Findings obtained in the article prove the need of maintaining the high level of faculty staff activity at the universities.

The problem of loss of universities operating effect is associated with the level decline of scientific research conducted by the employees, and with decline of high skilled staff training quality through postgraduate and doctoral training systems.

To provide further insight into existing tendencies of level dynamics of human resources at the university a phenomenological model was worked out which allows to analyze tendencies and influence at the potential of development of different factors typical to the modern university system [1–2].

Let us develop and research a mathematical model which allows studying at qualitative level possible variants of university development taking into account its operating effect and change in accumulated academic and faculty staff.

Assume each faculty staff age category N(T) is characterized by some functions φ(T, t) and a(T), where T is an age value of a considered faculty staff category, N(T) is a number in this category. Write a(T) for activity of this age category. Assume activity is a value ranging from 0 to 1. Zero means that a given age category is out of university activities, one means that a given category works with total efficiency. Activity can be assessed by holding faculty staff ranking at the university. Write φ(T, t) for potential of the faculty staff age category. Potential shall be understood to mean a combination of knowledge, skills and abilities which are possessed by representatives of the corresponding age group at a time t. Contribution of a given age category to the university work is

Φ(T, t) = a(T) φ(T, t). (1)

Measuring of value Φ(T, t) concerns quantity and quality of scientific paper issued by employees in a given age category, their training development papers, study guides, contribution to conference management, number of grants. System of faculty staff ranking introduced at the universities in some way reflects scope of their contribution.

Integral result of university activity at a time t is determined from the formula

(2)

Potential φ(T, t) of each age group changes with time. In a year age group potential will be defined by the formula

(3)

where k and r are constants of proportionality. Value N(T) is determined from the formula

N(T, t) = N(T – 1, t – 1), (4)

with neglect of staff flow from the university.

When reaching the maximum age Tmax the employees are no more considered, and the new vacancies are occupied by newcomers aged Tmin:

N(Tmin, t) = N0. (5)

Flow of faculty staff from each age group happens constantly, but at a first approximation this fact can be neglected, it does not affect quality results. Graduates join faculty staff, their initial potential is φ(Tmin, t) determined by general condition of the university Φ(t). University graduates potential is determined by the following expression

φ(Tmin, t) = cΦ(t), (6)

where the coefficient c < 1.

After performing certain calculations we need to use the formulas of dynamics potential (3), of number in each age group (4), graduates potential (6) and university general level (2).

Studies showed that the value Φ(t) charactering university condition rises in time. This corresponds to normal university development, growth of its scientific, methodical and teaching potentials. At the same time university graduates potential also grows, some of which later become university employees.

Assume what happens if faculty staff activity coefficient a(T) is less than 1. Analyze how half decrease in faculty staff activity coefficient will result. Calculations performed within the model present that in 25 years after half decrease in activity coefficient, university development level will decrease five times.

The developed model of university operating efficiency and quality analysis proved that it is important to consider in estimating state of university such factors as activity and accumulated potential of faculty staff.

**References**

- Dobrynina N.F. Stokhasticheskiye modeli v vysshem professional’nom obrazovanii // sb. statey 5 Mezhdunarodnoy nauchno-tekhnicheskoy konferentsii «Analiticheskiye i chislennyye metody modelirovaniya yestestvennonauchnykh i sotsial’nykh problem». [Stochastic Models in the higher Professional Education System // 5th International Scientific And Technical Conference Proceedings «Analytical and numerical methods of scientific and social problems modeling»]. Publishing house PSU (Penza state university). – Penza, 2011. – P. 174–178.
- Boikov I.V., Suzdaleva I.A. Ob odnoy modeli obrazovaniya // Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Yestestvennyye nauki. [About one educational model // News of Higher Educational Institutions. Povolzhsky Region. Technical Sciences]. – Penza: Information publishing center of the PSU, 2006. – № 6. – P. 3–12.

The work is submitted to the International Scientific Conference «Strategy for science education», Israel (Tel Aviv), April 25 – Мay 2, 2014, came to the editorial office оn 31.03.2014.