Keywords

1 Introduction

The results of research of the queuing system with infinite number of servers can be found in articles of A.V. Pechinkin [1–3], A.A. Nazarov, P. Abaev, R. Razumchik [4], B. D’Auria [5], D. Baum and L. Breuer [6, 7], J. Bojarovich and L. Marchenko [8], E.A. van Doorn and A.A. Jagers [9], N.G. Duffield [10], C. Fricker and M. R. Jaïbi [11], E. Girlich [12], A. K. Jayawardene and O. Kella [13], M. Parulekar and A. M. Makowski [14] and others.

Numerous studies of real flows in various subject areas, in particular, telecommunication flows and flows in economic systems led to the conclusion about the inadequacy of the classic models of flows of random events to real data. There is an interest in investigation of flows, in which the customers are not identical and therefore require fundamentally different services [23, 24]. The queuing systems with heterogeneous devices include systems of parallel service, which can be found in articles of G.P. Basharin, K.E. Samuylov [15], A. Movaghar [16], M. Kargahi [17], J.A. Morrisson, C. Knessl [18], D.G. Down [19], N. Bambos, G. Michalidis [20] and others. In these works, all systems have a Poisson input and exponential service time. In the papers [21, 22], systems with parallel service of MMPP and renewal arrivals with paired customers are investigated.

In this paper, we study a queueing system with renewal arrival process and heterogeneous service. The main difference between the system in the paper from the previously considered ones is that when the customer comes in the system it is marked by i-th(\(i=1,\dots ,n\)) type in order to given probabilities. Service times for customers of different types has different arbitrary distribution function.

2 Statement of the Problem

Consider the queuing system with infinite number of servers of n different types and arbitrary service time. Incoming flow is a renewal arrival process with n types of customers. Recurrent incoming flow is determined by the distribution function A(x) of the lengths of the intervals between the time of occurrence of renewal arrival process. At the time of occurrence of the event in this stream only one customer comes in the system. The type of incoming customer is defined as i-type with probability \(p_i\) \((i=1,\dots ,n)\). It is servicing during a random time having an arbitrary distribution function \(B_i\) corresponding to the type of the customer.

Set the problem of analysis of n-dimensional stochastic process \(\{l_1 (t),\) \(l_2 (t),\dots ,l_n(t)\}\) of the number of busy servers of each type at the moment t. Incoming stream is not Poisson, therefore the n-dimensional process \(\{l_1 (t),l_2 (t),\dots ,l_n(t)\}\) is non-Markov. Consider a \((n+1)\)-dimensional Markov process \(\{z(t),l_1(t),l_2(t),\dots ,l_n(t)\}\),where z(t) —the remaining time from t until the occurrence of the following event of renewal arrival process.

Denote: \(\{r_1(T),\ldots ,r_n(T)\}\) —the number of customers who have not completed service at time T and enrolled in at the time \(t,\;t<T\);

\(S_i(t)=P\{\tau _k^{(i)}>T-t\}=1-B_i(T-t)\) —the probability of non-completion of the service application type \(i,\;(i=1,\ldots ,n)\);

\(1-S_i(t)\) —the probability of completion of the service application type \(i,\;(i=1,\ldots ,n)\).

Let at the initial moment of time \(t_0<T\) the system is empty, i.e.\(l_1(t_0)=\ldots =l_n(t_0)=0.\) Then \(l_1(T)=r_1(T),\ldots ,l_n(T)=r_n(T)\). Thus to study the process \(\{l_1(t),\ldots ,l_n(t)\}\) it is necessary to investigate the n-dimensional process \(\{r_1(t),\ldots ,r_n(t)\}\) at any point of time \(t_0\le t\le T\) and put \(t=T\).

