Upper bounds on the rate of convergence for constant ...

Stat Papers https://doi.org/10.1007/s00362-018-1014-0 REGULAR ARTICLE

Upper bounds on the rate of convergence for constant retrial rate queueing model with two servers Yacov Satin1 · Evsey Morozov2 · Ruslana Nekrasova2 · Alexander Zeifman1,3 · Ksenia Kiseleva1 · Anna Sinitcina1 · Alexander Sipin1 · Galina Shilova1 · Irina Gudkova4

Received: 1 January 2018 / Revised: 29 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract The paper deals with a Markovian retrial queueing system with a constant retrial rate and two servers. We present the detailed description of the model as well as establish the sufficient conditions for null ergodicity and strong ergodicity of the corresponding process and obtain the upper bounds on the rate of convergence for both situations. Keywords Retrial queueing system · Constant retrial rate · Two-server model · Rate of convergence Mathematics Subject Classification 60 K 25

1 Introduction In this paper, we consider a M/M/2/2-type retrial queueing system with a constant retrial rate denoted as Σ. The exogenous (primary) customers arrive to the system according to a Poisson process with rate λ. The system has two stochastically identical

The research of EM and RN was supported by RFBR, Projects 18-07-00147, 18-07-00146, the research of YS, AZ, KK, AK, AS, GS was supported by RFBR. Project 18-47-350002. The research of IG has been prepared with the support of the “RUDN University Program 5-100”.

B

Alexander Zeifman [email protected]

1

Vologda State University, Vologda, Russia

2

IAMR, KarRC RAS and Petrozavodsk State University, Petrozavodsk, Russia

3

IPI FRC CSC RAS; VolSC RAS, Moscow, Russia

4

Peoples’ Friendship University of Russia (RUDN University), IPI FRC CSC RAS, Moscow, Russia

123

Y. Satin et al.

servers with i. i. d. exponential service times, generic service time S ,and rate μ := 1/E S. The system has the following special feature: if a primary customer finds both servers busy it goes to some kind of infinite-capacity repository (the so-called orbit that can be considered as a FIFO queue). Let us also assume that the orbit works as a single FIFS server, in which a head line (the oldest) secondary customer tries to enter the arbitrary server after an exponentially distributed time with (orbit) rate μ0 . (It returns to the orbit if both servers are busy). Since each secondary costumer has (potentially) an infinite number of attempts to enter servers, the system has no losses. Note that the orbit size (number of secondary customers) does not affect to orbit rate μ0 . This is the main difference with classical retrial models with the orbit rate proportional to the orbit size where all orbital customers make attempts independently (in general). It follows that the orbit in the system Σ can be interpreted as a single-server •/M/1-type queue with service rate μ0 , in which the jobs rejected from the primary system (servers) constitute an additional input to the primary system. Note also that due to the fact that we assume bufferless systems, the only subject of the (potential) instability of such a system is an infinite growth of the orbit size. This model is applied to analyze the multi-access protocol ALOHA (with some restrictions). We overview some related papers to motivate the system Σ. First, a single-server retrial queue with constant retrial rate was suggested by Fayolle (1986) and used to model a telephone exchange system. Then this model was extended in Artalejo (1996), Artalejo et al. (2001) to the system with multiple servers and waiting places. The closed retrial models have been studied in Efrosinin and Sztrik (2011a, b, 2016). We note that in the paper (Artalejo et al. 2001), the performance and stability analysis of the model was preformed by means of matrix-analytical method for c ≥ 2 servers, and the stability criterion was derived in an explicit form depending on input parameters λ, μ, μ0 . Recently, the retrial model with constant retrial rate has been further studied in much more general way including multiserver case with general service times and multiple classes of customers, see Avrachenkov et al. (2014, 2016), Morozov and Phung-Duc (2017). Nevertheless, the above mentioned researches assume the positive recurrence of the process (in general regenerative, not necessary Markovian) and as a result the existence of the stationary distribution. The fact that the speed of convergence of the considered model can be estimated by the exponent follows from the book of Thorisson (Chap. 10, Sect. 7, Th. 7.4). However, none of these papers analyze the rate of convergence to stationarity. Unlike the mentioned works, this paper presents the rate of convergence to stationarity of the corresponding Markov process describing the two-server retrial model, and thus extends the similar analysis developed for the simplest M/M/1-type retrial model in Zeifman et al. (2016). This result is the main contribution of the present paper. The retrial queueing systems with constant retrial rate could be applied for for a wide range of applications. For example, it has been used to model TCP traffic originated from short HTTP connections in Avrachenkov and Yechiali (2008, 2010), or unslotted Carrier Sense Multiple Access with Collision Detection (CSMA/CD) protocol in Choi et al. (1992). The model also appears to be relevant for the optical-electrical hybrid contention resolution scheme for Optical Packet Switching (OPS) networks (Wong et al. 2009;Yao et al. 2002), and it is also applicable in logistics (Lillo 1996).

