This paper provides the first exact analysis of a preemptive M/M/c queue with two priority classes having different service rates. To perform our analysis, we introduce a new technique to reduce the two-dimensionally infinite Markov chain (MC), representing the two class state space, into a one-dimensionally infinite MC, from which the generating function (GF) of the number of low-priority jobs can be derived in closed form. (The high-priority jobs form a simple M/M/c system and are thus easy to solve.) We demonstrate our methodology for the c =3D 1, 2 cases; when c > 2, the closed-form expression of the GF becomes cumbersome. We thus develop an exact algorithm to calculate the moments of the number of low-priority jobs for any c >=3D 2. Numerical examples demonstrate the accuracy of our algorithm and generate insights on (i) the relative effect of improving the service rate of either priority class on the mean sojourn time of low-priority jobs; (ii) the performance of a system having many slow servers compared with one having fewer fast servers; and (iii) the validity of the square root staffing rule in maintaining a fixed service level for the low-priority class. Finally, we demonstrate the potential of our methodology to solve other problems using the M/M/c queue with two priority classes, where the high-priority class is completely impatient.

The asymmetric unit of the title salt, C8H14N2 (2+)·2NO3 (-), contains two independent dications and four independent nitrate anions. The crystal structure consists of discrete nitrate ions, three of which stack in layers parallel to (001) at z = 0 and 0.5. These layers are connected via m-xylylenediaminium dications. The fourth anion is sandwiched by the two independent organic cations in the asymmetric unit. In the crystal, the ions are connected by a large number of bifurcated and non-bifurcated N-H⋯O(O) hydrogen bonds, forming sheets parallel to (100). These sheets are connected by C-H⋯O hydrogen bonds, forming a three-dimensional network.

We say that two sets (A and B) are qualitatively independent if none of A n B, (B) over bar n B, (A) over bar n B, (A) over bar n (B) over bar is empty, which means that they divide the underlying set into four parts. We show a short proof of the old result determining the maximum size of a pairwise qualitatively independent family on an n-element set, using the cyclic method of Katona. The cyclic method will be used for the general case, too, when we would like to determine the maximum size of a family on an n-element set for which it is required that there are two qualitatively independent among every m sets. However, it gives the exact result only when m is odd. For the case of even m our estimate is somewhat weaker than the expected exact value that we state in a form of a conjecture. Since our result is a special case of a difficult theorem of Gerbner (2013), this article can be considered as a new short proof for a special case.