UDC 517.516

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR L. A. LYUSTERNIK

ON A PROBLEM IN QUEUEING THEORY AND A RELATED GENERALIZATION OF THE CYLINDRICAL FUNCTIONS \(J_k\)

1. It will be convenient for us to use coordinates \(y_j,\ j=0,1,\ldots,n\), in the \(n\)-dimensional Euclidean space \(E_n\), which we shall call barycentric (abbreviated b.c.) (in what follows \(\sum_j\) and \(\prod_j\) denote \(\sum_{j=0}^n\), \(\prod_{j=0}^n\)). Let the unit vectors \(e_j,\ j=0,1,\ldots,n\), be the vertices of a regular \(n\)-dimensional simplex \(D_n\), whose center lies at the origin \(O\). Every vector \(Y \subset E_n\) will be represented in the form \(Y=\sum_j y_j e_j=[y_j]\), where \(y_j\) are b.c. of the vector \(Y\). Since, for any \(y\), \([y_j]=[y_j+y]\), one may fix one of the b.c. or their sum. For \(\sum_j y_j=1\) we obtain the usual b.c. Setting
\[ y_{\mathrm{cp}}=\frac{1}{n+1}\sum_j y_j, \]
we obtain the canonical b.c. \(y'_j\) of the vector \([y_i]=[y'_j]\),
\[ y'_j=y_j-y_{\mathrm{cp}},\qquad \sum_j y'_j=0. \]
Denoting \(\bar y=\min_j y_j,\ \bar y_j=y_j-\bar y;\ \bar y_j\ge 0,\ \min \bar y_j=0\), we shall call \(\bar y_j\) the second canonical b.c. of the vector \([y_j]=[\bar y_j]\). For \(Y=[y_i]=[y'_j]\) we have
\[ \|Y\|^2=\frac{n+1}{n}\sum_j y_j^2-\frac{1}{n}\left(\sum_j y_j\right)^2=\frac{n+1}{n}\sum_j y_j'^2= \]
\[ =\frac{n+1}{n}\sum_{k=1}^n \frac{k}{k+1}\left[y_k-\frac{1}{k}\sum_{i=0}^{k-1}y_i\right]^2. \tag{1} \]

By \(\Omega_n\) we shall denote the lattice of integer vectors \(K=[k_j]\) in \(E_n\), where \(k_j\) are integers; \(W_n\) is the Dirichlet region of the lattice \(\Omega_n\), its volume
\[ |W_n|=\sqrt{(n+1)^{\,n-1}/n^n}. \]
For a function \(f(Y)=f([y_j])\) defined in \(E_n\),
\[ \sum_j \frac{\partial f}{\partial y_j}=0;\qquad \sum_j \frac{\partial^2 f}{\partial y_j^2}=\frac{n+1}{n}\Delta f=\frac{n+1}{n}\sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2} \tag{2} \]
(\(x_i\) are Cartesian coordinates).

2. The assembly problem. A generalization of the problem of taxis and passengers \((^1)\) is the assembly problem. Given are \((n+1)\) streams \(P_j,\ j=0,1,\ldots,n\). The probability that by time \(t\) there arrive \(k\) elements of the stream \(P_j\) is equal to \(p_j(k,t)\). We shall regard the streams as Poisson:
\[ p_j(k,t)=e^{-\lambda_j t}\frac{(\lambda_j t)^k}{k!}\qquad \left(\frac{1}{k!}=0\ \text{for } k<0\right). \tag{3} \]

Abstracting from the assembly time, we assume that if by time \(t\) there have appeared \(k_j\) elements of the stream \(P_j,\ j=0,1,\ldots,n\), then \(k=\min_j k_j\) elements of the output stream have been formed—\((n+1)\)-tuples, each of which consists of \(n+1\) elements of the streams \(P_j,\ j=0,1,\ldots,n\), one from each \(P_j\). At the same time queues have formed of \(\bar k_j=k_j-\bar k\) elements of the stream \(P_j\). All \(\bar k_j\ge 0,\ \min_j \bar k_j=0\). We shall say: the state vector \((k_j),\ j=0,1,\ldots,n\), of the system has generated the queue vector \((\bar k_j)\). Likewise, the queue vector gener-

are vector states \((k_j+m)\), \(m\) an arbitrary integer. We form the vector \(K=[\bar k_j]=[k_j]=[k_j+m]\subset \Omega_n\). The components \(\bar k_j\) of the vector-queue \((k_j)\) are the second canonical b.c., while the vector states generating it are the various integer b.c. of this vector. Therefore we shall say: a vector-queue \(K=[\bar k_j]=[k_j]\) has formed; the probabilities \(P(K,t)\) of formation at time \(t\) of such a queue are equal \(\left(\sum_m=\sum_{m=1}^{\infty}\right)\):

\[ P(K,t)=\sum_m\prod_j p_j(k_j+m,t) =\exp\left(-\sum_j \lambda_j t\right)U_K^{(n)}(\lambda_j t), \tag{4} \]

where

\[ U_K^{(n)}(x_j)=U_{(k_j)}^{(n)}(x_j) =U_{k_0,k_1,\ldots,k_n}^{(n)}(x_0,x_1,\ldots,x_n)= \]

\[ =U_{(k_j+m)}^{(n)}(x_j) =\sum_m\prod_j \frac{x_j^{m+k_j}}{(m+k_j)!}. \tag{5} \]

For \(x_0=x_1=\cdots=x_n=x\) we shall have a function of one variable \(x\):

\[ U_K^{(n)}(x)=U_{k_0,k_1,\ldots,k_n}^{(n)}(x) =\sum_m\prod_j \frac{x^{m+k_j}}{(m+k_j)!} =\sum_m \frac{x^{(n+1)m+k}}{\prod_j (m+k_j)!}, \qquad k=\sum_j k_j. \]

For \(n=1\), \(U_{k_0,k_1}^{(1)}(x)=J_{k_1-k_0}(2x)\).