123

Upper bounds on the rate of convergence

2 Model description Now we describe the model in details. For instant t, let ν(t) be the number of customers in the servers and N (t) be the number of customers in the orbit. That is, ν(t) = 0 if both servers are empty, ν(t) = 1 if only one server is busy, and ν(t) = 2 if both servers are busy, while N (t) = 0, 1, . . . . We introduce the basic two-dimensional process {ν(t), N (t), t ≥ 0} with the state space {0, 1, 2} × {0, 1, 2, . . . }. To describe the transitions between the system states we first enumerate the states as follows: each state {0, n} will be denoted as 3n + 1 for n ≥ 0, each state {1, n} will be denoted as 3n + 2for n ≥ 0, and each state {2, n} will be denoted as 3n + 3 for n ≥ 0. Let Q = qi j be the transition matrix corresponding to the given enumeration. Then it follows that q1,1 = −λ, q1,2 = λ, q2,1 = μ, q2,2 = −(λ + μ), q2,3 = λ,

and, for n ≥ 0, q3n+3,3n+2 = 2μ, q3n+3,3n+3 = −(λ + 2μ), q3n+3,3n+6 = λ, q3n+4,3n+2 = μ0 , q3n+4,3n+4 = −(λ + μ0 ), q3n+4,3n+5 = λ. q3n+5,3n+3 = μ0 , q3n+5,3n+4 = μ, q3n+5,3n+5 = −(λ + μ0 + μ), q3n+5,3n+6 = λ. The transition rate diagram associated with the system Σ is presented on Fig. 1, where the horizontal axis illustrates the number of customers in the servers and the vertical axis illustrates the orbit size.

3 Auxiliary notions Under the standard suppositions, the probabilistic dynamics of the process is represented by the forward Kolmogorov system:

123

Y. Satin et al. Fig. 1 Transition rate diagram

dp = Ap , dt

(1)

where A = (ai j )i,∞j=1 = (q ji )i,∞j=1 = Q T is the corresponding transposed intensity matrix: A⎛= −λ μ 0 0 0 0 0 ⎜ λ −(λ + μ) 0 0 0 2μ μ 0 ⎜ ⎜ 0 0 0 λ −(λ + 2μ) 0 μ0 ⎜ ⎜ 0 ) μ 0 0 0 0 −(λ + μ 0 ⎜ ⎜ 0 0 0 λ −(λ + μ0 + μ) 2μ μ0 ⎜ ⎜ 0 0 λ 0 λ −(λ + 2μ) 0 ⎜ ⎜ 0 0 0 0 0 0 −(λ + μ0 ) ⎜ ⎜ 0 0 0 0 0 0 λ ⎜ ⎜ 0 0 0 0 λ 0 ⎜ 0 ⎝ . . . . . . .. .. .. .. .. .. .. .

··· ··· ··· ··· ··· ··· ··· ··· ··· .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

and p = p(t) = ( p1 (t), p2 (t), . . .)T is the vector of state probabilities

for process -norm, i.e. x = | xi | and X (t). Throughout the paper , we denote by · the l 1

B = sup j i | bi j | for matrix B = (bi j )i,∞j=1 . Recall that: (i) Markov chain X (t) is called null ergodic if pi (t) → 0 as t → ∞ for any initial condition p(0) and any i,

123


(ii) Markov chain X (t) is called weakly ergodic if p∗ (t) − p∗∗ (t) tends to zero as t → ∞ for any acceptable initial conditions p∗ (0), p∗∗ (0), and it is called ergodic if there exists the corresponding constant probability distribution p∗∗ (t) = p∗∗ .

