19.8. Queuing system with failures. Erlang equations

Queuing systems are divided into two main types: a) systems with failures, b) systems with waiting.

In systems with refusals, an application that arrived at the moment when all service channels are busy immediately receives a refusal, leaves the system and does not participate in the further maintenance process.

In systems with waiting, an application that has made all channels busy does not leave the system, but becomes in a queue and waits until a channel becomes free. Present we will consider the system with failures as the simplest.

Let there be -channel queuing system with failures. Consider it as a physical system. with a finite set of states:

- all channels are free,

- exactly one channel is busy,

……………

- busy exactly channels,

……………

- all busy channels.

The scheme of possible transitions is given in fig. 19.8.1.

Fig. 19.8.1.

We set the task: to determine the probabilities of the system states for any point in time . The task will be solved under the following assumptions:

1) the flow of applications - the simplest, with a density

2) service time - indicative, with parameter

. . (19.8.1)

Note that the parameter in the formula (19.8.1) is completely analogous to the parameter exponential law of distribution of the gap between neighboring events of the simplest flow:

. (19.8.2)

Parameter makes sense of "flow density applications." Similarly, the value can be interpreted as the “release flux density” of a busy channel. Indeed, let us imagine a channel that is continuously occupied (uninterruptedly supplied with applications); then, obviously, in this channel there will be a simple stream of releases with a density .

Since both flows — applications and exemptions — are the simplest, the process going on in the system will be Markov.

Consider the possible system states and their probabilities.

. (19.8.3)

Obviously for any point in time

. (19.8.4)

Create differential equations for all probabilities (19.8.3), starting with . Fix a point in time and find the probability that at the moment the system will be able to (all channels are free). This can happen in two ways (Fig. 19.8.2):

Fig. 19.8.2.

- in the moment the system was able to and in time did not pass from it to (not a single application has arrived)

- in the moment the system was able to and in time the channel was released, and the system went into a state .

The ability to "jump" the system through the state (for example, from at through ) for a small period of time can be neglected, as the value of a higher order of smallness compared with and .

By the addition theorem, we have

. (19.8.5)

Find the probability of the event by the multiplication theorem. Probability that at the moment the system was able to equal to . Probability that for time will not come any application equal . Up to values ​​of the highest order of smallness

. (19.8.6)

Consequently,

.

We find . Probability that at the moment the system was able to equal to . Probability that for time the channel is free, equal to up to low order values

.

Consequently,

.

From here

.

Shifting to the left by dividing by and going to the limit at , we obtain the differential equation for :

. (19.8.7)

Similar differential equations can be compiled for other probabilities of states.

Take any and find the probability that at the moment the system will be able to (fig. 19.8.3).

Fig. 19.8.3.

This probability is calculated as the probability of the sum of not two, but three events (according to the number of arrows directed to ):

- in the moment the system was able to (busy) channels), and in time did not pass from him to any nor in (not a single application has been received, not a single channel has been released);

- in the moment the system was able to (busy) channels), and in time moved to (one application has arrived);

- in the moment the system was able to (busy) channels), and in time one of the channels is free.

We find . We first calculate the probability that in time No application will come and none of the channels will be released:

.

Neglecting small values ​​of higher orders, we have

,

from where

.

Similarly

,

and

.

From here we get the differential equation for :

.

Let's make the equation for the last probability (fig. 19.8.4).

Fig. 19.8.4.

We have

,

Where - the probability that in time no channel is free; - the probability that in time one application will come. We obtain the differential equation for :

.

Thus, the obtained system of differential equations for probabilities :

(19.8.8)

Equations (19.8.8) are called Erlang equations. Integration of the system of equations (19.8.8) with initial conditions

;

(at the initial moment all channels are free) gives dependence for anyone . Probabilities characterize the average load of the system and its change over time. In particular, there is a possibility that the application that came at the moment , will find all channels busy (will be rejected):

.

Magnitude called relative system bandwidth. For the moment this is the ratio of the average number of applications served per unit of time to the average number of applications submitted.

The system of linear differential equations (19.8.8) can be relatively easily integrated with any particular number of channels. .

Note that in the derivation of equations (19.8.8) we never used the assumption that and (density of application flow and release flow) are constant. Therefore, equations (19.8.8) remain valid for time-dependent , as long as the streams of events that translate the system from state to state remain Poissonian (without this, the process will not be Markovian).