In a queueing network with a single server and <mml:math alttext="$r$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math> service nodes, a non-preemptive 
non-idling policy chooses a node to service at each service completion epoch. 
Under the assumptions of independent Poisson arrival processes, fixed routing 
probabilities, and linear holding cost rates, we apply Whistle's method for Arm-acquiring bandits to show that for minimizing discounted cost or long-run 
average cost the optimal policy is an index policy. We also give explicit 
expressions for those priority indices.

Chuanshu Ji

Sequential scheduling of priority queues and Arm-acquiring bandits

The author extends his alternative theory for Schr&#246;dinger quantum 
mechanics by introducing the idea of energy reference strata over configuration 
space. It is then shown that the view from various such strata defines, the 
content of the system of interest and enables a variety of different descriptions of 
events in the same space time region. Thus according to &#147;the point of view&#148; or 
energy stratum chosen so the type of Schr&#246;dinger equation, linear or otherwise, 
appropriate to describe the system is determined. A nonlinear information 
channel between two dimensional fluid action in hyperspace into two 
dimensional energy hyperspace is shown to exist generally as a background to 
nonlinear Schr&#246;dinger structures. In addition it is shown how soliton solutions 
of the one dimensional Schr&#246;dinger equation are related to two dimensional 
vortex fields in hyperspace.

James G. Gilson

Classical fluid aspects of nonlinear Schr&#246;dinger equations and solitons

Using the integral representation of the solution of the boundary 
value problem for the equation with one time-dependent coefficient at 
the highest space-derivative three inverse problems are solved. 
Depending on the property of the coefficient we consider cases when the 
equation is of the parabolic type and two special cases of the 
degenerate/mixed type. In the parabolic case the corresponding inverse 
problem is reduced to the nonlinear Volterra integral equation for which 
the uniqueness of the solution is proved. For the special cases explicit 
formulae are derived. Both &#147;minimal&#148; and overspecified boundary data 
are considered.

Igor  Malyshev

On the inverse problem for a heat-like equation

This article analyzes discrete Markov processes that arise in some 
general control problems of a storage (or a dam) with an infinite 
capacity. Employing a special &#147;step-wise&#148; structure of the transition 
matrices of these processes, the author obtains conditions for their 
ergodicity and develops an implementable algorithm for finding the 
generating function of the corresponding distributions. This algorithm 
is applied to a dam with a multilevel state-dependent control strategy, 
according to which parameters of the input and output processes may 
depend on the amount of the water in the dam. The method is 
illustrated with a case of a single-level control strategy where explicit 
results are obtained.

Lev  Abolnikov

On some inventory control problems with state-dependent parameters

Galilei's fundamental principle is reformulated in terms of 
abstract geometry and it is generalized to nonlinear and relativistic
systems. As a consequence the notion of a general mechanical system is 
developed and shown to be a universal frame for particle dynamics.

Christian  G&#252;nther

The geometrization of Galilei's principle

A multi-channel loss queueing system is investigated. The input 
stream is a controlled point process. The service in each of <mml:math alttext="$m$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> parallel 
channels depends on the state of the system at certain moments of time 
when input and service may be controlled. To obtain explicitly the 
limiting distribution of the main process (<mml:math alttext="$Z_t $" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Z</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math>) (the number of busy
channels) in equilibrium, an auxiliary three dimensional process with 
two additional components (one of them is a semi-Markov process) is 
treated as semi-regenerative process. An optimization problem is 
discussed. Simple expressions for an objective function are derived.

Jewgeni Dshalalow

On a multi-channel transportation loss system with controlled input and controlled service

Topologies of finite spaces and spaces with countably many 
points are investigated. It is proven, using the theory of ordered 
topological spaces, that any topology in connected ordered spaces, with 
finitely many points or in spaces similar to the set of all integers, is an 
interval-alternating topology. Integer and digital lines, arcs, and curves 
are considered. Topology of N-dimensional digital spaces is described. A 
digital analog of the intermediate value theorem is proven. The 
equivalence of connectedness and pathconnectedness in digital and 
integer spaces is also proven. It is shown here how methods of 
continuous mathematics, for example, topological methods, can be 
applied to objects, that used to be investigated only by methods of 
discrete mathematics. The significance of methods and ideas in digital 
image and picture processing, robotic vision, computer tomography and 
system's sciences presented here is well known.

