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1.     Introduction

Understanding whether extreme events happen independently or in groups is crucial for forecasting them and minimizing their impact. Extreme value theory can accommodate clustering via so-called extremal index 𝜃 which, measures the size of the cluster and thus has an appealing physical meaning

[1–4].

When 𝜃 = 1, extremes behave like a Poisson process, occurring in isolation. However, when 𝜃 < 1, extremes occur in groups (form clusters), following a compound Poisson process. In this case, 1/𝜃 estimates the mean cluster size or, equivalently, the mean time spent above the threshold. The limiting distribution of the maximum value in a stationary sequence is directly shaped by 𝜃, revealing the local dependence within the data.The extremal index therefore gives a measure of the fraction of extremes that are approximately independent and identically distributed (i.i.d.) [5]. The extreme case of 𝜃 = 0 indicates total dependence, where exceedances form very wide clusters. In practice, this means a sufficiently high threshold may never be crossed. Conversely, independent sequences always have 𝜃 = 1, with high thresholds exceeded only by isolated events.

© 2026 Peshkova, I. V.

                               This work is licensed under a Creative Commons “Attribution-NonCommercial 4.0 International” license.

It is important to mention that there are other definitions of clustering and extremal index in the literature [6, 7].

From a practical point of view, the extremal index is useful for estimating the size of clusters or the average length of intervals between the exceedances. In the telecommunications the interest is the estimation of the risk to lose customers with maximum waiting times (deadlines) exceeding the threshold. The study of extremal metrics in queueing theory relies on applying extreme value theory to regenerative processes. A central objective is determining the limiting distribution for the maxima of waiting times, virtual waiting times, or queue lengths, see for example, [8–11]. The extreme value theory for independent and one-dependent regeneration processes is developed in [11]. The algorithm of computing the extremal index of the stationary waiting time of a stable 𝐺/𝐺/1 system with distribution belonging to the domain of attraction of Gumbel distribution is given in [9]. The study of the distribution of the cluster and inter-cluster sizes is also an actual problem (see, for example, results for the Lindley process in 𝐺𝐼/𝐺/1 in [12]).

For a stable 𝐺𝐼/𝐺/1 system, if a non-zero solution 𝛾 > 0 exists for the equation 𝑒^{𝛾(𝑆−𝜏)} = 1, the maximum waiting time converges to a Gumbel distribution. The extremal index 𝜃 in this case is often amenable to explicit or numerical computation.

For the case of subexponential distributions of service times distributions, the parameter 𝛾 = 0, and the asymptotics of extreme values are studied using alternative methods based on the tail behavior of the waiting times themselves.

The main contribution of this paper is to establish conditions under which the extremal indexes of two queuing systems can be compared. Another objective is to determine which monotonicity properties in terms of the extremal index can be established for the systems with mixed service times.

The paper is organized as follows. In Section 2, we present known results on the extremal index for 𝐺𝐼/𝑀/1 and 𝑀/𝐺/1 systems. In Section 3 Theorems 1 and 2 were proved. They establish the conditions for comparing the extremal indexes of stationary waiting times in 𝑀/𝐺/1 and 𝐺𝐼/𝑀/1, respectively. For 𝑀/𝐺/1 system the obtained result is extended to the case of a system with mixed service times with ordered components (Section 3.1). In Section 3.2 we extend results obtained for

𝐺𝐼/𝑀/1 system to multiserver 𝐺𝐼/𝑀/𝑐. In Section 4 we investigate the class of limiting distributions of the stationary waiting time in 𝐺𝐼/𝐺/1 system with service time determined by a finite mixture whose dominant component of the equilibrium distribution belongs to the subexponential distributions.

2.      The extremal index in GI/G/1 system

Let 𝐺𝐼/𝐺/1 system have i.i.d. service times, {𝑆_𝑖,𝑖 ⩾ 1} and i.i.d. interarrivals, {𝜏_𝑖,𝑖 ⩾ 1}. Consider a reflected random walk (Lindley process), given by the recursion

                                                               𝑊 _𝑖+1 = (𝑊 _𝑖 + 𝑋_𝑖)⁺,          𝑖 ⩾ 1,                                                                           (1)

where (𝑥)⁺ = max(0,𝑥) and 𝑋_𝑖 = 𝑆_𝑖 − 𝜏_𝑖 — i.i.d. non-lattice r.v.s with common distribution function (d.f.) F_𝑋, E𝑋 < 0. Also assume that there is a 𝛾 such that

                                                                             E𝑒^𝛾𝑋 = 1.                                                                                                 (2)

Let 𝑌_𝑛 = 𝑋₁+⋯+𝑋_𝑛−1, 𝑛 ⩾ 1 (𝑌₀ = 0). The actual waiting time 𝑊_𝑛 (or just waiting time) of customer 𝑛 is the time from arrival 𝑡_𝑛 to the system until service starts. This process {𝑊_𝑛,𝑛 ⩾ 1} is a Lindley process (1) generated by {𝑌_𝑛,𝑛 ⩾ 0} [8]. In particular, 𝑊_𝑛 =^𝑑 max 𝑌 _𝑘 and, if 𝜌 = E𝑆/E𝜏 < 1 (corresponds to

0⩽𝑘⩽𝑛

E𝑋 < 0), then a limiting steady-state distribution exists. By 𝑊 we denote a random variable having the steady-state distribution of actual waiting time process {𝑊_𝑛},

                                                                    𝑊_𝑛 ⇒ 𝑊,       𝑛 → ∞.

It is obvious that the behaviour of the tail of P(max 𝑌_𝑛 > 𝑥) is determined by the components of 𝑋_𝑖. Next we consider two cases: the real positive root of the equation (2) 𝛾 > 0. This case refers to light-tailed distributions of 𝑋_𝑖 and exponential asymptotic behaviour of waiting time tail. In contrast, for subexponential distributions of 𝑋_𝑖 we get E𝑒^𝛾𝑋 = ∞ for any 𝛾 > 0 [8].

Let F_𝑆(𝑥) = 1 − F_𝑆(𝑥) be the tail of d.f. F_𝑆(𝑥) of r.v. 𝑆. A d.f. F_𝑆 is called subexponential if

                                                                 for all 𝑛 ⩾ 2,

where F^∗𝑛_𝑆 (𝑥) is the tail of the 𝑛-fold convolution of the distributions F_𝑆(𝑥) with itself, i.e., F ^{1 𝑛} > 𝑥), where 𝑆^𝑖 is the stochastic copy of 𝑆, 𝑖 = 1,…,𝑛.

