UDC 519.218.3
Stationarity conditions for the control systems that
provide service to the conicting nonordinary
Poisson ows
∗
M. A. Rachinskaya , M. A. Fedotkin
∗
∗
Department of Software Engineering
Lobachevsky State University of Niznhi Novgorod
23 Prospekt Gagarina, Niznhi Novgorod, 603950, Russia
A class of the systems with several nonordinary Poisson input ows
is studied. It is assumed that the ows are conicting which means they cannot
be served simultaneously. A service device carries out the control function also.
A probabilistic model for the class of the systems is constructed. Easily veriable
conditions of stationarity are determined analytically for two control algorithms:
a cyclic algorithm for the homogeneous ows and a feedback algorithm for the
ows that diers in priority and intensity.
Abstract.
nonordinary Poisson ow, Markov chain, stationarity conditions,
control algorithm.
Keywords:
1.
Introduction
Many nowaday systems combine service operations and control functions for several conicting ows of customers. Some good example is a
trac intersection controlled with an automated trac light. For the sake
of trac safety no two input ows can be served (which means vehicles
are allowed to move across the intersection) simultaneously. The problem
is to nd the most ecient control algorithm. The eciency in the trac
systems may imply minimization of the mean number of waiting vehicles
or minimization of the mean waiting time for an arbitrary vehicle. An
optimization process of this kind includes two possible steps: to choose
a control algorithm among the various classes of algorithms and to determine the optimal parameters values for the chosen algorithm [1]. For
instance, the control parameters in trac intersections coordinated with
trac lights are the light timing lengths that can vary from several seconds
to several minutes or even more. In order to reduce the range of possible values for the control parameters, it is rstly necessary to determine
the values that are unable to guarantee the ecient control, e. g. that
result in unlimited growth of mean waiting time. In such cases stationarity conditions are usually taken into account: the stationarity ensures
constancy of mean system characteristics. The present paper is devoted to
the problem of determination of such stationarity conditions. Two control
algorithms are considered: cyclic algorithm and feedback algorithm with
prolongations. Due to the stochastic nature of the input ows, the stated
probabilistic problem is solved basically by the methods of queuing theory
and the theory of Markov chains.
Π1
𝑂1
𝛿1
...
Π𝑚
𝑂𝑚 𝛿𝑚
Π1′
𝑠 Γ ...
′
Π𝑚
Figure 1. General scheme of the control system for the conicting ows
A general scheme of the queueing system under consideration is presented in Fig. 1. There are m ≥ 2 independent conicting input ows. It
is supposed that each input ow can be approximated as a nonordinary
Poisson ow Πj (unless otherwise noted herein j ∈ J = {1, 2, . . . , m})
with the following parameters: λj is an intensity of batches, pj , qj and
sj = 1 − pj − qj are the probabilities that an arriving batch consists of one,
two and three customers. The customers are accumulated into batches
due to inuence of external environment [2]. So the ows have the same
physical nature but can dier in intensity or priority of the customers. The
customers of the ow Πj which are arrived to the system and cannot be
served immediately are forced to wait for service in a FIFO queue Oj . A
service device can be in one of the possible states from the set Γ. All of
the states can be divided into two types: service states for each ow and
readjusting states that provide safe switching between service of dierent
ows. No service operations are carried out in any readjusting state. The
service state activates an extreme service strategy δj , i. e. when the service
device is in the service state for the ow Πj , as many present customers as
possible are served but their number cannot exceed system service capacity lj . The service device carries out control function as well. To this end,
certain state change algorithm s(Γ) is specied. The customers of the ow
Πj that are already served compose an output ow Π0j . The task of the
paper is to construct a model of the system and determine the conditions
of system stationary mode existence.
2.
Cyclic control algorithm
Let Γ = {Γ(1) , Γ(2) , . . . , Γ(2m) }. The state Γ(2j−1) is a service state
for the ow Πj . Such state is followed by the readjusting state Γ(2j) .
The cyclic algorithm s(Γ) with a graph presented in Fig. 2 is considered.
The service device stands in each state Γ(k) , k ∈ M = {1, 2, . . . , 2m}, for
a xed period ofPtime with duration of Tk . The full state change cycle
duration is T = k∈M Tk . With the help of the embedded Markov chain
method it is possible to observe the system in discrete moments without
signicant loss of information. Let τi (herein i ∈ I = {0, 1, . . .}) be a
random moment of the ith service device state change. The increasing
sequence {τi ; i ∈ I} divides the time axis [0, ∞) into the halfopen intervals
∆−1 = [0, τ0 ), ∆i = [τi , τi+1 ), i ∈ I . The service device state in interval
∆i is a random element denoted as Γi ∈ Γ. Let also ηj,i ∈ {0, 1, . . .} and
0
ξj,i
∈ Yj = {0, 1, . . . , lj } be the random variables that count the number of
the customers of the ow Πj which arrive to the system and depart from
the system in the interval ∆i correspondingly. The number of waiting
customers in the queue Oj in the moment τi is also a random variable
denoted as κj,i ∈ X = {0, 1, . . .}.
Г(1)
Г(2)
…
Г(2m-1)
Г(2m)
Figure 2. Graph of the cyclic control algorithm
The following results derived in [3] may be noted.
Lemma 1. For each j ∈ J and given initial distribution of the vector
0
0
(Γ0 , κj,0 , ξj,−1
) the sequence {(Γi , κj,i , ξj,i−1
); i ∈ I} is a homogeneous vector Markov chain with a countable state space Γ × X × Yj which consists
of a class of transient states and a class of recurrent states with period 2m.
0
The Markov chain {(Γi , κj,i , ξj,i−1
); i ∈ I} describes the system dynamics only for the ow Πj . Such separation is possible due to independence
of the input ows and determinacy of the algorithm. The stationarity
conditions are also splitted into independent conditions for each ow Πj .
