Ergodicity and Perturbation Bounds for $M_t/M_t/1$ Queue with Balking, Catastrophes, Server Failures and Repairs

In this paper, we display methods for the computation of convergence and perturbation bounds for $M_t/M_t/1$ system with balking, catastrophes, server failures and repairs. Based on the logarithmic norm of linear operators, the bounds on the rate of convergence, perturbation bounds, and the main limiting characteristics of the queue-length process are obtained. Finally, we consider the application of all obtained estimates to a specific model.


Introduction
Recently, there has been a noticeable interest from researchers to study nonstationary queueing systems because they represent the actual reality of many applications in our life. Nevertheless, we find few works around these systems, as studying these systems needs new, unrecognized mechanisms to analyze their behavior.
In the current paper, we deal with nonstationary M/M/1 queue with balking, catastrophes, server failures and repairs. We investigate the bounds on the rate of convergence, and the perturbation bounds for the corresponding queue-length process. Such kinds of bounds give us the possibility for finding the limiting bounds for the class of close to this queue Markovian models. We apply the approach based on the notion of logarithmic norm of an operator function, see detailed description in our recent papers [2,14]. The motivation of the proposed system comes from having wide and many applications and contributions in many fields, one of them is the field of communication network systems. The applicability of this model can be seen in communication network systems. If there are numerous packets lined up in the system, local packets are always accepted and remote packets are less than a threshold value packets, waiting in the node for process. Then a new arrival either decides not to join the system or departs after joining the system. If this network was infected with a virus, this could lead to the loss of some packets as a result of restarting the network again, or transferring theses packets to another network. Also, in computer systems where there are several clients (data) lined up in the system until a certain threshold value, a new arrival may decide not to enter the system after that value. Additionally, if a virus-infected data will annihilate or transfer it to other processors. These systems can be described as queueing models with catastrophes and balking. These systems can be represented as proposed queueing model.
Most of the literature on the subject of the present paper focused solely on the study of stationary behavior, although this behavior is a special case, as well as the parameters in many applications are varying with time in our daily life. An example of some works that has discussed queueing systems related to the subject of this paper, we find that in [1] the author discussed the stationary behavior of a two-processor heterogeneous system with catastrophes, server failures and repairs. Kumar et al. [5] analyzed the stationary behavior for an M/M/1 queue with catastrophes, server failures and repairs. In [8] the author has extend work which has been done by Kumar et al. [5] for an M/M/1 queue with balking, catastrophes, server failures and repairs where balking occurs if and only if the system size equals or exceeds a threshold value k. Also, Suranga et al. [7] considered an M/M/1 queue with reneging, catastrophes, server failures and repairs, they obtained the explicit expression for the stationary probabilities. While in [3] Crescenzo et al. studied the stationary behavior of a double-ended queue with catastrophes and repairs.
On the other hand, we find some of the works that discussed the nonstationary behavior as in [2] Ammar et al. explored the nonstationary of a two-processor heterogeneous system with catastrophes, server failures and repairs. In [4] DiCrescenzo et al. construed the a time-non-homogeneous for double-ended queue subject to catastrophes and repairs, as this is an extension of their previous work in [3].
We note from the previous literature that no paper has discussed the behavior of the proposed model and based on this observation, in this paper we examine convergence bounds for a non-stationary behavior of the proposed system. In case of stat constant parameters, our results are consistent with those found by Tarabia [8].
The paper is organized as follows. In Section 2 and Section 3, description of the model and basic notions are introduced. In Section 4 and Section 5, general theorems on the rate of convergence and perturbation bounds are considered, respectively. Finally, in Section 6, a specific queueing example is studied.