Efim  Khalimsky

Topological structures in computer science

This paper isolates and studies a class of Markov chains with a 
special quasi-triangular form of the transition matrix [so-called <mml:math alttext="$\Delta _{m,n} \left( {\Delta '_{m,n} } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>&#916;</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:msup><mml:mi>&#916;</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>-matrix]. Many discrete stochastic processes encountered in
applications (queues, inventories and dams) have transition matrices 
which are special cases of a <mml:math alttext="$\Delta _{m,n} \left( {\Delta '_{m,n} } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>&#916;</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:msup><mml:mi>&#916;</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>-matrix. Necessary and 
sufficient conditions for the ergodicity of a Markov chain with 
transition <mml:math alttext="$\Delta _{m,n} \left( {\Delta '_{m,n} } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>&#916;</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:msup><mml:mi>&#916;</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>-matrix are determined in the article in two 
equivalent versions. According to the first version, these conditions are 
expressed in terms of certain restrictions imposed on the generating 
functions <mml:math alttext="$A_i \left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> of the elements of the i-th row of the transition matrix,
<mml:math alttext="$i = 0,1,2, \ldots $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>&#8230;</mml:mo></mml:math>; in the other version they are connected with the 
characterization of the roots of a certain associated function in the unit 
circle of the complex plane. Results obtained in the article generalize, 
complement, and refine similar results existing in the literature.

L.  Abolnikov

A.  Dukhovny

Necessary and sufficient conditions for the ergodicity of Markov chains with transition <svg style="vertical-align:-4.39398pt" height="23.174999" version="1.1" viewBox="0 0 75.237503 23.174999" width="75.237503" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,17.65)"><path id="x394" d="M600 0h-557v24l268 633l28 8l261 -641v-24zM497 50l-194 489l-196 -489h390z" /></g><g transform="matrix(.012,-0,0,-.012,10.963,21.725)"><path id="x1D45A" d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6&#xA;q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" /></g><g transform="matrix(.012,-0,0,-.012,20.351,21.725)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.012,-0,0,-.012,23.064,21.725)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,29.65,17.65)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,35.532,17.65)"><use xlink:href="#x394"/></g><g transform="matrix(.012,-0,0,-.012,46.438,9.488)"><path id="x2032" d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" /></g><g transform="matrix(.012,-0,0,-.012,50.587,21.725)"><use xlink:href="#x1D45A"/></g><g transform="matrix(.012,-0,0,-.012,59.976,21.725)"><use xlink:href="#x2C"/></g><g transform="matrix(.012,-0,0,-.012,62.689,21.725)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,69.288,17.65)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg>-matrix

J. Dshalalow

On a multi-channel transportation loss system with controlled input and controlled service (Erratum)