We denote the class of subexponential distributions by 𝒮. We also denote by 𝑆_𝑒 the stationary residual service time given by the probability density function F_𝑆(𝑥)/E𝑆, and let F_𝑆𝑒(𝑥) be the d.f. of 𝑆_𝑒.

It is known [8, 10] that if 𝜌 < 1 the system is stationary, and equation (2) has a real positive solution 𝛾 > 0 [13], then:

• if E[𝑋𝑒^𝛾𝑋] < ∞, then the tail d.f. of max 𝑌_𝑛 is asymptotically (up to certain constant 𝐾 > 0)

𝑛≥0 equivalent to an exponential function, namely:

P(max 𝑌_𝑛 > 𝑥)

                                                                lim 𝑛⩾0−𝛾𝑥  = 𝐾.

                                                               𝑥→∞             𝑒

• if E[𝑋𝑒^𝛾𝑋] = ∞, then lim 𝑃(max 𝑌_𝑛 > 𝑥) = o(𝑒^−𝛾𝑥), as 𝑥 → ∞, where 𝑏 = o(𝑎) means lim 𝑏/𝑎 = 0.

                                           𝑥→∞       𝑛⩾0

If 𝑆_𝑒 ∈ 𝒮, then the the stationary waiting time 𝑊 is also subexponential, 𝑊 ∈ 𝒮, and [13]

                                                                   P(max𝑛⩾0 𝑌_𝑛 > 𝑥⁾           _𝜌

                                                         𝑥→∞lim      P(𝑆𝑒 > 𝑥)           = 1 − 𝜌.                                                                            (3)

Denote by 𝑀_𝑛 = max(𝑊₁,…,𝑊_𝑛) the largest waiting time among customers 1,…,𝑛. If 𝜌 ⩽ 1 and there exists a real positive solution to equation (2), then [14–16]

                                                          lim P(𝛾𝑀_𝑛 − log(𝑏𝜃𝑛) ⩽ 𝑥) = 𝛬(𝑥),                                                                       (4)

𝑛→∞

where 𝛬(𝑥) = exp(−𝑒^−𝑥) is the Gumbel distribution and 𝑏 is a constant. Moreover, 𝑀_𝑛/ log 𝑛 converges to 1/𝛾 whenever possible [15] for all 𝜖 > 0:

                                                 P (^|||log^𝑀𝑛𝑛 − 𝛾^1||| > 𝛾^𝜖 )→0 as 𝑛 → ∞.

For the 𝐺𝐼/𝑀/1 system with 𝜌 < 1, equation (2) gives a unique positive solution 𝛾 [8, Theorem 5.8]. Moreover, the distribution of the actual waiting time 𝑊 is a mixture of an atom at zero and an exponential distribution with parameter 𝛾 and mixture proportions 𝛾/𝜇 and 1 − 𝛾/𝜇, respectively, [8, Theorem 5.1]:

                                                         F_𝑊(𝑥) = 1 − (1 − 𝛾/𝜇)𝑒^−𝛾𝑥,           𝑥 > 0.

Theorem 7.5 in [15] implies that the extremal index of the process {𝑊_𝑛,𝑛 ⩾ 1} for a 𝐺𝐼/𝑀/1 system is calculated by the formula:

                                                                              𝑑                    1

                                                                   𝜃 = 𝛾(   𝜓_𝜏(𝛾) +    ),                                                                                  (5)

                                                                             𝑑𝛾                  𝜇

where 𝜓_𝜏(𝛾) = E𝑒^−𝛾𝜏 is the Laplace-Stieltjes transform of interarrival times, 𝜏, and, in addition,

𝑏 = 1 − 𝛾/𝜇 in (4).

For a stationary 𝑀/𝐺/1 system, equation (2) has the following form [8]:

                                                                         E𝑒^𝛾𝑆 = 1 + ^𝛾.                                                                                            (6)

𝜆

Furthermore, if equation (6) has a real positive solution 𝛾, then the formula for the extremal index 𝜃_𝑊 becomes [9]:

𝜃 = P(𝑊 = 0)(1 − 𝜓_𝜏(𝛾)) = ^{𝛾(1 − 𝜌)}. (7) 𝛾 + 𝜆

3.       Comparison of extremal indexes in 𝐺𝐼/𝐺/1 systems

Consider two queuing systems 𝛴⁽¹⁾ and 𝛴⁽²⁾ of type 𝐺𝐼/𝐺/1 (we assign index 𝑖 to quantities related to the 𝑖-th system). Let 𝑆⁽_𝑛^𝑖) be the service time of the 𝑛-th customer and 𝜏_𝑛^(𝑖) be the interval between the arrivals of the 𝑛-th and 𝑛 + 1-th requests in the 𝑖-th system, E𝜏^(𝑖) = 1/𝜆, 𝑖 = 1,2. Let 𝑊_𝑛^(𝑖) be the actual

waiting time of 𝑛-th customer, 𝑖 = 1,2. Let us denote (if they exist) the distribution limits

                                                         𝑊_𝑛^(𝑖) ⇒ 𝑊 ^(𝑖),       𝑛 → ∞,        𝑖 = 1,2.

Theselimitsexist, inparticular, iftheinterarrivaltimes𝜏^(𝑖), 𝑖 = 1,2arenon-latticeand𝜌_𝑖 = 𝜆_𝑖E𝑆^(𝑖) < 1, 𝑖 = 1,2 [8].

Let us compare the extremal indexes 𝜃⁽¹⁾ and 𝜃⁽²⁾ of stationary waiting time processes {𝑊_𝑛⁽¹⁾} and {𝑊_𝑛⁽²⁾} in the systems 𝛴⁽¹⁾ and 𝛴⁽²⁾, respectively. Further, to compare the r.v.s, we will need the stochastic order and the order in failure rate. We say that the r.v. 𝑍₁ is less that r.v. 𝑍₂ in stochastic order, 𝑍₁ ⩽ 𝑍₂ if

𝑠𝑡

                                                                F_𝑍1(𝑥) ⩽ F_𝑍2(𝑥),         𝑥 ∈ R.

Let 𝑟_𝑍(𝑥) ∶= 𝑓_𝑍(𝑥)/F_𝑍(𝑥) be failure rate function of r.v. 𝑍, where 𝑓_𝑍(𝑥) is density function. We say that r.v. 𝑍₁ is less than r.v. 𝑍₂ in failure rate order, 𝑍₁ ⩽ 𝑍₂, if

𝑟

                                                                𝑟_𝑍1(𝑥) ⩾ 𝑟_𝑍2(𝑥),        ^{𝑥 ∈} R.