4 Null ergodicity Let {di } be a sequence of positive numbers, d1 = dk4 , d2 = 1, d3 , and d3m+l = k m dl where l = 1, 2, 3, m ≥ 1, and k is a positive number such that 0 < k < 1. Consider the corresponding diagonal matrix D with diagonal elements {di }: ⎞ ⎛ d1 0 0 · · · ⎜ 0 d2 0 · · · ⎟ ⎟ D=⎜ ⎝ 0 0 d3 · · · ⎠ ··· Let l1D be the space of sequences: l1D = p = ( p1 , p2 , . . .)T : p1D ≡ Dp < ∞ . Then D⎛AD −1 = d1 μ 0 0 0 0 0 −λ d2 d2 2μ d2 μ0 ⎜ d2 λ −(λ + μ) 0 0 0 ⎜ d1 d3 d4 ⎜ d3 μ0 d3 λ ⎜ 0 −(λ + 2μ) 0 0 0 d d 2 5 ⎜ d4 μ ⎜ 0 0 0 0 0 −(λ + μ0 ) ⎜ d5 ⎜ d5 2μ d5 μ0 d5 λ 0 0 −(λ + μ + μ) ⎜ 0 0 d4 d6 d7 ⎜ d6 λ d6 λ ⎜ 0 0 0 −(λ + 2μ) 0 d3 d5 ⎜ ⎜ 0 0 0 0 0 0 −(λ + μ0 ) ⎜ d8 λ ⎜ 0 0 0 0 0 0 ⎜ d7 ⎜ d9 λ ⎜ 0 0 0 0 0 0 d 6 ⎝ .. .. .. .. .. .. .. . . . . . . .

··· ··· ··· ··· ··· ··· ··· ··· ··· .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

and one have the respective expression for logarithmic norm (see detailed consideration for instance in Doorn et al. (2010), Granovsky and Zeifman (2004), Zeifman et al. (2006, 2016): ⎛ γ (A)1D = γ (D AD −1 )1 = sup ⎝aii + i

dj j=i

di

⎞ |ai j |⎠ .

Moreover, one can use tthe bound of logarithmic norm for obtaining the following inequality of solutions of the linear differential equation dz/dt = Az in a Banach space B: z(t)B ≤ etγ (AB ) z(0)B ,

(2)

for any t ≥ 0.

123

Y. Satin et al.

Let the inequality 2μ2 μ0 . d2 λ d2 μ + (d2 − d3 )λ Now choose k such that d4 λ + (d4 − d2 )μ0 d4 μ k, and dk4 > 1. μ0 )d4 + λkd2 < 0, d4 λ+(dd42−d λ Hence we have from (2) the inequality z(t)1D =

i≥1

and the following statement holds.

123

di pi (t) ≤ e−βt

i≥1

di pi (0),


Theorem 1 Let the inequality (3) hold. Then the process X (t) is null ergodic and N

i=1

pi (t) ≤

dm −βt e , d N∗

(4)

for any t ≥ 0, any initial condition X (0) = m, and any natural N where d N∗ = min1≤i≤N (di ).

5 Ergodicity

Let c be a positive number such that 0 < c < μ. j≥1 p j (t) = 1 for any t ≥ 1. Hence the first equation Kolmogorov system can be written in the following

of the forward form ddtp1 = j≥1 a1 j (t) − c p j + c. Then the forward Kolmogorov system takes the following form: dp(t) = Ap(t) + f, dt

(5)

where f = (c, 0, 0, · · · )T , and ⎛

−λ − c μ−c −c −c −c −c ⎜ λ 0 0 −(λ + μ) 2μ μ0 ⎜ ⎜ 0 λ −(λ + 2μ) 0 μ0 0 ⎜ ⎜ 0 μ 0 0 0 −(λ + μ0 ) ⎜ ⎜ 0 2μ 0 0 λ −(λ + μ0 + μ) ⎜ = ⎜ 0 A 0 λ 0 λ −(λ + 2μ) ⎜ ⎜ 0 0 0 0 0 0 ⎜ ⎜ 0 0 0 0 0 0 ⎜ ⎜ 0 0 0 0 λ ⎜ 0 ⎝ .. .. .. .. .. .. . . . . . .