A random \((n+1)\)-dimensional process \(\{z(t),r_1(t),\ldots ,r_n(t)\}\) is a \((n+1)\)-dimensional non-stationary Markov chain. Write the system of Kolmogorov differential equations for the joint probability distribution \(P\{z,r_1,\ldots ,r_n,t\}\)

$$\begin{aligned}&\quad \, \frac{\partial P(z,r_1,\dots ,r_n,t)}{\partial t}=\frac{\partial P(z,r_1,\dots ,r_n,t)}{\partial z}+\frac{\partial P(0,r_1,\dots ,r_n,t)}{\partial z}(A(z)-1)\\&+\frac{\partial P(0,r_1-1,\dots ,r_n,t)}{\partial z}p_1S_1(t)A(z)+\ldots +\frac{\partial P(0,r_1,\dots ,r_n-1,t)}{\partial z}p_nS_n(t)A(z) \end{aligned}$$
$$\begin{aligned} -\frac{\partial P(0,r_1,\dots ,r_n,t)}{\partial z}A(z)\sum _{i=1}^{n}p_iS_i(t). \end{aligned}$$
(1)

Introduce the characteristic function of the form:

$$\begin{aligned} H(z,u_1,\dots ,u_n,t)=\sum \limits _{r_1=0}^{\infty }\dots \sum \limits _{r_n=0}^{\infty }e^{ju_1r_1}\times \dots \times e^{ju_nr_n}P(z,r_1,\dots ,r_n,t),\nonumber \end{aligned}$$

where \(j=\sqrt{-1}\) – imaginary unit.

Using (1) write the system of differential equations for the characteristic function \(H(z,u_1,\dots ,u_n,t)\)

$$\begin{aligned} \frac{\partial H(z,u_1,\ldots ,u_n,t)}{\partial t}=\frac{\partial H(z,u_1,\ldots ,u_n,t)}{\partial z}+\frac{\partial H(0,u_1,\ldots ,u_n,t)}{\partial z}(A(z)-1) \end{aligned}$$
$$\begin{aligned} +\frac{\partial H(0,u_1,\ldots ,u_n,t)}{\partial z}A(z)\sum _{i=1}^{n}p_iS_i(t)(e^{ju_i}-1), \end{aligned}$$
(2)
$$\begin{aligned} H(z,u_1,\dots ,u_n,t_0)=R(z),\nonumber \end{aligned}$$

where R(z) - stationary probability distribution of the stochastic process z(t).

3 Method of the Asymptotic Analysis

3.1 Asymptotics of the First Order

We will solve the basis equation for the characteristic function (2) in the asymptotic condition that service time on appliances growths equivalently to each other, viz. \(b_i\rightarrow \infty ,\) where \(b_i=\int _{0}^{\infty }(1-B_i(x))dx,\;i=1,\dots ,n\) —the average value of the service time customer such as the i-th.

Denote

$$\begin{aligned} t\varepsilon =\tau ,\;t_0\varepsilon =\tau _0,\;b_i=\frac{1}{q_i\varepsilon },\;u_i=\varepsilon x_i, \end{aligned}$$
(3)
$$S_i(t)=\tilde{S}_i(\tau ),\;i=1,\ldots ,n,\;H(z,u_1,\ldots ,u_n,t)=F_1(z,x_1,\ldots ,x_n,\tau ,\varepsilon ).$$

Taking into account (3) we can write (2) as

$$\begin{aligned} \varepsilon \frac{\partial F_1(z,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial \tau }=\frac{\partial F_1(z,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z} \end{aligned}$$
(4)
$$\begin{aligned} +\frac{\partial F_1(0,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}(A(z)-1)+\frac{\partial F_1(0,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}A(z)\sum _{i=1}^{n}p_i\tilde{S}_i(\tau )(e^{j\varepsilon x_i}-1).\nonumber \end{aligned}$$

Lemma 1

Limit value function \(F_1(z,x_1,\ldots ,x_n,\tau ,\varepsilon )\) at \(\varepsilon \rightarrow 0\) has the form

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}F_1(z,x_1,\dots ,x_n,\tau ,\varepsilon )=F_1(z,x_1,\dots ,x_n,\tau )\nonumber \end{aligned}$$
$$\begin{aligned} =R(z)\exp \left\{ j\lambda \sum _{i=1}^np_ix_i\int _{\tau _o}^{\tau }\tilde{S}_i(w)dw\right\} , \end{aligned}$$
(5)

where \(\lambda =\frac{\partial R(0)}{\partial z}.\)