The following theorem allows to compare the extremal indexes of stationary waiting times in two different 𝑀/𝐺/1 systems for which the real positive roots of equation (2) exist.

Theorem 1. Suppose that in two 𝑀/𝐺/1 systems 𝛴⁽¹⁾ and 𝛴⁽²⁾, 𝜌_𝑖 < 1, E[𝑆^(𝑖)𝑒^{𝛾𝑆(𝑖)}] < ∞,𝑖 = 1,2 for and any 𝛾 ⩾ 0 and relations

                                                 𝑊1(1) = 𝑊1(2) = 0,         𝜏(1) ⩾ 𝜏(2),         𝑆(1) ⩽ 𝑆(2),                                                           (8)

                                                                                       𝑟                           𝑟

are satisfied. Additionally, suppose that there exist real positive roots 𝛾₁ and 𝛾₂ of equation (2) for both systems. Then the extremal indices of the stationary waiting times are ordered,

                                                                           𝜃(1) ⩾ 𝜃(2).                                                                                               (9)

Proof. First of all, note that the ordering in failure rate implies the ordering of the exponential moments [17]. Consequently, from the relations (8) it follows that

                                         E𝑒𝛾𝑆⁽¹⁾ ⩽ E𝑒𝛾𝑆⁽²⁾,             E𝑒−𝛾𝜏⁽¹⁾ ⩽ E𝑒−𝛾𝜏⁽²⁾,             for any 𝛾 > 0.

Let 𝛾_𝑖 denote the positive real root of equation (2) for the system 𝛴^(𝑖), 𝑖 = 1,2. If there are several such roots, then, according to Theorem 7.2 in [15], the smallest of them is chosen. Then the equalities

E𝑒𝛾1(𝑆(1)−𝜏(1)) ₌ E𝑒𝛾2(𝑆(2)−𝜏(2)) = 1

are satisfied if

𝛾₁ ⩾ 𝛾₂.

Since the intervals between arrivals are ordered in failure rate, 𝜏⁽¹⁾ ⩾ 𝜏⁽²⁾, then 𝜆₁ < 𝜆₂. Consequently,

𝑟

𝜆₁/𝛾₁ ⩽ 𝜆₂/𝛾₂. Moreover, by 𝑆⁽¹⁾ ⩽ 𝑆⁽²⁾, we also have E𝑆⁽¹⁾ ⩽ E𝑆⁽²⁾ and 1 − 𝜌₁ ⩾ 1 − 𝜌₂. Substituting

𝑟 these inequalities into the expression for the extremal index (7), we obtain

                                                       𝜃(1) =       1 − 𝜌1      ⩾      1 − 𝜌2       = 𝜃(2).

                                                                  1 + 𝜆₁/𝛾₁              1 + 𝜆₂/𝛾₂

To illustrate the statement of Theorem 1 we consider the simple example. It’s easy to show that, for the 𝑀/𝑀/1 system, the extremal index of the stationary waiting time has the form

𝜃 = (1 − 𝜌)². Consider two 𝑀/𝑀/1 systems in which the service times are exponential with parameters 𝜇₁ and 𝜇₂, respectively, and 𝜇₁ ⩾ 𝜇₂ > 0. Assume that 𝜆₁ ⩽ 𝜆₂ and 𝜌_𝑖 = 𝜆_𝑖/𝜇_𝑖 < 1, 𝑖 = 1,2. In this case, the conditions (8) are satisfied and

𝛾1 = 𝜇1 − 𝜆1 ⩾ 𝛾2 = 𝜇2 − 𝜆2,𝜃(1) = (1 − 𝜌1)2 ⩾ (1 − 𝜌2)2 = 𝜃(2),

i.e., the inequality (9) is satisfied.

Now consider 𝑀/𝑊𝑒/1 system with Weibull service time d.f.

                                                     F_𝑆(𝑥) = 1 − 𝑒^{−(𝑥/𝛼)𝛽},           𝛼, 𝛽 > 0,       𝑥 ⩾ 0.

The exponential moments E𝑒^𝛾𝑆 for the Weibull distribution exist only for 𝛽 ⩾ 1, so in this case the equation (6) takes form

                                                    ∞ (𝛼𝛾)𝑘          𝑘                           𝛾

                                                   ∑                𝛤 (        + 1) = 1 +    ,        for 𝛽 ⩾ 1.

                                                          𝑘!           𝛽                           𝜆

𝑘=0

Now we compare the extremal indexes of waiting times in two 𝑀/𝑊𝑒/1 systems. For example, let

𝜆 = 2, 𝛼₁ = 0.25, 𝛽₁ = 1.5 in the first system and 𝜆₂ = 2, 𝛼₂ = 0.4, 𝛽₂ = 1.2. With these parameters we have 𝑆⁽¹⁾ < 𝑆⁽²⁾. By numerical calculation we obtain a single roots 𝛾₁ = 3.7 and 𝛾₂ = 0.79, respectively,

𝑟 therefore, 𝜃⁽¹⁾ = 0.35 > 𝜃⁽²⁾ = 0.07.

Now we prove a statement similar to the Theorem 1 for 𝐺𝐼/𝑀/1 systems.

Theorem 2. Let the stationarity conditions 𝜌_𝑖 < 1, E[(𝑆^(𝑖)𝑒^{𝛾𝑆(𝑖)}] < ∞ for any 𝛾 ⩾ 0, 𝑖 = 1,2, and the relations (8) be satisfied for two 𝐺𝐼/𝑀/1-type systems 𝛴⁽¹⁾ and 𝛴⁽²⁾. Let there exist real positive roots 𝛾₁ and 𝛾₂ of the equation (2) for these systems and the following inequality holds:

                                                               E[𝜏(1)𝑒−𝛾₁𝜏⁽¹⁾] ⩽ E[𝜏(2)𝑒−𝛾₂𝜏⁽²⁾].                                                                            (10)

Then the extremal indexes of the stationary waiting times are ordered as

𝜃(1) ⩾ 𝜃(2).

Proof. In the proof of Theorem 1 it is shown that the roots 𝛾_𝑖 of equation (2) for systems 𝛴^(𝑖) are related by the inequality

𝛾₁ ⩾ 𝛾₂,

and by (8) the exponential moments are also ordered in the same way. Therefore,

𝜓_𝜏(1)(𝛾₁) ^{≤ 𝜓}_𝜏(2)(𝛾₂).