Model Description
Consider a single server queueing system with balking, catastrophes and server failure and repairs. The arrival process of customers is Poisson process with mean arrival rate λ during times that the server is working. Assume that the customers are served on first-come, first-served discipline with the service time following an exponential distribution with mean 1/µ . On arrival a customer either decides to join the queue with probability one if the number of customers in the system is less than a threshold value k. If there are k customers or more ahead of him, then he joins the queue with probability β and may balk with probability 1 − β . The capacity of the system is infinite. When the system is idle or busy, catastrophes occur at the service station according to Poisson process of rate γ . Whenever a catastrophe occurs at the busy server all customers in the system are destroyed immediately and the server gets inactivated, i.e. the server is subject to failure and has to repair it. The repair times of failed server are i.i.d, according to an exponential distribution with parameter η . After repairing the server, the server becomes ready to serve new customers. Let r(t) be the probability that the server is under repair at the instant t with r(0) = 0.
It is easy to see that the given model can be described by Markov process X(t), t > 0 where X(t) denotes the number of customers in the system at time t (queue-length process). Denote by p n (t) = P (X(t) = n), n = 0, 1, 2, 3, . . . .
From the above assumptions the resulting behavior of the state probabilities are described by a forward Kolmogorov system: Now we consider the corresponding nonstationary situation. Namely, we suppose that the queue-length process {X(t), t ≥ 0} be an inhomogeneous continuous-time Markov chain. All possible transition intensities, say q ij (t), are suppoesed to be non-random functions of time. We suppose that all intensity functions are nonnegative and locally integrable on [0, ∞).

Basic Notions
Denote by p (t) = (r(t), p 0 (t) , p 1 (t) , . . . ) T the vector of state probabilities at the We will supposed through the paper that the following assumption of boundedness holds: for almost all t ≥ 0.
Denote by · the l 1 -norm of vector, , and denote by Ω the set of all vectors from l 1 with nonnegative coordinates and unit norm. We have A(t) = 2 sup k |a kk (t)| ≤ 2L for almost all t ≥ 0.
Then we can consider the forward Kolmogorov system (2.1)-(2.5) as a differential equation in the space of sequences l 1 , where A (t) is a bounded for almost all t ≥ 0 linear operator from l 1 to itself and it is generated be the corresponding transposed intensity matrix: Recall the following definitions: In this situation any p 1 (t) can be considered as a quasi-stationary distribution of the chain X (t).
Recall also that the logarithmic norm of operator function from l 1 to itself is calculated by the formula holds for the Cauchy operator of the corresponding differential equation 4 Bounds on the rate of convergence As we noted earlier, our method based on the notion of logarithmic norm and the corresponding bounds for the Cauchy operator. Moreover, for the considered model we can use the both approaches of [2,14]. Describe briefly these approaches.
First approach, see [2]. Rewrite the forward Kolmogorov system (3.2) as , and The solution of this equation can be written in the form where U * (t, s) is the Cauchy operator of the corresponding homogeneous equation Then the queue-length process X (t) is weakly ergodic in the uniform operator topology and the following bound hold for any initial conditions p * (0) , p * * (0) and any t ≥ 0.
Proof. The statement follows from the equality hence we obtain for any initial conditions p * (0) , p * * (0) and any t ≥ 0.
Consider (4.1) as a differential equation in the space of sequences l 1D . We have hence the operator function A * (t) is bounded on the space l 1D and we can apply the same approach to equation (4.1) in the space l 1D . Now we obtain 12) and the following statement.
Then X(t) is weakly ergodic and the following bound on the rate of convergence holds: for any initial conditions p * (0), p * * (0) and any t ≥ 0.
Let l 1E = {p = (r, p 0 , p 1 , p 2 , . . .)} be a space of sequences such that p 1E = k≥0 k|p k | < Corollary 1. Let a sequence {d i } be such that (4.13) holds and W > 0. Then X(t) has the limiting mean, say φ(t) = E(t, 0), and the following bound holds: for any j and any t ≥ 0.
Consider bounds for our model in more details.
Then we have: Hence, in (4.12) we have and we obtain instead of estimates (4.14) and (4.15) the following bounds: and for the corresponding l 1D and W .
Consider now the reduced forward Kolmogorov system (4.1) in the form where f (t) = (η(t)r (t) , 0, 0, . . . ) T , z (t) = (p 0 (t), p 1 (t), . . . ) T , and The solution of equation (4.20) can be written in the form where U B (t, s) is the Cauchy operator of the corresponding homogeneous equation Note that the uniform estimate is completely analogous to Theorem 1, only with the replacement on the left side of p by z.
A significantly different situation with this approach arises when obtaining weighted estimates. Put Consider (4.20) as a differential equation in the space of sequences l 1D . We have hence the operator function B(t) is bounded on the space l 1D , and we can apply the same approach to equation (4.20) in the space l 1D . Now the equality implies the following statement.
Then X(t) is weakly ergodic and the following bound holds: for any initial conditions z * (0), z * * (0) and any t ≥ 0.
Corollary 2. Let a sequence {d i } be such that (4.27) holds, and W > 0. Then X(t) has the limiting mean, say φ(t) = E(t, 0), and the following bound holds: for any j and any t ≥ 0.
Consider bounds for our model in more details.
Then we have: Hence, in (4.26) we get and we obtain instead of (4.28) and (4.29) the following bounds: and Remark. In all our statements, we can replace the condition of monotonicity of the sequence {d k } by condition d = inf k d k > 0, with the corresponding change in the estimates; see, for example, [16].

