### Note on a construction of unbounded measures on a nonseparable Hilbert space quantum logic

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For a discrete modified $GI/GI/c/\infty $ queue, $1\le c<\infty $, where the service times of all customers served during any busy period are independent random variables with not necessarily identical distribution functions, the joint distribution of the busy period, the subsequent idle period and the number of customers served during the busy period is derived. The formulae presented are in a convenient form for practical use. The paper is a continuation of [5], where the $M/GI/c/\infty $ discrete modified queue has been studied.

This paper i a continuation of the first part under the same title. The author studies a joint distribution in $\sigma $-finite measures for noncompatible observables of a quantum logic defined on some system of $\sigma $-independent Boolean sub-$\sigma $-algebras of a Boolean $\sigma $-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system...

The busy period distribution of a discrete modified queue $M/GI/c/\infty $, with finitely or infinitely many severs , and with different distribution functions of customer service times is derived.

The notion of a joint distribution in $\sigma $-finite measures of observables of a quantum logic defined on some system of $\sigma $-independent Boolean sub-$\sigma $-algebras of a Boolean $\sigma $-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.

We give two variations of the Holland representation theorem for $\ell $-groups and of its generalization of Glass for directed interpolation po-groups as groups of automorphisms of a linearly ordered set or of an antilattice, respectively. We show that every pseudo-effect algebra with some kind of the Riesz decomposition property as well as any pseudo $MV$-algebra can be represented as a pseudo-effect algebra or as a pseudo $MV$-algebra of automorphisms of some antilattice or of some linearly ordered set.

We study states on unital po-groups which are not necessarily commutative as normalized positive real-valued group homomorphisms. We show that in contrast to the commutative case, there are examples of unital po-groups having no state. We introduce the state interpolation property holding in any Abelian unital po-group, and we show that it holds in any normal-valued unital $\ell $-group. We present a connection among states and ideals of po-groups, and we describe extremal states on the state space of...

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