Note, that E𝜏𝑒^−𝛾𝜏 < ∞ for any r.v. 𝜏 ≥ 0 and any 𝛾 ≥ 0 and that

                                                     𝑑 𝜓𝜏(𝑖)(𝛾𝑖) = −E(𝜏(𝑖)𝑒−𝛾𝑖𝜏(𝑖)),          𝑖 = 1,2.

𝑑𝛾𝑖

Thus, it follows from condition (10), that

                                                               𝑑                             𝑑

𝑑𝛾₁ 𝜓_𝜏(1)(𝛾₁) ^⩾ 𝑑𝛾₂ ^𝜓_𝜏(2)(𝛾₂).

The equation (2) for the systems under consideration is equivalent to

                                                             𝜓_𝜏(_𝑖)(𝛾_𝑖) + 𝛾_𝑖/𝜇_𝑖 = 1,           𝑖 = 1,2.

Therefore, expression (5) can be rewritten as

𝜃

                             (𝑖) = 𝛾𝑖(𝑑𝛾𝑑 𝜓𝜏(𝑖)(𝛾𝑖) + 1/𝜇𝑖) = (1 − 𝜓𝜏(𝑖)(𝛾𝑖))(𝜇𝑖 𝑑𝛾𝑑 𝑖 𝜓𝜏(𝑖)(𝛾𝑖) + 1),              𝑖 = 1,2.

𝑖

From the ordering in failure rate of service times, it follows that 𝜇₁ ⩾ 𝜇₂. Substituting the obtained inequalities into the expression for the extremal index (5), we obtain the required inequality

𝜃

(1) = (1 − 𝜓𝜏(1)(𝛾1))(𝜇1 𝑑𝛾𝑑1 𝜓𝜏(1)(𝛾1) + 1) ⩾ (1 − 𝜓𝜏(2)(𝛾2))(𝜇2 𝑑𝛾𝑑2 𝜓𝜏(2)(𝛾2) + 1) = 𝜃(2).

It is worth noting that the set of queueing systems satisfying the inequality (10) is not empty. In particular it holds for the popular 𝑀/𝑀/1 system because

𝑑                   𝜆𝑖 𝜆𝑖 𝜓𝜏(𝑖)(𝛾𝑖) = − = − , 𝑑𝛾𝑖 (𝜆𝑖 + 𝛾𝑖)2 𝜇2𝑖

and if 𝜆₁ ⩽ 𝜆₂,𝜇₁ ⩾ 𝜇₂, then condition (10) holds. As another example, consider two systems 𝛴⁽¹⁾ and

𝛴⁽²⁾, with two-component hyperexponential interarrival times, where d.f. tail has form

                       F𝜏(𝑖)(𝑥) = 𝑝𝑒−𝜆(1𝑖)𝑥 + (1 − 𝑝)𝑒−𝜆(2𝑖)𝑥,            𝜆(𝑗𝑖) > 0,        0 < 𝑝 < 1,        𝑥 ⩾ 0,        𝑖, 𝑗 = 1,2.

Let 𝑝 = 0.5,𝜆⁽¹⁾₁ = 1, 𝜆⁽¹⁾₂ = 3, 𝜆⁽²⁾₁ = 2, 𝜆⁽²⁾₂ = 3. Suppose that the service time is exponential with parameter 𝜇₁ = 4 in 𝛴⁽¹⁾ and with parameter 𝜇₂ = 3 in 𝛴⁽²⁾. Then numerical analysis shows that only positive (numerical) solution of (2) is 𝛾₁ = 2.24, for which 𝜃_𝑊(1) = 0.33. Similarly, the unique positive solution 𝛾₂ = 0.58 can be obtained by solving numerically the equation (2), for which 𝜃_𝑊(2) = 0.0385.

With these parameters, the conditions (8) and (10) are satisfied, since

                                         𝑟𝜏(1)(𝑥) = 1 < 𝑟𝜏(2)(𝑥) = 2;                  𝑟𝑆(1)(𝑥) = 4 > 𝑟𝑆(2)(𝑥) = 3,

and

                                                                   𝛾1 > 𝛾2,        𝜃(1) > 𝜃(2).

3.1. 𝑀/𝐺/1 system with mixed service times

Consider a single-server 𝑀/𝐺/1-type system 𝛴 with service time 𝑆 given by an 𝑚-component

mixture d.f.                                            _{𝑚                                               𝑚}

                                        F_𝑆(𝑥) = ∑ 𝑝_𝑖F_𝑆(_𝑖)(𝑥),              ∑ 𝑝_𝑖 = 1,         𝑝_𝑖 ⩾ 0,         𝑖 = 1,…,𝑚.                                           (11)

                                                  𝑖=1                             𝑖=1

Assume that the r.v.’s 𝑆^(1),…,𝑆^(𝑚) are independent and 𝑆^(𝑖) has d.f. F_𝑆(_𝑖)(𝑥), 𝑖 = 1,…,𝑚. Denote by

                                                                               𝑚                  𝑚

𝜌 = 𝜆E𝑆 =𝑝_𝑖𝜌_𝑖

the traffic intensity of the system 𝛴 where 𝜌_𝑖 = 𝜆/𝜇_𝑖, 𝜇_𝑖 = 1/E𝑆^(𝑖), 𝑖 = 1,…,𝑚. Assume that the components 𝑆^(𝑖) of the service time 𝑆 are ordered in failure rate

𝑆(1) ⩽ ⋯ ⩽ 𝑆(𝑚).

                                                                             𝑟          𝑟

Consider two queuing systems 𝛴⁽¹⁾ and 𝛴^(𝑚) with inputs 𝜏⁽¹⁾, 𝜏^(𝑚), respectively. Let 𝜏 be the input process in the original system 𝛴, and E𝜏^(𝑖) = 1/𝜆_𝑖, E𝜏 = 1/𝜆, 𝑖 = 1,𝑚. (As usual, the index 𝑖 relates to the 𝑖-th system.) The service time 𝑆^(𝑖) is given by the d.f. F_𝑆(_𝑖)(𝑥), 𝑖 = 1,𝑚.

Theorem 3. Assume the stationarity conditions 𝜌_𝑖 < 1, 𝑖 = 1,𝑚; 𝜌 < 1 hold and the following relations are satisfied:

𝑊1(1) = 𝑊1(𝑚) = 𝑊1 = 0.

𝜏(1) ⩾ 𝜏 ⩾ 𝜏(𝑚).

                                                                              𝑟       𝑟

Suppose, that the components of the service time mixture are ordered

𝑆(1) ⩽ ⋯ ⩽ 𝑆(𝑚).

