This paper analyzes a discrete-time $Geo/Geo/c$ queueing system with multiple working vacations and reneging in which customers arrive according to a geometric process. As soon as the system gets empty, the servers go to a working vacations all together. The service times during regular busy period, working vacation period and vacation times are assumed to be geometrically distributed. Customer...

for any collection of positive integers s1, s2, . . . , sl. By definition, Lis(1) = ζ(s), s ∈ Z, s1 ≥ 2, s2 ≥ 1, . . . , sl ≥ 1. (4.2) Taking, as before for multiple zeta values, Lixs(z) := Lis(z), Li1(z) := 1, (4.3) let us extend action of the map Li : w 7→ Liw(z) by linearity on the graded algebra H (not H, since multi-indices are coded by words in H). Lemma 4.1. Let w ∈ H be an arbitrary non...

1. The following example shows that in some physical situations, non-uniqueness is natural and obvious, not pathological. Consider a water bucket with a hole in the bottom. If you see an empty bucket with a puddle beneath it, can you figure out when the bucket was full? No, of course not! It could have finished emptying a minute ago, ten minutes ago, or whatever. The solution to the correspondi...

Throughout, R represents a ring and (R) denotes the commutator ideal of R. For any x,y in R, [x,y] = xy −yx is the usual commutator. A word w(x,y) is a product in which each factor is x or y . The empty word is defined to be 1. We now state formally the definition of a subweakly periodic ring. Definition 1. A ring R is called subweakly periodic if every x in R\J can be written in the form x = a...