Proof

If \(\varepsilon \rightarrow 0\) in (4), then obtain:

$$\begin{aligned} \frac{\partial F_1(z,x_1,\ldots ,x_n,\tau )}{\partial z}+\frac{\partial F_1(0,x_1,\ldots ,x_n,\tau )}{\partial z}(A(z)-1)=0. \end{aligned}$$
(6)

Then we look for \(F_1(z,x_1,\dots ,x_n,\tau )\) as

$$\begin{aligned} F_1(z,x_1,\dots ,x_n,\tau )=R(z) \varPhi _1(x_1,\dots ,x_n,\tau ), \end{aligned}$$
(7)

where \(\varPhi _1(x_1,\dots ,x_n,\tau )\) - the desired function.

If \(z\rightarrow \infty \) in (4), then obtain:

$$\begin{aligned} \varepsilon \frac{\partial F_1(\infty ,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial \tau }=\frac{\partial F_1(0,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}\sum _{i=1}^{n}p_i\tilde{S}_i(\tau )(e^{j\varepsilon x_i}-1). \end{aligned}$$
(8)

Expand exponents in the Eq. (8) into a Taylor series, divide the left and right side of it by \(\varepsilon \), substitute into the received expression the function \(F_1(z,x_1,\dots ,x_n,\tau )\) in the form (7) and let \(\varepsilon \rightarrow 0\):

$$\begin{aligned} \frac{\partial \varPhi _1(x_1,\ldots ,x_n,\tau )}{\partial \tau }=j\frac{\partial R(0)}{\partial z}\varPhi (x_1,\ldots ,x_n,\tau )\sum _{i=1}^n p_i\tilde{S}_i(\tau )x_i. \end{aligned}$$
(9)

Taking into account the initial condition \(\varPhi _1(x_1,\dots ,x_n,\tau _0)=1\) we obtain the following expression

$$\begin{aligned} \varPhi _1(x_1,\dots ,x_n,\tau )=\exp \left\{ j\lambda \sum _{i=1}^np_ix_i\int _{\tau _0}^{\tau }\tilde{S}_i(w)dw\right\} . \end{aligned}$$
(10)

Thus,

$$\begin{aligned} F_1(z,x_1,\ldots ,x_n,\tau )=R(z)\exp \left\{ j\lambda \sum _{i=1}^np_ix_i\int _{\tau _0}^{\tau }\tilde{S}_i(w)dw\right\} .\nonumber \end{aligned}$$

   \(\square \)

Taking into account Lemma 1 and substitutions (3) we can write the asymptotic approximate equality (\(\varepsilon \rightarrow 0\)):

$$\begin{aligned} H(z,u_1,\dots ,u_n,t)=F_1(z,x_1,\dots ,x_n,\tau ,\varepsilon )\approx F_1(z,x_1,\dots ,x_n,\tau ) \end{aligned}$$
$$\begin{aligned} =R(z)\exp \left\{ j\lambda \sum _{i=1}^np_iu_i\int _{t_0}^{t}S_i(w)dw\right\} . \end{aligned}$$
(11)

For the characteristic function of process \(\{l_1(t),\dots ,l_n(t)\}\) at \(t=T=0\) denote

$$\begin{aligned} h_1(u_1,\dots ,u_n)=\exp \left\{ j\lambda \sum _{i=1}^np_iu_i\int _{-\infty }^{0}(1-B_i(-w))dw\right\} \nonumber \end{aligned}$$
$$\begin{aligned} =\exp \left\{ j\lambda \sum _{i=1}^np_iu_ib_i\right\} . \end{aligned}$$
(12)

The function \(h_1(u_1,\dots ,u_n)\) will be called the asymptotics of the first order for the system \( GI| GI|\infty \) with heterogeneous service.