                                                                             𝑟          𝑟

Assume that real positive roots of equation (2) exist for all three systems. Then

𝜃(1) ⩾ 𝜃 ⩾ 𝜃(𝑚).

The proof follows from Theorem 1 and the monotonicity property of waiting times (see Theorems 5 and 6 in [18]).

To illustrate the statement of Theorem 3, consider an 𝑀/𝐻_𝑚/1 system as the original system where service times have 𝑚-component mixture d.f.

                                                   𝑚                                               𝑚

                                       F_𝑆𝑝_𝑖𝑒^{−𝜇𝑖𝑥},         𝜇_𝑖       = 1,𝑝_𝑖 ⩾ 0,         𝑥 ⩾ 0.

We consider two systems 𝛴⁽¹⁾ and 𝛴^(𝑚) (of type 𝑀/𝑀/1), in which the service times 𝑆^(𝑖) have exponential distribution, F_𝑆(_𝑖)(𝑥) = 𝑒^{−𝜇𝑖𝑥}, 𝑖 = 1,𝑚. The interarrival times in all three systems have exponential distribution with parameter 𝜆. Assume that the stationarity conditions are satisfied in all systems:

𝑚

                                               𝜌_𝑖 = 𝜆/𝜇_𝑖 < 1,          𝑖 = 1,𝑚,           𝜌 = 𝜆 ∑ 𝑝_𝑖/𝜇_𝑖 < 1.

𝑖=1

The service time failure rate function 𝑟_𝑆 in the original system 𝛴 is equal to

𝑚

∑ 𝑝_𝑖𝜇_𝑖𝑒−𝜇^𝑖𝑥

                                                        𝑟_𝑆(𝑥) ∶= ^𝑖=1_𝑚                  ,                   𝑥 ⩾ 0.

−𝜇_𝑖𝑥

The service time failure rate functions 𝑆⁽¹⁾ and 𝑆^(𝑚) in the systems 𝛴⁽¹⁾ and 𝛴^(𝑚) are, respectively, equal to 𝑟_𝑆(1)(𝑥) = 𝜇₁, 𝑟_𝑆(_𝑚)(𝑥) = 𝜇_𝑚. It is easy to verify that for

                                                                         𝜇₁ ⩾ ⋯ ⩾ 𝜇_𝑚                                                                                                                                        (12)

the failure rate functions are ordered as follows:

                                                         𝑟_𝑆(1)(𝑥) ⩾ 𝑟_𝑆(^{𝑥) ⩾ 𝑟}_𝑆(_𝑚)(𝑥),         𝑥 ⩾ 0,

and, therefore, the service times in these systems are ordered in the failure rate as

𝑆(1) ⩽ 𝑆 ⩽ 𝑆(𝑚).

                                                                              𝑟        𝑟

The extremal index in the original system 𝛴 can be calculated by the formula (7). Moreover, from the condition for the parameters (12) and the equation (2) it follows that

                                                 𝜆 + 𝛾         𝑚     𝑝𝑖𝜇𝑖           𝑚         𝑝𝑖                   𝜇1

                                                            = ∑                    ⩾ ∑                       =                 ,

                                                   𝜆             𝑖=1 𝜇_𝑖 − 𝛾           _𝑖=1 1 − 𝛾/𝜇₁                𝜇₁ − 𝛾

and, therefore,

𝛾(𝜇₁ − 𝜆 − 𝛾) ⩾ 0

and 𝛾 ⩽ 𝜇₁ − 𝜆 = 𝛾₁. Further, since 𝜌₁ ⩽ 𝜌, then

                                                          1 − 𝜌             1 − 𝜌^{1                                           2} = 𝜃⁽¹⁾.

                                                 𝜃 =                    ⩽                       = (1 − 𝜌₁)

                                                         1 + 𝜆/𝛾         1 + 𝜆/𝛾₁

Similarly, it can be shown that

                                                            𝜆 + 𝛾         𝑚     𝑝𝑖𝜇𝑖               𝜇𝑚

                                                                       = ∑                  ⩽                  ,

                                                               𝜆             𝑖=1 𝜇_𝑖 − 𝛾           𝜇_𝑚 − 𝛾

and 𝛾 ⩽ 𝜇_𝑚 − 𝜆 = 𝛾_𝑚, and therefore,

                                                        1 − 𝜌             1 − 𝜌^{𝑚                                              2} = 𝜃^(𝑚).

𝜃 = ⩾ = (1 − 𝜌_𝑚) 1 + 𝜆/𝛾 1 + 𝜆/𝛾_𝑚

3.2.      Extension of Theorem 2 to multiserver systems

The extremal behaviour of multiserver system 𝐺𝐼/𝑀/𝑐 with 𝑐 working in parallel servers and service intensity 𝜇 is the same as the in 𝐺𝐼/𝑀/1 system with service intensity 𝑐𝜇 with identical interarrival distribution F_𝜏(𝑥) [9]. Moreover, if 𝜌 = (𝜇E𝜏)⁻¹ < 𝑐, then waiting time process has (total variation) limit which is a mixture of an atom at zero and exponential distribution with intensity 𝛾, where 𝛾 is unique solution of the equation [8][Theorem 3.2]

.

From (5) one can also easily obtain the relation for extremal index of stationary waiting time in 𝐺𝐼/𝑀/𝑐 system by substituting 𝑐𝜇 instead 𝜇, namely,

                                                                             𝑑                     1

𝜃 = 𝛾( 𝜓_𝜏(𝛾) + ). 𝑑𝛾              𝑐𝜇

The last formula allows us to compare the extremal indexes of waiting times in two 𝐺𝐼/𝑀/𝑐_𝑖 systems by adding the condition 𝑐₁ ⩾ 𝑐₂ in Theorem 2. We formulate the obtained result as the following statement.

Theorem 4. Let the stationarity conditions, 𝜌_𝑖 = 𝜆₁E𝑆^(𝑖) < 𝑐_𝑖, 𝑖 = 1,2, and the relations (8), be satisfied for two systems 𝛴⁽¹⁾ and 𝛴⁽²⁾ of type 𝐺𝐼/𝑀/𝑐_𝑖. Let there exist real positive roots of the equation (2) for these systems and

𝑐₁ ⩾ 𝑐₂.

Then the extremal indexes of the stationary waiting times are ordered as

𝜃(1) ⩾ 𝜃(2). In particular, it is easy to check that for 𝑀/𝑀/𝑐 system

                                                                                                𝜆    2

                                                                𝛾 = 𝑐𝜇 − 𝜆,𝜃 = (1 −            ) .