Perturbation bounds
Consider here the application of general perturbation bounding (see the recent review in [15]) for the models under study. Consider a "perturbed" queue-length processX(t), t ≥ 0 with the corresponding transposed intensity matrixĀ(t), where the "perturbation" matrix A(t) = A(t) −Ā(t) is small in a sense. Namely, we assume that the perturbed queue is of the same nature as the original one. Hence, the perturbed intensity matrix also has the same structure, with the corresponding perturbed intensitiesη(t),γ(t),λ(t),μ(t),β(t). Let Hence |λ(t)β(t) −λ(t)β(t)| ≤ λ(t)|β(t)| +β(t)|λ(t)| ≤ (L + 1)ˆ . Firstly we formulate the perturbation bounds for the vector of state probability in the situation of Theorem 1.
The next statement follows immediately from Theorem 1 [15] (see also the first corresponding homogeneous result in [6] and for inhomogeneous situation in [9]). for any perturbed queue with the respectively closed intensities satisfying to (5.1).
Stability bound from Theorem 2 is based on results of [11,12].
Then Theorem 4 from [12] imply the next statement.
Theorem 5. Let under assumptions of Theorem 2 the following estimates hold: for some positive N * * , γ * * 0 , and H = sup . (5.10) Finally, we obtain perturbation bounds based on the ergodicity estimates of Theorem 3.

Theorem 6.
Let under assumptions of Theorem 3 the following estimates hold: . . (5.14) Proof. It is sufficient to note that and Then our claim follows from Theorem 2 [15].

Numerical Example
In this section, we will review a numerical example to support the results obtained as well as clarify the nature of the behavior of the proposed system. Where through this example, we will apply the analytical results obtained in the theories corollaries in the previous sections.
Apply all our bounds for this specific situation.
For Theorem 1 and the respective "stability" Theorem 4 we need L, N and γ 0 . Obviously we have L ≤ 25.5. Consider now hence one can put N = 2 and γ 0 = 2 in (5.4).
Now we obtain the following bounds on the rate of convergence: from Theorem 1; from Theorem 2 and Corollary 1, and almost the same from Theorem 3 and Corollary 2.
The corresponding perturbation bounds are: One can note that for this model bounds from Theorems 3 and 6 are worse because of the matrices B(t) and B * (t) have very special structure.
Further we follow the method that was described in [10,13] in detail. Namely, we choose the dimensionality of the truncated process (200 in our case), the interval on which the desired accuracy is achieved ([0, 20]) in the example under consideration) and the limit interval itself (here it is [19, 20]), expose the plots of the expected number of customers in the system and some most probable states. Figures 1-4 shows us the behavior of the probability of the empty queue and the mean respectively. In Figures 5-6 one can see the perturbation bounds for the corresponding limiting characteristics withˆ = 10 −3 for bound (6.4) andˆ = 10 −6 for (6.6).