··· ··· ··· ··· ··· ··· ··· ··· ··· .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Put p(t) = p∗ (t) − p∗∗ (t), for any t ≥ 0 and any initial conditions p∗ (0), p∗∗ (0). Then we have d p p(t) = A (6) dt Let {δi } be a sequence of positive numbers, δ1 = 1, δ2 , δ3 , and δ3m+l = k m dl where l = 1, 2, 3, m ≥ 1, and k > 1 is a positive number. Consider the corresponding diagonal matrix D: ⎛ ⎞ 1 0 0 0 0 0 0 ··· ⎜ 0 δ2 0 0 0 0 0 ···⎟ ⎜ ⎟ ⎜ 0 0 δ3 0 0 0 0 ···⎟ ⎜ ⎟ ⎜0 0 0 k 0 0 0 ···⎟ ⎜ ⎟ = ⎜ 0 0 0 0 kδ . D 0 0 ···⎟ 2 ⎜ ⎟ ⎜ 0 0 0 0 0 kδ3 0 · · · ⎟ ⎜ ⎟ ⎜0 0 0 0 0 0 k2 · · · ⎟ ⎝ ⎠ .. .. .. .. .. .. .. . . . . . . .

123

Y. Satin et al.

Let now l1 D be the space of sequences:

T < ∞}. l1 D = {p = ( p1 , p2 , . . .) : p1 D ≡ Dp

A D −1 = One has B=D ⎛

μ−c −λ − c − δc − kc − kδc − kδc δ2 3 2 3 ⎜ 2μδ2 μ0 δ2 −(λ + μ) 0 0 ⎜ δ2 λ δ3 k ⎜ μ0 δ3 λδ3 ⎜ 0 −(λ + 2μ) 0 0 ⎜ δ2 kδ2 ⎜ 0 μ 0 0 −(λ + μ ) 0 ⎜ 0 δ2 ⎜ 2μδ2 ⎜ 0 −(λ + μ0 + μ) 0 0 λδ2 δ3 =⎜ ⎜ λδ3 −(λ + 2μ) 0 λk 0 ⎜ 0 δ 2 ⎜ ⎜ 0 0 0 0 0 0 ⎜ ⎜ 0 0 0 0 0 0 ⎜ ⎜ 0 0 0 0 0 λk ⎝ .. .. .. .. .. .. . . . . . .

···

⎞

⎟ ··· ⎟ ⎟ ··· ⎟ ⎟ ··· ⎟ ⎟ ⎟ ··· ⎟ ⎟. ⎟ ··· ⎟ ⎟ ··· ⎟ ⎟ ··· ⎟ ⎟ ··· ⎟ ⎠ .. .

Then one has the following expression for the logarithmic norm of B: γ ( B)1 D

μ−c λδ3 = sup −λ − c + δ2 λ, − (λ + μ) + , δ δ2 2 i c 2μδ2 + − (λ + 2μ) + λk, δ3 δ3 μ0 δ2 c c μ0 δ3 + − (λ + μ0 ) + λδ2 , + + k k kδ2 kδ2 μ λδ3 = −θ. − (λ + μ0 + μ) + δ2 δ2

Let now

μ2 μ0 > 1. 2λ2 μ0 + λ2 μ + 2λ3

(7)

In addition, we suppose that μ2 μ0 < 1. 2λ2 μ0 + μ3 + 3λμ2 + 3λ2 μ + 2λ3 Put now k =

λ+μ λ ,

λ2 + 2λμ + μ2 + (μ + λ) μ0 , λμ0 + μ2 + λμ + λ2 λ3 + 3λ2 μ + 2λμ2 + μ2 + 2λμ + λ2 μ0 δ3 = . λ2 μ0 + λμ2 + λ2 μ + λ3

δ2 =

123

(8)


Then we have for δ = μ − c the following equality δ=−

−2λ3 μ − 3λ2 μ2 − 3λμ3 − μ4 + μ3 − 2λ2 μ μ0 , 2λ2 + λμ μ0 + μ3 + 3λμ2 + 3λ2 μ + 2λ3

(9)

and (8) implies δ > 0. Put now e=−

−2λ4 μ − 3λ3 μ2 − λ2 μ3 + μ4 + λμ3 − 2λ2 μ2 − 2λ3 μ μ0 , 2λ3 + λ2 μ μ0 + λμ3 + 3λ2 μ2 + 3λ3 μ + 2λ4

then (7) implies inequality e < 0. Moreover, one has: ⎧ δ2 λ − λ + δ − μ = ke ⎪ ⎪ ⎪ ⎪ ⎨ −δ2 (λ + μ) + δ3 λ + δ = ke 2μδ2 + δ3 (−λ − 2μ) + λkδ3 − δ + μ = e . ⎪ ⎪ δ + k − μ k − δ + μ = e μ + λδ (−λ ) ⎪ 0 2 0 2 ⎪ ⎩ μ0 δ3 + kδ2 (−λ − μ − μ0 ) + kμ + λδ3 k − δ + μ = e