𝑐𝜇

Therefore, if 𝜆₁ ⩽ 𝜆₂, 𝜇₁ ⩾ 𝜇₂, 𝑐₁ ⩾ 𝑐₂, then 𝜃⁽¹⁾ ⩾ 𝜃⁽²⁾.

4.      GI/G/1 system with subexponential service times

In this section we consider the systems with the subexponential service times. In contrast to the light-tailed case, the stationary waiting time 𝑊 is also subexponential and the relation (3) holds. Moreover, the maximum stationary waiting time 𝑀_𝑛 = max(𝑊₀,…,𝑊_𝑛) has the same asymptotics as max(𝑋₀,…,𝑋_𝑛), as 𝑛 → ∞, with 𝑋_𝑖 = 𝑆_𝑖 − 𝜏_𝑖 [19]. Moreover, if 𝑆 ∈ 𝒮, then the extremal index of the stationary waiting time for systems with subexponential service time is zero [19], i.e.,

𝜃 = 0.

Now consider 𝐺𝐼/𝐺/1 system with 𝑚-component mixture service times with d.f. given by (11) with a dominant component.

We say that a component F^(𝑗), 𝑗 ∈ {1,…,𝑚} is asymptotically 𝑟-dominant for an 𝑚-component

 

mixture of distributions

F(𝑥) = 𝑝₁F⁽¹⁾⁽𝑥) + ⋯ + 𝑝_𝑚F^(𝑚)(𝑥),

if

𝑚

∑ 𝑝_𝑖 = 1,

𝑖=1

 

                                                                       = 𝑟_𝑖,         𝑖 ∈ {1,…,𝑚},         𝑖 ≠ 𝑗,

where 𝑟 = (𝑟₁,…,𝑟_𝑚), 𝑟_𝑗 = 1, and 0 ⩽ 𝑟_𝑖 < 1, for 𝑖 ≠ 𝑗, 𝑥_𝑅 is the right endpoint of F(𝑥).

The following theorem states that, if in a 𝐺𝐼/𝐺/1 system, the service time is determined by a finite mixture (11) whose 𝑟-dominant component of the equilibrium distribution belongs to the class of

subexponential distributions, F_𝑆𝑒(_𝑗) ∈ 𝒮, then the tail of the limit distribution of the stationary waiting time is equivalent to the tail of this distribution up to a constant,

lim P(𝑊 > 𝑥) =_{∶ 𝛿.}

𝑥→∞ F𝑆𝑒(𝑗)(𝑥)

Furthermore, the limit distribution of the maximum of the stationary waiting time belongs to the maximum domain of attraction (MDA) of the distribution of extreme values of the same type as the maximum of the r.v. defined by the d.f. F_𝑆𝑒(_𝑗), and the extremal index of the stationary waiting time is obviously 0.

Theorem 5. Let the original system 𝛴 be stationary, 𝜌 < 1. Let the service time be defined by an 𝑚component mixture of distributions (11) and let there exist a set of numbers

                                                𝑟 = (𝑟₁,…,𝑟_𝑚),          0 ⩽ 𝑟_𝑖 < 1,        𝑖 ≠ 𝑗,        𝑟_𝑗 = 1,

such that the equilibrium distribution of the 𝑗-th component is 𝑟-dominant in the mixture and belongs to the class of subexponential distributions, ^F_𝑆𝑒(_𝑗) ∈ 𝒮. Then

• The equilibrium distribution of service time belongs to the class of subexponential distributions, F_𝑆𝑒 ∈ 𝒮.
• The tails of the equilibrium distributions F_𝑆𝑒 and F_𝑆𝑒(_𝑗) are equivalent up to a constant,

                                                                           F𝑆𝑒(𝑥)          𝑚

lim= ∑ 𝑞_𝑖𝑟_𝑖

𝑥→∞

F𝑖=1

where

                                        𝑟_𝑖 ∶= lim,                    0 ⩽ 𝑟_𝑖 < 1,          𝑖 = 1,…,𝑚,         𝑖 ≠ 𝑗,         𝑟_𝑗 = 1;

                                         𝑥→∞            (𝑥)

𝑆

𝑝 E

𝑞_𝑖 =       𝑖E_𝑆𝑆(𝑖) .

• The tails of the distribution of the stationary waiting time, P(𝑊 > 𝑥), and the equilibrium distribution

F_𝑆𝑒(_𝑗) are equivalent up to a constant 𝛿,

𝑚

𝜆 ∑ 𝑝𝑖𝑟𝑖E𝑆(𝑖)

                                                      lim P(𝑊 > 𝑥) =             𝑖=1 𝑚                     ∶= 𝛿.

                                                   𝑥→∞       F𝑆𝑝𝑖E𝑆(𝑖)

• If ^F_𝑆𝑒(_𝑗) ∈ 𝑀𝐷𝐴(𝐺), then d.f. stationary waiting time in the original system F_𝑊 ∈ 𝑀𝐷𝐴(𝐺^𝛿). 5) The extremal index of the stationary waiting time in the original system is zero, 𝜃_𝑊 = 0.

Proof. 1) Find the equilibrium distribution of service time in the original system

∞

𝑚

     F𝑆𝑒                      E𝑆                    𝑆            (𝑦) + …𝑝𝑚F𝑆(𝑚)(𝑦))𝑑𝑦 = 𝑖=1∑ E𝑝𝑆𝑖 ∫ F𝑆(𝑖)(𝑦)𝑑𝑦 =

                          𝑥                                                                                            𝑥

                                                                                                              𝑚 𝑝_𝑖E𝑆(𝑖)                           𝑚       F𝑆𝑒(𝑖)(𝑥).

= ∑𝑞𝑖

                                                                                                            𝑖=1       E𝑆

𝑚

Obviously, ∑ 𝑞_𝑖 = 1. Thus, the equilibrium service time distribution in the original system is

𝑖=1 a mixture of the equilibrium distributions of the components with mixture proportions 𝑞_𝑖,

𝑚

                                                      𝑆_𝑒 ,         ∑ 𝐽_𝑖 = 1

𝑖=1

where the indicator 𝐽_𝑖 takes the value 1 with probability 𝑞_𝑖, 𝑖 = 1,…,𝑚. It is known that if at least one component of the final mixture has a subexponential distribution, then the mixture of distributions belongs to the class of subexponential distributions [20]. Therefore, point 1) of the theorem is proved.