Defenition 1

The functions

$$\begin{aligned} {h_1}^{(i)}(u_i)=Me^{ju_il_i(t)}=h_1(0,\dots ,u_i,\dots ,0)=\exp \{j\lambda p_iu_ib_i\},\; i=1,\dots ,n\nonumber , \end{aligned}$$

will be called the asymptotics of the first order for the characteristic function of the busy servers of any type in system \(GI|GI|\infty \) with heterogeneous service.

Consider the asymtotics of the second order for more accurate approximation.

3.2 Asymptotics of the Second Order

Consider the function \(H(z,u_1,\dots ,u_n,t)\) in the form of

$$\begin{aligned} H(z,u_1,\ldots ,u_n,t)=H_2(z,u_1,\dots ,u_n,t)\exp \left\{ j\lambda \sum _{i=1}^np_iu_i\int _{t_0}^{t}S_i(w)dw\right\} . \end{aligned}$$
(13)

Using (13) in (2) obtain the expression for \(H_2(z,u_1,\dots ,u_n,t)\):

$$\begin{aligned} \frac{\partial H_2(z,u_1,\ldots ,u_n,t)}{\partial t}+H_2(z,u_1,\ldots ,u_n,t)j\lambda \sum _{i=1}^{n}p_iS_i(t)u_i\nonumber \end{aligned}$$
$$\begin{aligned} =\frac{\partial H_2(z,u_1,\ldots ,u_n,t)}{\partial z}+\frac{\partial H_2(0,u_1,\ldots ,u_n,t)}{\partial z}(A(z)-1) \end{aligned}$$
(14)
$$\begin{aligned} +\frac{\partial H_2(0,u_1,\ldots ,u_n,t)}{\partial z}A(z)\sum _{i=1}^np_iS_i(t)(e^{ju_i}-1),\nonumber \end{aligned}$$

where \(\lambda =\frac{\partial R(0)}{\partial z}\).

Substitute the following in (14):

$$\begin{aligned} t\varepsilon ^2=\tau ,\;t_0\varepsilon ^2=\tau _0,\;b_i=\frac{1}{q_i\varepsilon ^2},\;u_i=\varepsilon x_i, \end{aligned}$$
(15)
$$S_i(t)=\tilde{S}_i(\tau ),\;i=1,\ldots ,n,\;H_2(z,u_1,\ldots ,u_n,t)=F_2(z,x_1,\ldots ,x_n,\tau ,\varepsilon )$$

and obtain:

$$\begin{aligned} \varepsilon ^2\frac{\partial F_2(z,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial \tau }+F_2(z,x_1,\ldots ,x_n,\tau ,\varepsilon )j\lambda \varepsilon \sum _{i=1}^np_ix_i\tilde{S}_i(\tau )\nonumber \end{aligned}$$
$$\begin{aligned} =\frac{\partial F_2(z,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}+\frac{\partial F_2(0,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}(A(z)-1) \end{aligned}$$
(16)
$$\begin{aligned} +\frac{\partial F_2(0,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}A(z)\sum _{i=1}^np_i\tilde{S}_i(\tau )(e^{j\varepsilon x_i}-1).\nonumber \end{aligned}$$

Theorem 1

Limit value function \(F_2(z,x_1,\ldots ,x_n,\tau ,\varepsilon )\) at \(\varepsilon \rightarrow 0\) has the form

$$\begin{aligned} \begin{aligned}&\qquad \qquad \qquad \lim \limits _{\varepsilon \rightarrow 0}F_2(z,x_1,\ldots ,x_n,\tau ,\varepsilon )=F_2(z,x_1,\ldots ,x_n,\tau ) \\&\qquad \qquad \qquad \quad =R(z)\exp \left\{ j^2\left[ \lambda \sum _{i=1}^np_i\frac{x_i^2}{2}\int _{\tau _0}^{\tau }\tilde{S}_i(w)dw\right. \right. \\&\left. \left. +\sum _{i=1}^np_i^2x_i^2\frac{\partial f_i(0)}{\partial z}\int _{\tau _0}^{\tau }\tilde{S}_i^2(w)dw+\sum _{i=1}^{n}\sum _{g=1,g\ne i}^{n}p_ip_gx_ix_g\int _{\tau _0}^{\tau }\tilde{S}_i(w)\tilde{S}_g(w)dw\right] \right\} , \end{aligned} \end{aligned}$$
(17)

where \(\lambda =\frac{\partial R(0)}{\partial z}\) and functions \(f_i(z)\) are defined by the following system of equations

$$\begin{aligned} \frac{\partial f_i(z)}{\partial z}+\frac{\partial f_i(0)}{\partial z}(A(z)-1)+\lambda A(z)=\lambda R(z),\;i=1,\dots ,n. \end{aligned}$$
(18)