(10)

Now we obtain from (2) the following bound: 1 z(t)1 = p∗ (t) − p∗∗ (t) ≤ max p∗ (t) − p∗∗ (t)1 D i di ≤ eγ ( B)t p∗ (0) − p∗∗ (0)1 D , and the respective statement. Theorem 2 Under assumptions (7) and (8) the process X (t) is ergodic and the following bound holds: ∗ p (t) − p∗∗ ≤ e−θt p∗ (0) − p∗∗

1D

,

(11)

for any t ≥ 0 and any initial condition p∗ (0), where p∗∗ is the corresponding stationary distribution of X (t).

6 Examples In this section, we present some numerical examples that illustrate the above developed analysis and demonstrate the rate of convergence for the corresponding models. Examples 1–3 treat the null-ergodicity case, while Examples 4–6 describe the ergodic situation. As it easy to check, the values of parameters in Examples 1–3 satisfy nullergodicity condition (2), while they satisfy conditions (6), (7) for the ergodic Examples 1–3.

123

Y. Satin et al.

Null ergodicity Example 1 In this example, we take the following parameters: λ=9 μ = 10 μ0 = 3

d3 = 0.92 ∈ (0.881; 1) d4 ≈ 0.855 k = 0.8 ∈ (0.798; 0.807)

γ (A) ≈ − min (0.58, 0.03, 0.06, 0.07) γ (A) = −β ≈ −0.03 < 0

N dm −0.03t . i=1 pi (t) ≤ d ∗ e

d3 = 0.6 ∈ (0.470; 1) d4 ≈ 0.789 k = 0.45 ∈ (0.36; 0.56)

γ (A) ≈ − min (3.87, 1.34, 0.95, 1.203) γ (A) = −β ≈ −0.95 < 0

N dm −0.95t . i=1 pi (t) ≤ d ∗ e

N

Example 2 λ=9 μ=3 μ0 = 10

N

Example 3 λ = 10 μ=6 μ0 = 7

d3 = 0.8 ∈ (0.680; 1) d4 ≈ 0.824 k = 0.65 ∈ (0.6; 0.7)

γ (A) ≈ − min (2.107, 0.398, 0.5, 0.607) γ (A) = −β ≈ −0.398 < 0

N dm −0.398t . i=1 pi (t) ≤ d ∗ e N

These results demonstrate the strong sensitivity of decay parameter β from intensities μ and μ0 for the same λ-s. Now we turn to the ergodic models Ergodicity Example 4 λ=3

δ2 =

23 13

δ=

μ = 10

δ3 =

199 39

e = − 65 6

μ0 = 10

k=

13 3

135 26

γ ( B)1 D = max −2.5, −

65 46 ,

−

845 398 ,

= − 65 46 . Thus we have the following bound ∗ p (t) − p∗∗ (t) ≤ e− 65 46 t p∗ (0) − p∗∗ (0)

1D

for any t ≥ 0 and any initial conditions.

123

− 2.5, −

65 46


Example 5 λ=1

δ2 =

7 5

δ=

μ=6

δ3 =

21 5

e = −7

μ0 = 2

k=7

23 5

5 γ ( B)1 D = max −1, − 7 , −

35 21 ,

− 1, −

5 7

= − 57 .

Then we have for any t ≥ 0 and any initial conditions: ∗ p (t) − p∗∗ (t) ≤ e− 57 t p∗ (0) − p∗∗ (0)

1D

.

Example 6 8 λ = 1 δ2 = 27 13 δ = 13 69 μ = 2 δ3 = 13 e = − 12 13 12 μ0 = 6 k = 3 γ ( B)1 D = max − 39 , −

4 27 ,

−

4 23 ,

−

12 39 ,

−

4 27

4 . = − 27

Thus we have for any t ≥ 0 and any initial conditions: ∗ p (t) − p∗∗ (t) ≤ e− 274 t p∗ (0) − p∗∗ (0)

1D

.