The functions introduced are only loosely connected with Akimov’s generalized Bessel functions \((^2)\).

3. Generating function. Let \(\prod_j t_j=1\); for \(K=[k_j]\), \(t^K=\prod_j t_j^{k_j}\); then

\[ \exp\left(\sum_j t_j x_j\right) = \sum_{K\subset \Omega_n} t^K U_K^{(n)}(x_j). \tag{6} \]

From this follow the addition theorem (7) and formulas (8)—(10):

\[ U_K^{(n)}(x_j+y_j) = \sum_{K'\subset \Omega_n} U_{K'}^{(n)}(x_j)\,U_{K-K'}^{(n)}(y_j); \tag{7} \]

\[ \frac{\partial}{\partial x}U_K^{(n)}(x) = \frac{1}{n+1}U_{K-e_j}^{(n)}(x); \tag{8} \]

\[ \sum_j U_{K-e_j}^{(n)}(x_i)\,x_j e_j = U_K^{(n)}(x_i)\,K; \tag{9} \]

\[ \sum_j x_j e_j = \sum_{K\subset \Omega_n} K U_K^{(n)}(x_i). \tag{10} \]

Let \(Y\subset [y_j]=[y'_j]\), \(K=[k_j]\); \(KY=\sum_j k_j y_j\); \(a_K(Y)=\exp(2\pi iKY)\). The system \(\{a_K(Y)\}_{K\subset\Omega_n}\) is orthogonal in \(Y\) in \(W_n\), complete in the class of functions periodic with periods \(e_j\) and with integrable squares on \(W_n\). Every such function \(f(Y)\) is representable by the Fourier series

\[ f(Y)=\sum_{K\subset\Omega_n} c_K a_K(Y), \qquad c_K=\frac{1}{|W_n|}\int_{W_n}\cdots\int f(Y)a_K(Y)\,d\omega_n. \tag{11} \]

Putting in (6) \(t_j=\exp(2\pi i y'_j)\), \(\sum_j y'_j=0\), \(Y=[y'_j]\), we obtain:

\[ \exp\left(\sum_j x_j \exp(2\pi i y'_j)\right) = \sum_{K\subset\Omega_n} U_K(Y)a_K(Y). \tag{12} \]

Hence, from (11) follows the integral representation

\[ U_{(k_j)}^{(n)}(x_j)=U_K^{(n)}(x_j)= \frac{1}{|W_n|}\int_{W_n}\cdots\int \exp\left[\sum_j\left(\exp(2\pi i y_j')-2\pi i k_j y_j\right)\right]\,d\omega_n . \tag{13} \]

4. Asymptotics. Let \(x\gg 1\), \(k_j=O(\sqrt{x})\), \(K=[k_j]\subset \Omega_n\),

\[ U_K^{(n)}(x)= \frac{1}{\sqrt{(n+1)(2\pi x)^n}} \exp\left[-\frac{n\|K\|^2}{2(n+1)x}\right] \left[1+O\left(\frac{1}{\sqrt{x}}\right)\right]. \tag{14} \]

Let

\[ x\gg 1;\qquad \lambda_j>0;\qquad z_j=\lambda_j-k_jx=O(\sqrt{x}); \]

\[ S_k=\sum_{i=0}^{k}\frac{1}{\lambda_i},\qquad k=0,1,\ldots,n;\qquad z^{(k)}=\frac{1}{S_{k-1}}\sum_{i=0}^{k-1}\frac{z_i}{\lambda_i}, \qquad k=1,2,\ldots,n+1; \]

\[ \Phi(z_j)= \sum_j \frac{z_j^2}{\lambda_j} -\frac{1}{S_n}\left(\sum_j\frac{z_j}{\lambda_j}\right)^2 \equiv \sum_j\frac{1}{\lambda_j}\left(z_j-z^{(n+1)}\right)^2 \equiv \]

\[ \equiv \sum_{k=1}^{n}\frac{S_{k-1}}{\lambda_k S_k} \left(z_k-z^{(k)}\right)^2; \]

\[ U_K^{(n)}(z_j)= \left[\prod_j \lambda_j/S_n(2\pi x)^n\right]^{1/2} \exp\left[-\frac{1}{2x}\Phi(z_j)\right] \left[1+O\left(\frac{1}{\sqrt{x}}\right)\right]; \tag{15} \]

\[ O\left(\frac{1}{\sqrt{x}}\right)\sim \sum_{p=1}^{\infty}\frac{a_p}{(\sqrt{x})^p}; \]
\(a_p\) is a polynomial of degree \(p\) in \(k_i/\sqrt{x}\) in (14), and in \(z_i/\sqrt{x}\) in (15).

Moscow State University
named after M. V. Lomonosov

Received
15 IX 1967

REFERENCES

¹ Kh. Sh. Margulis, Collection Cybernetics in the Service of Communism, 2, 1964, p. 266. ² M. I. Akimov, On Bessel Functions of Many Variables and Their Applications in Mechanics, Petrograd, 1922.