16 Pulsing process

Distribution of residual time. A population mean and SCV(coefficient of variation- оэффициент вариации случайной величины) residual time. The law of preservation of rate. Local balance of a network.

If time of interval Х between consecutive employment as authentic event, tell, the call has arrived, is independent and identically distributed (iid) - independently and identically distributed - with function of distribution F (x), process {X} is called as pulsing process. Puassonovsky process is a special case of pulsing process in which function of distribution F (x) экспоненциально is distributed.

For pulsing process (fig. 9 see) the interval of time X*ÎX from any way observable point X* till the moment of following employment is called as residual (lives) time or the future time of repetition (forward). Thereupon the interval generated by employment X is called as life time.

Time from last employment to an observable point of an epoch is called as a century or last time of repetition (back). It is known that the century, as well as residual time X *, is distributed according to distribution function formula (13)

Where m = E {X} - there is a population mean of time of life X.

Basically believe that F (x) should be function of distribution of casual continuous variable Х ³ 0. Then define Laplace-Stieltjes transform (LST) from F (x) as formula (14)

Where formula - is function of density Х.

Transformations LST of function of distribution F (x) it is equivalent to transformations of Laplasa (LT) to density function f (x). Transformations LST of distribution of residual time R (t) are expressed as formula (15)

Where formula f * (θ) - is LST time of life of distribution F (x).

From here we receive average residual time formula (16)

Where formula E {X 2} - is the second moment Х,

σ2 - is dispersion Х,

Is square-law factor of a variation (SCV).

Let's consider system in which calls arrive in pulsing process with rate l, and we will demand экспоненциального a holding time with average size m-1. Believing that Pj the probability of that j calls exists in system in a steady condition, and Пj - probability just before call receipt, we have the law of preservation of rate: formula (17)

For systems with losses the law is interpreted as follows. We will designate a presence condition in system j calls symbol Sj. As Пj-1 there is a probability from condition Sj-1 just before call receipt, and l there is a rate of receipt of a call the left-hand side of the law (17) represents transition of rate from Sj-1 ® Sj. On the other hand, as jm there is a rate of clearing (an end rhythm) call, and Pj there is a probability of existence j calls the right party of the law (17) represents transition of rate from Sj ® Sj-1. Thus, both parties balance in a steady condition, providing local balance of a network: rate-up = rate-down.

The same interpretation is applied and to systems with delay unless rate of clearing (an end rhythm) call accepts value sm at j> s as only s served calls can come to the end.

17-18 Models of Poisson loading of port with any holding time

It is known that the multiserver of systems with losses and any holding time M/G/s (0) is equivalent to Markovsky model M/M/s (0), and the probability of blocking here is set by the formula of Erlanga of V.Krome togo, probability of blocking for system with the limited number of inputs from n sources M (n)/G/s (0) set by losses under the formula of Engseta. The given properties concern to робастности service time.

It is known that system M/G/1 has a steady condition, if and only if offered transport loading a = r = lh <1 erl where h there is an average holding time, and r - system operating ratio. It can be understood intuitively because the server can serve 1 erl, as a maximum.

Let's choose some call and we will mark it as a test call (fig. 10 see). Probability of that the server is occupied, when the test call will arrive, on property of 3 loadings and PASTA(Poisson arrivals see time average,т.е это Пуассоновское поступление вызовов, наблюдаемое за среднее время) is equal and. Time until will come to the end call service, is residual service time. From here, having designated an average residual holding time a symbol, average of expecting calls - and an average waiting time - we have the following dependence at an order of service of calls FIFO: formula (18)

On the right side of expression (18) the first product corresponds to average time for a call in service if some has to be finished, and the second product corresponds to service of those expecting calls which stand in a queue ahead of a test call.

Using the formula of Littla = l and solving dependence (18), we will receive an average waiting time formula (19)

From expression (16) we will receive average residual time formula (20)

Where Cs2 = σs2/h2 - is SCV service time (service); σs2 - Is a dispersion of service time (service).

Substituting average residual time (20) in an average waiting time (19), we will receive the formula of Poljacheka-Hinchina: formula (21)

In stochastic process time moment in which property марковости keeps, called as a pulsing point. For system M/G/1 the epoch of deportation (clearing) in which the call comes to the end and leaves system, becomes a pulsing point.

Markovsky process with discrete space of conditions is called as the Markovsky chain. In the Markovsky chain all times of epoch in which a condition changes, become pulsing points. On the other hand, stochastic process is called as the introduced (enclosed) Markovsky chain if pulsing points take root or put during special time of epoch, such as deportation (clearing) of a call in systems of type M/G/1.

In the introduced Markovsky chain (fig. 11 see) a condition of probability Пj* just after clearing (call deportation) to equally condition of probability Пj just before call receipt in a steady condition. From PASTA(Poisson arrivals see time average,т.е это Пуассоновское поступление вызовов, наблюдаемое за среднее время) follows that if in system M/G/1 exists j calls, then formula

Distribution of waiting time W (t) in system M/G/1 define under the formula of Benesha. Can be shown that the equation of integral of Volterra satisfies to distribution W (t). For system M/D/1 analytical expression for function of distribution of waiting time W (t) is received, and for system M/G/1 (m) the formula of calculation of an average waiting time is received.

19-20 A Poisson model loading of port with a constant holding time

It is known that in system M/D/s there is a steady condition, if and only if a <s.

Generating function g (z) from probability Pj is defined as formula (22)

The average waiting time pays off under the formula of Krommelina-Poljacheka.

Function of distribution of a waiting time is set as formula (23)

Where b (ks +s - 1) - the probability of that ks +s - 1 exists in system in the range of time (t0 + x). It is deduced from function Qr at t0 ® ¥.

Function (23) is set as follows. We will choose any call and we will mark it as a test call. We will assume that the test call arrives during time t0 in a steady condition, and (ks +s - 1) calls exist in the range of time (t0 + x). As s calls release system during each time h (holding time) after time kh the number of the occupied servers becomes (s - 1). Then the test call is entered into service.

Function of distribution of a waiting time of one-linear system with a constant holding time of Poisson loading of port M/D/1 is defined as formula (24)

Average waiting time in system M/D/1vychisljaetsja as formula (25)

It is necessary to note, the average waiting time in turn (25) twice is less, than in system M/M/1.

Thus, the one-linear system with constant holding time M/D/1 is twice more effective in operation, than one-linear system with экспоненциальным server M/M/1.