M. Ishaq Zahid

A preliminary report on Turbo Pascal for the Macintosh

In applications, considerations on stochastic models often 
involve a Markov chain <mml:math alttext="$\left\{ {\zeta _n } \right\}_0^\infty  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>&#950;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mn>0</mml:mn><mml:mi>&#8734;</mml:mi></mml:msubsup></mml:math> with state space in <mml:math alttext="$R_ +  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mo>+</mml:mo></mml:msub></mml:math>, and a transition 
probability <mml:math alttext="$Q$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math>. For each <mml:math alttext="$x\varepsilon R_ +  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi><mml:mi>&#8201;&#8201;</mml:mi><mml:msub><mml:mi>R</mml:mi><mml:mo>+</mml:mo></mml:msub></mml:math> the support of <mml:math alttext="$Q\left( {x,.} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mo>.</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> is <mml:math alttext="$\left[ {0,x} \right]$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math>.
This implies that <mml:math alttext="$\zeta _0  \geqslant \zeta _1  \geqslant  \ldots $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#950;</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&#8805;</mml:mo><mml:msub><mml:mi>&#950;</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#8805;</mml:mo><mml:mo>&#8230;</mml:mo></mml:math>. Under certain regularity assumptions 
on <mml:math alttext="$Q$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> we show that <mml:math alttext="$Q^n \left( {x,B_u } \right) \to 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>u</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8594;</mml:mo><mml:mn>1</mml:mn></mml:math> as <mml:math alttext="$n \to \infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>&#8594;</mml:mo><mml:mi>&#8734;</mml:mi></mml:math> for all <mml:math alttext="$u &#62; 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>u</mml:mi><mml:mo>&#62;</mml:mo><mml:mn>0</mml:mn></mml:math> and
that <mml:math alttext="$1 - Q^n \left( {x,B_u } \right) \leqslant \left[ {1 - Q^n \left( {x,B_u } \right)} \right]^n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mo>&#8722;</mml:mo><mml:msup><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>u</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8804;</mml:mo><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#8722;</mml:mo><mml:mi>Q</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>u</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> where <mml:math alttext="$B_u  = \left[ {0,u} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>B</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>. Set <mml:math alttext="$\tau _0  = \max \left\{ {k;\zeta _k  = \zeta _0 } \right\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#964;</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mo>max</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi>&#950;</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>&#950;</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$\tau _n  = \max \left\{ {k;\zeta _k  = \zeta _{\tau _{n - 1}  + 1} } \right\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#964;</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>max</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi>&#950;</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>&#950;</mml:mi><mml:mrow><mml:msub><mml:mi>&#964;</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math> and write <mml:math alttext="$X_n  = \zeta _{\tau _{n - 1}  + 1} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>&#950;</mml:mi><mml:mrow><mml:msub><mml:mi>&#964;</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>, 
<mml:math alttext="$T_n  = \tau _n  - \tau _{n - 1} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>&#964;</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>&#964;</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>. We investigate some properties of the imbedded
Markov chain <mml:math alttext="$\left\{ {X_n } \right\}_0^\infty  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mn>0</mml:mn><mml:mi>&#8734;</mml:mi></mml:msubsup></mml:math> and of <mml:math alttext="$\left\{ {T_n } \right\}_0^\infty  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mn>0</mml:mn><mml:mi>&#8734;</mml:mi></mml:msubsup></mml:math>. We determine all the marginal 
distributions of <mml:math alttext="$\left\{ {T_n } \right\}_0^\infty  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mn>0</mml:mn><mml:mi>&#8734;</mml:mi></mml:msubsup></mml:math> and show that it is asymptotically stationary
and that it possesses a monotonicity property. We also prove that 
under some mild regularity assumptions on <mml:math alttext="$\beta \left( x \right) = 1 - Q\left( {x,B_x } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#946;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>&#8722;</mml:mo><mml:mi>Q</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>,
<mml:math display="block" alttext="$\sum\limits_1^n {{{\left( {T_i  - a} \right)} \mathord{\left/  {\vphantom {{\left( {T_i  - a} \right)} {b\sqrt n }}} \right.  \kern-\nulldelimiterspace} {b\sqrt n }}} \xrightarrow{d}Z \sim N\left( {0,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:munderover><mml:mo>&#8721;</mml:mo><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:munderover><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>&#8722;</mml:mo><mml:mi>a</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mi>b</mml:mi><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt></mml:mrow></mml:mrow></mml:mrow><mml:mover><mml:mo>&#8594;</mml:mo><mml:mi>d</mml:mi></mml:mover><mml:mi>Z</mml:mi><mml:mo>&#8764;</mml:mo><mml:mi>N</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>.

P.  Todorovic

Remarks on a monotone Markov chain

For a star like network (central server system) with state 
dependent branching we compute the cycle time distribution. The 
Laplace-Stieltjes transform of this distribution is of product form. This 
allows to define a recursive algorithm for evaluation of cycle time 
moments of any order.

Hans  Daduna

Cycle times in a starlike network with state-dependent routing

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International Journal of Stochastic Analysis - Table of contents 1987

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International Journal of Stochastic Analysis

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International Journal of Stochastic Analysis