Theorem 1. For any j ∈ J the stationary mode for the ow Πj exists
if and only if the system parameters meet the following inequality
λj T (2sj + qj + 1) − lj < 0.
(1)
The criterion (1) is close to the stationarity conditions for the classical
queueing systems. The value λj T (2sj + qj + 1) characterizes the mean
number of customers of the ow Πj that arrive to the system during the
full state change cycle. The inequality (1) means that the mean number of
arriving customers must be less than the mean number of served customers.
3.
Feedback control algorithm
Suppose now that the input ows dier signicantly in intensity and
priority. The intensity of arrivals of the ow Π1 is quite small though
the priority of its customers is the highest. The ow Πm has the highest
intensity of arrivals but low customer priority. The ows Π2 , Π3 , . . . , Πm−1
are lowintensity lowpriority ows. Based on the dierences between the
ows the feedback algorithm is proposed. The decision about service device
state change is made according to the number of waiting customers in the
queue O1 . The algorithm assumes that service for the highintensity ow
Πm may be prolonged. For this reason there are two service states for
the ow Πm : Γ(2m−1) with duration of T2m−1 and Γ(2m) with duration of
T2m < T2m−1 . The service intensity is the same for both of these states
and equals µm . This means the service capacity in these states is measured
0
= [µm T2m ] ≤ lm . The readjusting
by the variables lm = [µm T2m−1 ] and lm
(2m+1)
state for the ow Πm is Γ
. It is proposed to prolong service for the
ow Πm in case the number of waiting customers of the highpriority ow
is less than certain threshold value h1 . The graph of this feedback control
algorithm is presented in Fig. 3. Here is a random variable η1,i that counts
the number of customers of the ow Π1 that arrive to the system during the
(1)
(2)
(2m+1)
interval
}, M = {1, 2, . . . , 2m + 1},
P ∆i . Thus, Γ = {Γ , Γ , . . . , Γ
T = k∈M Tk , and all of the other variables and denotions are the same
as in the section 2. It should be noted that in case h1 = 0 the feedback
algorithm becomes a pure cyclic algorithm.
Г(1)
Г(2m + 1)
Г(2)
κ1,i + η1,i ≥ h1
Г(3)
Г(2m)
κ1,i + η1,i < h1
κ1,i + η1,i ≥ h1
Г(2m – 1)
…
Г(2m – 2)
κ1,i + η1,i < h1
Figure 3. Graph of the feedback control algorithm
The following statements are proved for the system controlled with the
feedback algorithm based on threshold priority.
0
0
Lemma 2. For any vector (Γ0 , κ1,0 , κm,0 , ξ1,−1 , ξm,−1 ) distribution the
sequence
0
0
{(Γi , κ1,i , κm,i , ξ1,i−1
, ξm,i−1
); i ∈ I}
(2)
is a homogeneous multidimensional Markov chain with a countable state
space Γ × X × X × Y1 × Ym which consists of a class Dm of transient states
and a class Em of recurrent aperiodic states.
Lemma 3. For any initial distribution of (2) either the limiting equal0
0
ity lim P (Γi = Γ(k) , κ1,i = x1 , κm,i = xm , ξ1,i−1
= y1 , ξm,i−1
= ym ) = 0
i→∞
takes place for each k ∈ M , x1 , xm ∈ X , y1 ∈ Y1 , ym ∈ Ym and no
stationary distribution exists or
p(Γ(k) , x1 , xm , y1 , ym ) =
0
0
= lim P (Γi = Γ(k) , κ1,i = x1 , κm,i = xm , ξ1,i−1
= y1 , ξm,i−1
= ym )
i→∞
and an only stationary distribution of (2) exists. In the second case
p(Γ(k) , x1 , xm , y1 , ym ) is positive for (Γ(k) , x1 , xm , y1 , ym ) ∈ Em and equal
to null for the transient states (Γ(k) , x1 , xm , y1 , ym ) ∈ Dm .
Theorem 2. The stationarity criterion for the ow Π1 consists in the
inequality λ1 (T −T2m−1 )(3s1 +2q1 +p1 )−l1 < 0 for the system parameters.
Theorem 3. No stationary mode for the ow Πm exists if the system
parameters meet conditions
0
λm T2m (3sm + 2qm + pm ) − lm
> 0, λm T2m−1 (3sm + 2qm + pm ) − lm > 0
or conditions
0
λm (T −T2m )(3sm +2qm +pm )−lm < 0, λm T (3sm +2qm +pm )−lm −lm
> 0.
4.
Conclusion
The stationarity conditions derived in sections 2 and 3 can be easily
veried for the real systems. Some simulation computer researches show
that the feedback algorithm can degenerate in the cyclic one that in some
cases may be even more ecient than the feedback control. In case of feedback control algorithm it is not possible to nd the stationarity criterion
for the highintensity ow. However, the conditions stated in Theorem 3
give an idea of the desired behavior of the system: on average, the system must serve more customers than arrive to the system for any possible
pathes on the graph presented in Fig. 3.
References
1.
Conicting nonordinary Poisson
ows cyclic control process simulation model // Bulletin of the Volga
State Academy of Water Transport, 2016. No. 47. P. 4351.
2. Fedotkin M., Rachinskaya M. Parameters estimator of the probabilistic model of moving batches trac ow // Distributed Computer and
Communication Networks / Ser. Communications in Computer and Information Science. 2014. Vol. 279. P. 154168.
3. Rachinskaya M. A., Fedotkin M. A. Construction and investigation of
a probabilistic model for the cyclic control of lowintensity ows //
Vestnik of UNN, 2014. No 4(1). P. 370376.
Fedotkin M. A., Rachinskaya M. A.