• Since F_𝑆𝑒 is an 𝑚-component mixture with proportionality coefficients 𝑞_𝑖, where the 𝑗-th

𝑚

component is 𝑟-dominant, then F_𝑆𝑒 ^{and F}_𝑆𝑒(_𝑗) have equivalent tails up to the constant ∑ 𝑞_𝑖𝑟_𝑖 [21].

𝑖=1

• By relation 3 and point 2) of this theorem,

                                               𝜌                             𝜌       ^𝑚

                               P(𝑊 > 𝑥) ∼ 1 − 𝜌F_𝑆𝑒(𝑥) ∼ 1 − 𝜌 ∑ 𝑞_𝑖𝑟_𝑖^F_𝑆𝑒(_𝑗)(𝑥) = 𝛿^F_𝑆𝑒(_𝑗)(𝑥),                  as 𝑥 → ∞,

𝑖=1

where 𝑎 ∼ 𝑏 means 𝑎/𝑏 → 1 and

𝑚

𝜆 ∑ 𝑝

                                                                       𝑚                                  𝑖𝑟𝑖E𝑆(𝑖)

                                                                𝜌                                 𝑖=1

                                                    𝛿 = 1 − 𝜌 ∑ 𝑞_𝑖𝑟_𝑖 = _𝑚         .

                                                                       𝑖=1                   1 − 𝜆 ∑ 𝑝 E𝑆(𝑖)

_𝑖

𝑖=1

The point 4) follows from the point 3) of this theorem and [21, Theorem 8].

5) Since F_𝑆𝑒 ∈ 𝒮 then, for 𝐺𝐼/𝐺/1 system with subexponential service, the extremal index of the stationary waiting time is zero [19].    

Corollary 1. Assume that the system 𝛴 is stationary, 𝜌 < 1. Let the service time be given by an 𝑚-component mixture of distributions (11) with components ordered by the failure rate

                                                                       𝑆(1) ⩽ ⋯ ⩽ 𝑆(𝑚).                                                                                       (13)

                                                                             𝑟          𝑟

Suppose that 𝑆. Then all statements of Theorem 5 are true.

Proof. It suffices to show that the distribution F⁽_𝑒^𝑚) is 𝑟-dominant for F_𝑆𝑒. Since the ordering by failure rate of service times (13) implies the stochastic ordering of the r.v. 𝑆⁽_𝑒^𝑖),

                                                                     𝑆(1)_𝑒 ⩽ ⋯ ⩽ 𝑆(_𝑒𝑚),

                                                                             𝑠𝑡        𝑠𝑡

that

F𝑆𝑒(𝑖)(𝑥)

                                                                              ⩽ 1,        for all 𝑥.

^F_𝑆𝑒(𝑚)(𝑥)

Obviously, there exist 𝑟_𝑖 such that

F𝑆𝑒(𝑖)(𝑥) lim= 𝑟_𝑖,           0 ⩽ 𝑟_𝑖 ⩽ 1,                  𝑖 = 1,…,𝑚 − 1,            𝑟_𝑚 = 1.

𝑥→𝑥𝑅 ^F_𝑆𝑒(𝑚)(𝑥)

In this case, the distribution F⁽_𝑒^𝑚) is asymptotically 𝑟-dominant for the mixture distribution F_𝑆𝑒,

𝑟 = (𝑟₁,…,𝑟_𝑚), 𝑟_𝑚 = 1, and the conditions of Theorem 5 are satisfied for 𝑗 = 𝑚^.  .

As an example, we consider a stationary 𝐺𝐼/𝐺/1 system (i.e., 𝜌 = 𝜆E𝑆 < 1) with the service times having an exponential-Pareto distribution [22] with parameter 𝛼 > 1

^{𝑥                  𝛼} F𝑆(𝑥) = 1 − 𝑝𝑒          ^𝑆                            − (1 − 𝑝)( ⁰                   ) ,                𝜆_𝑆 > 0,       𝛼 > 1,        𝑥₀ > 0,       𝑥 ⩾ 0.

^{−𝜆 𝑥}

𝑥₀ + 𝑥

In this case the equilibrium distribution function of 𝑆_𝑒 has the form

𝑥

F𝑆𝑒(𝑥) = 𝜇∫ F𝑆(𝑡)𝑑𝑡 = 1 − 𝜇(𝑝𝑒𝜆−_𝑆𝜆𝑆𝑥 + (𝛼 − 1)(𝑥(1 − 𝑝)₀ +𝑥0𝛼𝑥)𝛼−1 ),

0

where 𝜇 = 1/E𝑆. Note that

F_𝑆𝑒(𝑥) = 𝑞₁^F_𝑆𝑒(1)(𝑥) + 𝑞₂^F_𝑆𝑒(2)(𝑥)

and

                                                     𝑞₁ = 𝜇𝑝/𝜆_𝑆,           𝑞₂ = 𝜇(1 − 𝑝)𝑥₀/(𝛼 − 1).

By Theorem 5 the limiting distribution of the maximum stationary waiting time 𝑀_𝑛 is a Frechet distribution of the form

                                                                                           (1 − 𝑝)𝜇𝜆𝑥₀           1−_𝛼

                                                                                      − _𝑥

lim P(𝑀_𝑛 ⩽ 𝑢_𝑛(𝑥)) = 𝑒 (𝛼 − 1)(𝜇 − 𝜆)

𝑛→∞

with the normalizing sequence (for 𝑥 > 0)

                                                  𝑢𝑛(𝑥) = 𝑎𝑛𝑥 + 𝑏𝑛 = 𝑥0𝑥𝑛1/(𝛼−1) − 𝑥0,            𝑛 ⩾ 1.

Indeed, it is obvious that the second component of the distribution is 𝑟-dominant, with 𝑟₁ = 0,𝑟₂ = 1,

                                                      F𝑆𝑒(1)(𝑥)                                𝑒−𝜆𝑆𝑥

𝑥→∞lim ^F_𝑆𝑒(2)(𝑥) = 𝑥→∞lim 𝑥𝛼−1/(𝑥0 + 𝑥)𝛼−1 = 0 = 𝑟1.

0

Now we find maximum domain of attraction of dominant component 𝑆⁽²⁾_𝑒 .

𝑥

                                                                                       𝛼                                          𝛼−1

                                       F𝑆𝑒(2)(𝑥) = 𝛼 − 1𝑥0 ∫ (𝑥0𝑥+0 𝑦) 𝑑𝑦 = 1 − (𝑥0𝑥+0 𝑥)        .