Proof

Desirable solution of the Eq. (16) should be like the following:

$$\begin{aligned} \begin{aligned}&F_2(z,x_1,\dots ,x_n,\tau ,\varepsilon )=\varPhi _2(x_1,\dots ,x_n,\tau ) \\&\times \left\{ R(z)+j\varepsilon \sum _{i=1}^np_ix_if_i(z)\tilde{S}_i(\tau )\right\} +O(\varepsilon ^2). \end{aligned} \end{aligned}$$
(19)

Using (19) in (16), obtain:

$$\begin{aligned} R(z)j\varepsilon \lambda \sum _{i=1}^np_ix_i\tilde{S}_i(\tau )=\frac{\partial R(z)}{\partial z}+\frac{\partial R(0)}{\partial z}(A(z)-1) \end{aligned}$$
(20)
$$\begin{aligned} +j\varepsilon \sum _{i=1}^np_ix_i\tilde{S}_i(\tau )\left\{ \frac{\partial f_i(z)}{\partial z}+(A(z)-1)\frac{\partial f_i(0)}{\partial z}+\lambda A(z)\right\} +O(\varepsilon ^2).\nonumber \end{aligned}$$

Hence taking into account \(\frac{\partial R(z)}{\partial z}+\frac{\partial R(0)}{\partial z}(A(z)-1)=0\) may earn the following system of equations for the functions \(f_i(z),\,i=1,\dots ,n\) when \(\varepsilon \rightarrow 0\):

$$\begin{aligned} \frac{\partial f_i(z)}{\partial z}+\frac{\partial f_i(0)}{\partial z}(A(z)-1)+\lambda A(z)=\lambda R(z),\nonumber \end{aligned}$$

which coincides with (18).

Expand exponents in the Eq. (16) into a Taylor series:

$$\begin{aligned}&\quad \varepsilon ^2\frac{\partial F_2(z,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial \tau }=(j\varepsilon )^2A(z)\sum _{i=1}^{n}p_i\frac{x_i^2}{2}\tilde{S}_i(\tau )\frac{\partial F_2(0,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}\\&+(j\varepsilon )\left[ A(z)\sum _{i=1}^{n}p_ix_i\tilde{S}_i(\tau )\frac{\partial F_2(0,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}-\lambda \sum _{i=1}^{n}p_ix_i\tilde{S}_i(\tau )F_2(z,x_1,\ldots ,x_n,\tau ,\varepsilon )\right] \\&\quad \quad +\frac{\partial F_2(z,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}+(A(z)-1)\frac{\partial F_2(0,x_1,\ldots ,x_n,\tau ,\varepsilon )}{\partial z}+O(\varepsilon ^3). \end{aligned}$$

Substitute into received expression (19). Since \(\frac{\partial R(z)}{\partial z}+\frac{\partial R(0)}{\partial z}(A(z)-1)=0\) we can write

