denote by the mean number of customers joining the class- orbit in the time interval , if , and from (18)–(21) we obtain

(29)
(30)

Our further analysis is based on Theorem A presented in the Appendix, a statement from [FMM]. Note, that in the general case, Theorem A is applicable under some additional technical conditions (see Appendix A), which hold automatically when the input is Poisson. Denote the total load coefficients by

Now we are in a position to state our central result.


Theorem 1. Two-class retrial system with constant retrial rates, Poisson inputs, general service times and exponential retrials is ergodic if and only if

(31)
Proof.

Note that the defined above drifts , correspond to the drifts used in the statement of Theorem A (see the Appendix).

First, let us consider the conditions for the case (a) of Theorem A. Specifically, the condition takes the following form

(32)

while the condition takes the form

(33)

The inequalities (32) and (33) can be further rewritten as

Next, the first condition in (77) becomes, after a tedious algebra (see Appendix B for details),

(34)

while the second condition in (77) can be transformed to

(35)

Now we can express in terms of load coefficients, the three ergodic cases (a.1), (b.1) and (c.1) of Theorem A (see inequalities (77), (79) and (81) in Appendix A) as follows:


Case (a.1)

(36)

Case (b.1)

(37)

Case (c.1)

(38)

Now our goal is to simplify ergodicity conditions (36)–(38). Towards this goal, we rewrite the system (36) in terms of functions of and , assuming other parameters fixed. The first pair of inequalities in (36) can be transformed to

(39)
(40)

while the pair of inequalities becomes

(41)

For fixed values , such that , the right hand sides of (39), (40) are the increasing linear functions of with a common point , where

(42)

The ergodic case (a.1), described by system (36), corresponds to the values of such that

(43)
(44)

Let us now show that the set of corresponding to (43) and (44) is non-empty as long as . Assume on contrary that

(45)

thus (43) is violated. The inequality (45) for linear increasing functions under conditions implies a similar relation for the coefficients in front of (see (39),(40)), that is

(46)

Multiplying both parts of (46) by , we obtain

which is equivalent to and yields a contradiction.

Now, similarly, we describe the stability regions (b.1) and (c.1), presented in (37) and (38), in terms of the functions and as follows:

Case (b.1):
Case (c.1):

Next, by combining the three cases, we conclude that the embedded MC is ergodic, if and , which is equivalent in fact to . Thus, the conditions (36)–(38) can be written as (31).

Moreover, we can also delimit the transience regions in terms of (see Theorem A in the Appendix). Figure 1 illustrates stability/transience regions in different cases for a fixed .

Figure 1: Stability/transience regions.

Note, if (31) is violated, then the basic MC is transient by Theorem A. In this case, using the proof by contradiction and regenerative approach, one can show that at least one component of this two-dimensional vector goes to infinity in probability, see for instance, [MorozovDelgado].

Thus (31) is a sufficient stability (ergodicity) condition. To show that this condition is also necessary, we refer to the paper [AMNS] where it is shown, in the adopted notation and with , that if -class retrial system with Poisson inputs is ergodic then

(47)

(Indeed, in paper [AMNS], we apply an equivalent notion positive recurrence in the framework of the regenerative approach, see Lemma 1 above.) We can rewrite (47) (for =2) as

implying

(48)

Thus (48) coincides with (31) and condition (31) is also the necessary stability condition. ∎

Remark. It follows from (48) that if the two-class retrial system under consideration is ergodic then


3.3 Stability of a system with balking

We can assume an extra feature in the system under consideration as follows. If a primary class- customer meets busy server, he joins the corresponding orbit with a given (balking) probability and leaves the system with probability . In this case, the stability condition of Theorem 1 transforms to

(49)

This is an immediate extension. Namely, taking into account balking policy, we redefine the transition probabilities (5), (6), and the statement (49) is then proved by the same arguments as Theorem 1. We note that this modification of the system is useful to model two-way communication systems, for more details see e.g., [Tuan, PEVA].

4 Partial stability

Let us now discuss an effect of partial stability to the best of our knowledge first discovered in [Questa2015]. In the case of two classes of customers, the statement of Theorem 4 from [Questa2015] can be qualitatively formulated as follows: under some (given below) conditions, class-1 orbit size stays tight while class-2 orbit increases unlimitedly in probability. (Of course, by the symmetry, this can be formulated for the opposite case, when the 2nd orbit is tight while the 1st orbit increases.) By the evident reason, this statement can be regarded as the case of partial stability.

The purpose of this section is firstly to show that, in terms of the embedded MC , the partial stability corresponds to the transient case (c.2) of Theorem A, i.e., and condition .

Secondly, by establishing a relation with a single-orbit system, we shall show how to describe the long run behaviour of the stable orbit.

Note that the stability conditions which correspond to transience case (c.2) can be defined in terms of the load coefficients as follows:

(50)
(51)
(52)

It is important to note that (50) can be written as

(53)

Now we show that, provided conditions (50) and (52) hold, then they imply condition (51), which turns out to be redundant.

Now we consider in detail inequalities (50)–(52) in all three possible sub-cases when .

Sub-case 1: . In this case it is convenient to rewrite conditions (50)–(52) as follows:

(54)
(55)
(56)

respectively. Next assume that the following relation holds between the r.h.s. of conditions (54) and (55)

(57)

After some algebra, (57) transforms to the inequality

which contradicts (56). Thus, we have

(58)

and inequality (54) implies inequality (55); or equivalently, (50) implies (51). Thus, the latter condition is redundant.

Sub-case 2: . In this case, by rewriting condition (51) as

(59)

we see that, since the right hand side is negative, the condition (51) always holds and hence is redundant in this sub-case.

Sub-case 3: . In this case conditions (54), (55) remain unchanged, while condition (56) becomes

(60)

As in sub-case 1, it is easy to check that, provided inequality (60) holds, then condition (58) holds as well, and thus condition (55); or equivalently, condition (51) is redundant again.

Consequently a pair of conditions and (or its analogues (50) and (52)) define the transience case (c.2). Before we formulate our next results, let us recall the definition of the failure rate

of a non-negative absolutely continuous distribution with density , defined for all such that . We say that a distribution belongs to class if its failure rate satisfies . (Some, fairly common, distributions satisfying this requirement can be found in [Questa2015].)


Theorem 2. If, in the initially empty system, conditions

(61)

hold and distribution of service time of class-k customers belongs to class then the 1-st orbit is tight and the 2-nd orbit increases in probability, that is .

Proof.

Recall notation

and also denote

(62)

Following [Questa2015], we consider an auxiliary two-class system with two Poisson inputs with rates and the same service times as in the original system. In this new system, any class- customer meeting server busy becomes “colored” and joins a virtual orbit being a part of an infinite class- queue, which in turn is a source of the Poisson input with rate . (For details see [Questa2015].) Then is the stationary “loss” probability in this auxiliary system, that is the probability that a customer meets server busy. It easily follows from [Questa2015] that class-k orbit size (the number of colored customers) in the auxiliary system stochastically dominates class-k orbit size in the original system, provided . Moreover, it is shown in [Questa2015] (Theorem 4 there) that, if the system is initially empty, and the following conditions hold: