Random Search as a Means for Queuing Network Optimization

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Random Search as a Means for Queui= ng Network Optimization

= &nb= sp; = Ziny Flikop = &nb= sp; = UDC 519.95

Translated from Kibernetica, No.4 pp.75-79, July-August, 1969. Origin= al article submitted January 3, = 1968.

© 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 1= 7^th Street, New York, N.Y. 10011. All rights reserved. A copy of this article is available from the publisher for $15.00.

One of the problems in qu= euing theory is to find the optimal structure of a complex queuing network. The optimal network should offer minimal operational loss due to idle periods of the servers and served units. In most cases it is impossible to evaluate the operational quality of a branched multiphase network by analytical means. <= span class=3DGramE>Hence the wide popularity of the statistical modeling method,= which uses a stochastic model to determine quite accurately the network characteristics. The method also enables a simple random search algorithm to be used for optimization.

Consider a queuing network specified by a stochastic model and consisting of n servicing systems. The network state is defined by a vector

= &nb= sp; = &nb= sp; = &nb= sp; = (1)

where= is the numb= er of servers in the i-th system, i<= /span>=3D1,2,…,n.

Denote by A the set of available states; for each there is a corresponding value of the target function Q=3DQ(X). It will be assumed that Q(X) ha= s a j-th l= ocal minimum at

if j=3D1,2,…. = &nb= sp; = &nb= sp; = &nb= sp; (2)

for <= /span> , where is the neig= hborhood of the point . In view of the fact, that the must be pos= itive integers, by is meant th= e set of points for which

= = &nb= sp; = &nb= sp; = &nb= sp; (3)

If the network is not at = minimum, then for at leas= t one ,where is, this ca= se, the position of the network. If the points are selected equiprobably for checking, we can find one at which the value of the target= functions less then. After such a “successful” attempt, we trans= late the network to the newly found and confine= the check to its neighborhood. On finding with, we translate the network again, to = . This process continues till all attempts to find a poin= t in with prove fruit= less. The number of points in a neighborhood in the absence of constrains is , while the number of checked is primarily determined by the time for evaluating Q(X).

If a complete check is made of = , the exact coordinates of the local minimum will be foun= d. If only part of the neighborhood is investigated, we can speak of the netwo= rk being at a local minimum with some probability P, which depends on the number of available points in the neighborhood, the number of for which, and how many of are checked= .

An in= complete investigation of a neighborhood may be organized in two ways.

= 1.&n= bsp; After each unsuccessful attempt to find in a point wit= h, Bayes formula is used to ev= aluate P allowing each time for the reduction in the number of unchecked. The attempts cease of reaching, where some previo= usly assigned figure.

&n= bsp;

= 2.&n= bsp; We fi= rst find the number N of points the investigation of which

gives. For this purpose we use the empirical formula

&= nbsp;

= &nb= sp; N=3DP(S+1) -1 = &nb= sp; { (1/(S+1))P1} = (4) &n= bsp;

where S is the total number= of checked points.

For sampling the queue fr= om to , a testing vector is added, in which z components take the val= ues +1 and -1 equiprobably, while the remaining q=3Dn-z are equal to zero, and , where is the prob= ability of obtaining . Denote by the vector = enabling a = point with, to be discovered in.

In this type of neighborh= ood, the points are arranged in layers relative to _{<=
!--[if gte vml 1]>}. The distance from _{<=
!--[if gte vml 1]>} to the p-th l= ayer is given by

= &nb= sp; =

= &nb= sp; =3D = &nb= sp; = &nb= sp; = &nb= sp; = (5)

The number of points in t= he p-th layer [1] is:

= &nb= sp; = &nb= sp; = &nb= sp; = = (6)

The neighborhood is best = checked in layers, the number of points c= hecked in the p-t= h layer be dependent on and. The zero layer is inspected first. (If the test is succ= essful, the network is displaced over the greatest distance.) If, after investigati= on of points in t= he zero layer, no is discover= ed with , the first layer is examined , then if this fails, the second layer and so on, till allayers have been checked. To speed up the optimization, when transferring the network from to, the coordinates of the next check point must be found by adding to.

As an example, consider t= he process of finding the optimal number of servers in the stochastic model of= a queuing network consisting of a source with an l= imited number of demands and to serving systems connected in parallel. The network operation is organized in such a way that every demand, served in the syste= m, is a return to the source. In view of this, change in the operational mode = of one system changes the operating of conditions in other (the number of dema= nds present at the source depends on the operational quality of both systems).<= /p>

The model state is define= d by a vector where is the numb= er of servers in the first (second) system. The network operating quality will be assessed in terms of the total losses due to idle periods of the servers and served elements. The value of this target function depends on and, i.e., Q(X).=

The restriction is imposed = on the vector X (severs cannot be allo= wed absent in the system). The number of a available points= in a neighborhood is thus defined by the position of the network. When, the number of is a maximum (four points each in the zero and first laye= r). If one of the coordinates of is equal to= unity, there will be two points in the zero layer and three in the first. When, the number of available points is a maximum (one in the zero and two in the first layer).