$$\begin{aligned}&\qquad \qquad \varepsilon ^2\frac{\partial \varPhi _2(x_1,\ldots ,x_n,\tau )}{\partial \tau }R(z)=(j\varepsilon )^2\varPhi _2(x_1,\ldots ,x_n,\tau )\\&\quad \times \left[ A(z)\lambda \sum _{i=1}^{n}p_i\frac{x_i^2}{2}\tilde{S}_i(\tau )+A(z)\sum _{i=1}^{n}p_ix_i\tilde{S}_i(\tau )\sum _{g=1}^{n}p_gx_g\tilde{S}_g(\tau )\frac{\partial f_g(0)}{\partial z}\right. \\&\left. -\lambda \sum _{i=1}^{n}p_ix_i\tilde{S}_i(\tau )\sum _{g=1}^{n}p_gx_g\tilde{S}_g(\tau )f_g(z)\right] +j\varepsilon \varPhi _(x_1,\ldots ,x_n,\tau )\sum _{i=1}^{n}p_ix_i\tilde{S}_i(\tau )\\&\qquad \quad \times \left[ \lambda A(z)-\lambda R(z)+\frac{\partial f_i(z)}{\partial z}+(A(z)-1)\frac{\partial f_i(0)}{\partial z}\right] +O(\varepsilon ^3). \end{aligned}$$

Using (18) we obtain the following expression:

$$\begin{aligned} \begin{aligned}&\qquad \qquad \varepsilon ^2\frac{\partial \varPhi _2(x_1,\ldots ,x_n,\tau )}{\partial \tau }R(z)=(j\varepsilon )^2\varPhi _2(x_1,\ldots ,x_n,\tau )\\&\times \left[ A(z)\lambda \sum _{i=1}^{n}p_i\frac{x_i^2}{2}\tilde{S}_i(\tau )+A(z)\sum _{i=1}^{n}p_ix_i\tilde{S}_i(\tau )\sum _{g=1}^{n}p_gx_g\tilde{S}_g(\tau )\frac{\partial f_g(0)}{\partial z}\right. \\&\qquad \qquad \quad \left. -\lambda \sum _{i=1}^{n}p_ix_i\tilde{S}_i(\tau )\sum _{g=1}^{n}p_gx_g\tilde{S}_g(\tau )f_g(z)\right] +O(\varepsilon ^3). \end{aligned} \end{aligned}$$
(21)

Divide both sides of the expression (21) by \(\varepsilon ^2\) and pass to the limit provided \(\varepsilon \rightarrow 0\) and \(z\rightarrow \infty \):

$$\begin{aligned} \begin{aligned}&\qquad \qquad \frac{\partial \varPhi _2(x_1,\ldots ,x_n,\tau )}{\partial \tau }=j^2\varPhi _2(x_1,\ldots ,x_n,\tau )\\&\times \left[ \lambda \sum _{i=1}^{n}p_i\frac{x_i^2}{2}\tilde{S}_i(\tau )+\sum _{i=1}^{n}p_ix_i\tilde{S}_i(\tau )\sum _{g=1}^{n}p_gx_g\tilde{S}_g(\tau )\frac{\partial f_g(0)}{\partial z}\right] . \end{aligned} \end{aligned}$$
(22)

Solution of the differential Eq. (22) corresponding to the initial condition \(\varPhi _2(x_1,\ldots ,x_n,\tau _0)=1\) is the function \(\varPhi _2(x_1,\ldots ,x_n,\tau )\) of the form:

$$\begin{aligned} \varPhi _2(x_1,\ldots ,x_n,\tau )=&\exp \left\{ j^2\left[ \lambda \sum _{i=1}^np_i\frac{x_i^2}{2}\int _{\tau _0}^{\tau }\tilde{S}_i(w)dw +\sum _{i=1}^np_i^2x_i^2\frac{\partial f_i(0)}{\partial z}\int _{\tau _0}^{\tau }\tilde{S}_i^2(w)dw\right. \right. \nonumber \\&\left. \left. +\sum _{i=1}^{n}\sum _{g=1, g\ne i}^{n}p_ip_gx_ix_g\frac{\partial f_i(0)}{\partial z}\int _{\tau _0}^{\tau }\tilde{S}_i(w)\tilde{S}_g(w)dw\right] \right\} . \end{aligned}$$
(23)

   \(\square \)

Taking into account the approximate equations of the form

$$\begin{aligned} H_2(z,u_1,\dots ,u_n,t)=F_2(z,x_1,\dots ,x_n,\tau ,\varepsilon ) \end{aligned}$$
$$\begin{aligned} \approx F_2(z,x_1,\dots ,x_n,\tau )=R(z)\varPhi _2(x_1,\dots ,x_n,\tau ). \end{aligned}$$