The results of Examples 4–6 show the more complicated connection between decay parameter β and intensities λ, μ and μ0 in ergodic case.

7 Conclusions The paper deals with a class of constant retrial rate queueing models with two servers. We presented the detailed description of the model and establish the sufficient conditions for null ergodicity and strong ergodicity of the corresponding process. Moreover, we obtain the upper bounds on the rate of convergence for both situations. Note that the analysis of the convergence rate to stationarity of the retrial model with constant retrial rate turns out to be new topic, and the development of methodology to the respective inhomogeneous model seems to be a promising direction of research.

References Artalejo JR (1996) Stationary analysis of the characteristics of the M/M/2 queue with constant repeated attempts. Opsearch 33:83–95 Artalejo JR, Gómez-Corral A, Neuts MF (2001) Analysis of multiserver queues with constant retrial rate. Eur J Oper Res 135:569–581 Avrachenkov K, Yechiali U (2008) Retrial networks with finite buffers and their application to Internet data traffic. Probab Eng Inf Sci 22:519–536 Avrachenkov K, Yechiali U (2010) On tandem blocking queues with a common retrial queue. Comput Oper Res 37(7):1174–1180

123

Y. Satin et al. Avrachenkov K, Morozov E, Nekrasova R, Steyaert B (2014) Stability analysis and simulation of N-class retrial system with constant retrial rates and Poisson inputs. Asia-Pac J Oper Res 31(02):1440002 Avrachenkov K, Morozov E, Steyaert B (2016) Sufficient stability conditions for multi-class constant retrial rate systems. Queuing Syst 82:149–171 Choi BD, Shin YW, Ahn WC (1992) Retrial queues with collision arising from unslotted CSMA/CD protocol. Queueing Syst 11:335–356 Choi BD, Park KK, Pearce CEM (1993) An M/M/1 retrial queue with control policy and general retrial times. Queueing Syst 14:275–292 Choi BD, Rhee KH, Park KK (1993) The M/G/1 retrial queue with retrial rate control policy. Probab Eng Inf Sci 7:29–46 Van Doorn EA, Zeifman AI, Panfilova TL (2010) Bounds and asymptotics for the rate of convergence of birth-death processes. Theory Probab Appl 54:97–113 Efrosinin D, Sztrik J (2011a) Stochastic analysis of a controlled queue with heterogeneous servers and constant retrial rate. Inf Process 11:114–139 Efrosinin D, Sztrik J (2011b) Performance analysis of a two-server heterogeneous retrial queue with threshold policy. Qual Technol Quant Manag 8(3):211–236 Efrosinin D, Sztrik J (2016) Optimal control of a two-server heterogeneous queueing system with breakdowns and constant retrials. In: Dudin A, Gortsev A, Nazarov A, Yakupov R (eds) Information technologies and mathematical modelling—queueing theory and applications. ITMM 2016. Communications in computer and information science. Springer, Cham Fayolle G (1986) A simple telephone exchange with delayed feedback. In: Boxma OJ, Cohen JW, Tijms HC (eds) Teletraffic analysis and computer performance evaluation. North-Holland Publishing Co., Amsterdam Granovsky BL, Zeifman AI (2004) Nonstationary queues: estimation of the rate of convergence. Queueing Syst 46:363–388 Lillo RE (1996) A G/M/1-queue with exponential retrial. Top 4:99–120 Morozov E, Phung-Duc T (2017) Stability analysis of a multiclass retrial system with classical retrial policy. Perform Eval 112:15–26 Wong EWM, Andrew LLH, Cui T, Moran B, Zalesky A, Tucker RS, Zukerman M (2009) Towards a bufferless optical internet. J Lightwave Technol 27:2817–2833 Yao S, Xue F, Mukherjee B, Yoo SJB, Dixit S (2002) Electrical ingress buffering and traffic aggregation for optical packet switching and their effect on TCP-level performance in optical mesh networks. IEEE Commun Mag 40(9):66–72 Zeifman A, Leorato S, Orsingher E, Ya Satin, Shilova G (2006) Some universal limits for nonhomogeneous birth and death processes. Queueing Syst 52:139–151 Zeifman A, Satin Y, Morozov E, Nekrasova R, Gorshenin A (2016) On the ergodicity bounds for a constant retrial rate queueing model. In: Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2016 8th International Congress, pp. 269–272, IEEE. http://ieeexplore.ieee. org/document/7765369/

123