Two types of memory, M and are required operation the optimizing program. The memory M stores all that previously checked X, and only the. The number of previously inspected is found by= means of, while M ena= bles the repatriation of checked points to be avoided. It is assumed that, not contained in, is selected= for investigation. If is stored in M, then. If is not present in M or, is evaluate= d by means of simulation.

Using Buslenko’s notation [2], the operator scheme of the optimization algorithm is:

Here is the source of demands, the selecti= on of serving system, the first s= erving system, this second= serving system, determinati= on of the instant when transfer into the net work terminates, evaluation = of the target function on the basis of the queuing system operational results, comparison = of the target function values obtained with the least value obtained, storage of = the least value of the target function and the corresponding vector, determinati= on in the light of the boundary conditions of the number of available points in t= he neighborhood, clearance of memory containing the coordinates of the points of the j-th neighborhood, storage of = the network coordinates in , clearance o= f the unsuccessful checks counter, adjustments= of the program for generating the test vector to obtaining , selection o= f the next point to be checked in the discovered “successful” directi= on, check of the bindery conditions, checked of = memory , so as to a void repeat. inspection of points of the j-th neighborhood, storage of = the coordinates of for which , the unsuccessful checks counter, the determi= nation of whether unchecked points are present in the j-th neighborhood, determinati= on of whether unchecked points are present in the zero layer, adjustment = of the program for generating the tasting vector to obtaining , adjustment = of the program generating the tasting vector to obtaining one coordinate equal to = zero where , obtains the testing vector, determines = the coordinates of the next check point, checks the = memory M, so that repeat inspection of= is avoided,= stores the coordinates of points checked by simulation, and prints the minimum value of the target function and the coordinates of the correspondi= ng point, and stops the algorithm.

The optimization process = will be discussed with help of described above algorithm.

Demands appearing at the source are directe= d by operator in accordan= ce with the accepted law to one of the serving systems (or), whence they are returned to. The nature of the process is estimated by which reali= zes the transfer to when the mo= del reaches a stationary state The operator evaluates <= sub> from the simulation data, while compares the value obtained for the target function with the previously found minimum [a= part from , which is compared with some figure known to be larger]. If , we store in and increas= e by one the content of the counter ( stores them number of checked for which <= sub>. Next, determines whether all. the availab= le points of the neighborhood has been inspected. If there are uninspected points in , checks the quality of investigation of the zero layer. If there are uninspected _{<=
!--[if gte vml 1]>} in the zero= layer, is adjusted= by for generat= ing of the testing vector with coordi= nates ; otherwise it is adjusted by to obtaining with =3D 0 or =3D 0. The testing vector is applied = to which evalu= ates. If, the point found for checking is not accepted by operato= r. In this c= ase generates <= sub> afresh,…(it is unreadable text here) …the check, a new point is selected by mean= s of the circuit . The resulting uninspected is checked = by which, if <= sub> is present = in M, carries out the transfer to and so on (= the point was previously checked and for it), wh= ile if is not in M, it transfers to which store= X in M. The value of Q(X) is found by m= eans of the model for the new unchecked X.

Sense of the condition has already= been considered; let us examine the program operation with. In this case and are stored by the operator (the network moved to a new point). Preparation is made for complete check of the neighb= orhood. For this 1) determines = the number in accordance with the boundary conditions; clears₌; 3) stores in; 4) the operator clears the unsucce= ssful attempts counter; 5) adjusts for obtaini= ng with. Next, evaluates t= he coordinates of the next point by adding to the next testing vector. The obtained go= es to the check circuit. If the does not sa= tisfy the conditions of operators and, another point is selected by means = of . During the passage of through and the operato= r either rega= rds the attempt as unsuccessful and transfers to or after st= oring, it obtains Q(X)= by means of the model. The optimization process terminates when generates a characteristic signal indicating that for all . After this prints and, and stops the process.

To reduce the search time= , the initial point should be as close as possible to is the o= ptimal; this is achieved by using experience gained with like queuing networks.

Figure 2 illustrated the optimization process for a queuing network model, consisting of two parallel systems (the algorithm of the model was developed by I.A. Luik [3]). The following network characteristics were chosen for our illustratio= n. The input flow is Poisson with intensity where g is the number of units present i= n the source and is the inte= nsity of input demands from one unit. The serving time is exponential. The servin= g intensity by one server of the first system is and the second system, . The cost of idling time per unit is 0.5 rubles/h per se= rver of the first system 2 rubles/h, and per server of the second system 10 rubles/h. Thet total number of units g=3D50, demand is directed equpr= obably to either of the serving system.

The initial state of the network wa= s assumed to be. As a result of optimization is obtained= . The BESM-3m computer required 40 min machine time for founding.

In conclusion, the author= thanks V.M. Faivyshevski for valuable advice.

LITERATURE CITED

B.A. Rosenfeld, Multidimensional Spaces [in Russia], izd Nauka, = Moscow, 1966.
N.P. Buslenko, Mathem= atical Modeling of Industrial Processes by Digital Computers [in Russian], izd Nauka, Moscow, 1964.
Recommendations on Mathematical Modeling of Machi= ne Stock Utilization Processes [in Russia<= /st1:place>], izd. NIISP Gosstroya UkrSSR, Kiev, 1966.

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