Using (15) write expression for the function \(H_2(z,u_1,\dots ,u_n,t)\):

$$\begin{aligned}&\qquad \qquad H_2(z,u_1,\ldots ,u_n,t)=R(z)\exp \left\{ j^2\left[ \lambda \sum _{i=1}^np_i\frac{u_i^2}{2}\int _{t_0}^{t}S_i(w)dw\right. \right. \\&\left. \left. +\sum _{i=1}^np_i^2u_i^2\frac{\partial f_i(0)}{\partial z}\int _{t_0}^{t}S_i^2(w)dw+\sum _{i=1}^{n}\sum _{g=1, g\ne i}^{n}p_ip_gu_iu_g\frac{\partial f_i(0)}{\partial z}\int _{t_0}^{t}S_i(w)S_g(w)dw\right] \right\} . \end{aligned}$$

Then using (13) we obtain:

$$\begin{aligned}&\qquad H(z,u_1,\ldots ,u_n,t)=R(z)\exp \left\{ j\lambda \sum _{i=1}^np_iu_i\int _{t_0}^{t}S_i(w)dw\right. \\&+\left. j^2\left[ \lambda \sum _{i=1}^np_i\frac{u_i^2}{2}\int _{t_0}^{t}S_i(w)dw+\sum _{i=1}^np_i^2u_i^2\frac{\partial f_i(0)}{\partial z}\int _{t_0}^{t}S_i^2(w)dw\right. \right. \\&\qquad \quad \left. \left. +\sum _{i=1}^{n}\sum _{g=1, g\ne i}^{n}p_ip_gu_iu_g\frac{\partial f_i(0)}{\partial z}\int _{t_0}^{t}S_i(w)S_g(w)dw\right] \right\} . \end{aligned}$$

Denote

$$\begin{aligned}&\int _{-\infty }^{0}S_i^2(w)dw=\int _{-\infty }^{0}(1-B_i(-w))^2dw=\int _{0}^{\infty }(1-B_i(w))^2dw=\beta _i,\\&\qquad \int _{-\infty }^{0}S_i(w)S_g(w)dw=\int _{-\infty }^{0}(1-B_i(-w))(1-B_g(-w))dw\\&\qquad \qquad \qquad =\int _{0}^{\infty }(1-B_i(w))(1-B_g(w))dw=\beta _{ig},\\&\qquad \qquad \qquad \qquad \quad i=1,\ldots ,n,\;g=1,\ldots ,n. \end{aligned}$$

Then for the characteristic function of the random process \(\{l_1(t),l_2(t),\dots ,\) \(l_n(t)\}\) \(h_2(u_1,\ldots ,u_n)=Me^{j\sum \limits _{i=1}^nu_ll_i(T)}=H(\infty ,u_1,\dots ,u_n,T)\) at \(t=T=0\) and \(t_0\rightarrow -\infty \) we obtain

$$\begin{aligned} \begin{aligned}&h_2(u_1,\ldots ,u_n)=\exp \left\{ j\lambda \sum _{i=1}^np_iu_ib_i+j^2\left[ \lambda \sum _{i=1}^np_i\frac{u_i^2}{2}b_i\right. \right. \\&+\left. \left. \sum _{i=1}^np_i^2u_i^2\frac{\partial f_i(0)}{\partial z}\beta _i+\sum _{i=1}^{n}\sum _{g=1, g\ne i}^{n}p_ip_gu_iu_g\frac{\partial f_i(0)}{\partial z}\beta _{ig}\right] \right\} . \end{aligned} \end{aligned}$$
(24)

The expression (24) will be called the asymptotics of the second order for the system \( GI| GI|\infty \) with heterogeneous service.

4 Conclusion

In this paper, we construct and investigate the mathematical model of the queuing system with the renewal arrival process and heterogeneous service. The system under consideration is studied using asymptotic analysis. Namely, the expression for the asymptotic of the first and the second order are obtained for the characteristic function of the busy servers of each type.