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Consider a dam model, $L^{upper}$ and $L^{lower}$ are upper and, respectively, lower levels, $L = L^{upper}-L^{lower}$ is large and if the level of water is between these bounds, then the dam is said to be in a normal state. Passage across lower or upper levels leads to damage. Let $J_1=j_1L$ and $J_2=j_2L$ denote the damage costs per time unit of crossing the lower and, correspondingly, upper level where $j_1$ and $j_2$ are given real constants. It is assumed that input stream of water is de...
In this paper continuity theorems are established for the number of losses during a busy period of the $M/M/1/n$ queue. We consider an $M/GI/1/n$ queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if furth...
Let $\mathcal{G}(\frak{g}_1,\frak{g}_2)$ be the class of all probability distribution functions of positive random variables having the given first two moments $\frak{g}_1$ and $\frak{g}_2$. Let $G_1(x)$ and $G_2(x)$ be two probability distribution functions of this class satisfying the condition $|G_1(x)-G_2(x)|1$ let us denote by $\gamma_{G_1}$ and $\gamma_{G_2}$ the least positive roots of the equations $z=\widehat{G}_1(\mu-\mu z)$ and $z=\widehat{G}_2(\mu-\mu z)$ respectively. In the pape...
The book of Lajos Tak\'acs \emph{Combinatorial Methods in the Theory of Stochastic Processes} has been published in 1967. It discusses various problems associated with $$ P_{k,i}=\mathrm{P}{\sup_{1\leq n\leq\rho(i)}(N_n-n)0$, and $\rho(i)$ is the smallest $n$ such that $N_n=n-i$, $i\geq1$. (If there is no such $n$, then $\rho(i)=\infty$.) (*) is a discrete generalization of the classic ruin probability, and its value is represented as $P_{k,i}={Q_{k-i}}/{Q_k}$, where the sequence $\{Q_k\}_{k\...
Under the assumption that the initial population size of a Galton-Watson branching process increases to infinity, the paper studies asymptotic behavior of the population size before extinction. More specifically, we establish asymptotic properties of the conditional moments (which are exactly defined in the paper).
A large dam model is the object of study of this paper. The
parameters Llower and Lupper
define its lower and upper levels,
L = Lupper - Llower is
large, and if the current level of water is between these bounds,
the dam is assumed to be in a normal state. Passage across one or
other of the levels leads to damage. Let J1 and
J2 denote the damage costs of crossing the lower
and, respectively, the upper levels. It is assumed that the input
stream of water is described...
The paper studies a single-server queueing system with autonomous service and $\ell$ priority classes. Arrival and departure processes are governed by marked point processes. There are $\ell$ buffers corresponding to priority classes, and upon arrival a unit of the $k$th priority class occupies a place in the $k$th buffer. Let $N^{(k)}$, $k=1,2,...,\ell$ denote the quota for the total $k$th buffer content. The values $N^{(k)}$ are assumed to be large, and queueing systems both with finite and...
The paper establishes necessary and sufficient conditions for stability of different join-the-shortest-queue models including load-balanced networks with general input and output processes. It is shown, that the necessary and sufficient condition for stability of load-balanced networks is associated with the solution of a linear programming problem precisely formulated in the paper. It is proved, that if the minimum of the objective function of that linear programming problem is less than 1, ...
The paper studies closed queueing networks containing a server station and $k$ client stations. The server station is an infinite server queueing system, and client stations are single-server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by a strictly stationary and ergodic sequence of random variables. The total number of units in the network is $N$. The expected times between departures in client stations are $...
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