0

Obviously, 𝑆⁽²⁾_𝑒             has a Pareto distribution with parameters 𝛼 − 1, 𝑥₀ and therefore belongs to the class of subexponential distributions [23]. Let 𝑣_𝑛(𝑥) = 𝑥₀𝑛^{1/(𝛼−1)𝑥} − 𝑥₀. Then for 𝑛 → ∞,

                   𝑛F𝑆𝑒(2)(𝑣𝑛(𝑥)) = (𝑥0 + 𝑥0𝑛1^𝑥/(𝛼−1)⁰                 𝑥 − 𝑥0 )𝛼−1 → 𝑥^−𝛼+1,     as 𝑛 → ∞,

which implies F_𝑆𝑒(2) ∈ 𝑀𝐷𝐴(𝛷_𝛼−1) [1] where 𝛷_𝛼(𝑥) = 𝑒^{−𝑥−𝛼}, 𝑥 ⩾ 0, is Frechet distribution.

Now we can calculate 𝛿

                                                   𝜆(𝑝𝑟₁E𝑆⁽¹⁾ + (1 − 𝑝)𝑟₂E𝑆⁽²⁾)               𝜇𝜆      (1 − 𝑝)𝑥₀

                                          𝛿 =                                                                 =                                    .

                                                    1 − 𝜆(𝑝E𝑆^(𝑖) + (1 − 𝑝)E𝑆⁽²⁾)             𝜇 − 𝜆       𝛼 − 1

Figure 1. left figure: P(𝑀_𝑛 ⩽ 𝑥) and 𝛷_𝛼−1(𝑥−𝑏_𝑛^̃ )/𝑎̃_𝑛, right figure: P(𝑀_𝑛 ⩽ 𝑎̃_𝑛𝑥+𝑏_𝑛^̃ ) and 𝛷_𝛼−1(𝑥) for group size 𝑛 = 1000

^{Since F}_𝑆𝑒(2) ∈ 𝑀𝐷𝐴(𝛷_𝛼−1), then by 4) of Theorem 5, F_𝑊 ∈ 𝑀𝐷𝐴(𝛷_𝛼−1^𝛿                         ).

Thus, the asymptotic behaviour of the maximum stationary waiting time is determined by the second component (the Pareto distribution).

         Note also, that since 𝐹_𝑤 ∈ 𝑀𝐷𝐴(𝛷_𝛼−1^𝛿                                                 ) with 𝑢_𝑛(𝑥) = 𝑎_𝑛𝑥 + 𝑏_𝑛, then 𝐹_𝑤 ∈ 𝑀𝐷𝐴(𝛷_𝛼−1) with

𝑢̃_𝑛(𝑥) = 𝑎̃_𝑛𝑥 + 𝑏_𝑛̃ where 𝑎̃_𝑛 = 𝑎_𝑛𝛿^1/(𝛼−1), 𝑏_𝑛̃ = 𝑏_𝑛.

To illustrate the conclusion of Theorem 5, we carried out a numerical simulation of the system for 100 replications. We compare the estimate of P(𝑀_𝑛 ⩽ 𝑥) with 𝛷_𝛼−1(𝑥 − 𝑏_𝑛̃ )/𝑎̃_𝑛 and the estimate of P(𝑀_𝑛 ⩽ 𝑎̃_𝑛𝑥 + 𝑏_𝑛̃ ) with 𝛷_𝛼−1(𝑥). We have run a Kolmogorov–Smirnov (K-S) test for goodness of fit. We find the empirical distribution function for 𝑀_𝑛 from observed maximum waiting times for 𝑘 = 100 groups of customers (each group of size 𝑛). Figure 1 demonstrates results for 𝑥₀ = 1, 𝛼 = 5, 𝑝 = 0,5, 𝜆_𝑠 = 4, 𝜆 = 0.5 and group size 𝑛 = 1000. K-S test statistic is 0.81 for left figure and 0.74 for right figure. Therefore, our hypothesis that the Frechet distribution with the normalizing constants, 𝑎_𝑛, 𝑏_𝑛 describes nicely the maxima of waiting times (at level 0.05) is confirmed.

5.     Results

The sufficient conditions for comparing the extremal indexes of stationary waiting time in the 𝑀/𝐺/1 and 𝐺𝐼/𝑀/1 systems are obtained. We have proven that if in both systems the equation e𝑒^{𝛾(𝑆−𝜏)} = 1 has a real positive roots and the interarrival intervals and the service times are ordered in failure rate then the extremal indexes are ordered. For 𝑀/𝐺/1 the obtained result is extended to the case of a system with mixed service times with ordered components. In the case of multiserver 𝐺𝐼/𝑀/𝑐 system we have shown that it is also possible to establish a comparison of the extremal indexes. We illustrate this results on examples with some special distributions. For 𝐺𝐼/𝐺/1 system with service time determined by a finite mixture whose dominant component of the equilibrium distribution belongs to the class of subexponential distributions,we have proven that the tail of the limiting distribution of the stationary

waiting time is equivalent to the tail of this distribution up to a constant, the form of which is obtained. Furthermore, the limiting distribution of the maximum of the stationary waiting time belongs to the maximum domain of attraction of the distribution of extreme values of the same type as the maximum of the random variables defined by the dominant component, while the extremal index of

waiting time is zero.

6.     Discussion

The numerical examples given in Sections 2 and 3 of the article demonstrate the correctness of the obtained statements. Note that applying the order in failure rate makes it possible to compare systems with different interarrival distributions and different service time distributions (not just with identical distributions but different parameters).

To demonstrate the assertion of Theorem 5, we considered a system with an exponential-Pareto service-time distribution. The results of our numerical experiments show that the asymptotic distribution of maximum waiting time works well when the traffic is light. Continuing the present work, we plan to extend the numerical experiments with different distributions and investigate the

sensitivity of the approximation scheme to the group size 𝑛 and the traffic intensity 𝜌.

About the authors

Irina V. Peshkova

Petrozavodsk State University; Institute of Applied Mathematical Research of the KarRC RAS

Author for correspondence.
Email: iaminova@petrsu.ru
ORCID iD: 0000-0002-1461-2425
Scopus Author ID: 57190062292
ResearcherId: P-4375-2015

Candidate of Physical and Mathematical Sciences, Senior researcher, SMITS Lab, Institute of Applied Mathematical Research of the Karelian Research Center of Russian Academy of Sciences; head of the Applied Mathematics and Cybernetics department, Petrozavodsk State University 

33 Lenina Pr, Petrozavodsk, 185910, Russian Federation; 11 Pushkinskaya St, Petrozavodsk, 185910, Russian Federation

References

Supplementary files