S. A. ALIEV, Y. I. YELEYKO, Y. V. ZHERNOVYI. STEADY-STATE DISTRIBUTIONS FOR CERTAIN MODIFICATIONS OF THE M/M/1/m QUEUEING SYSTEM

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1 Trasactios of Azerbaija Natioal Acadey of Scieces, Series of Physical-Techical ad Matheatical Scieces: Iforatics ad Cotrol Probles 009 Vol XXIX, 6 P S A ALIEV, Y I YELEYKO, Y V ZHERNOVYI STEADY-STATE DISTRIBUTIONS FOR CERTAIN MODIFICATIONS OF THE M/M// QUEUEING SYSTEM For the M/M// queueig syste with blockig of a iut flow fro the oet of the begiig of the secod service i a row to the oet of free syste such cases are exaied: ) breaks i work are iossible (roble I) ) after services i a row a break i work of syste occurs (roble II) For the roble I ad for the roble II as [, ] ergodic distributios of robabilities of states of syste ad of legth of queue are obtaied, ad for a algorith for fidig of ergodic distributio is costructed I the case of the M/M// syste coditios of existece of ergodic rocess for the roble I ad for the roble II as are defied A aother forulatio of the roble I whe blockig of a iut flow cotiues fro the start of the third service i a row is exaied Itroductio The queueig systes of tye M/G// with liited queue ad regeeratig level of iut flow are exaied i works [, ] If the legth of queue reaches uber the arrival of custoers i syste is blocked ad reews oly whe the legth of queue decreases u to soe threshold level l [0, ] I article [3] the M/G// queue which feature cosists that after services i a row the forced breaks i work of syste occur is studied Necessity of such ause ca be caused by techological or other reasos For exale, the techical device which cosistetly carries out hoogeeous oeratios, at desigig ca be calculated o the liited uber of oeratios which are carried out without iterrutio betwee oeratios Other odificatios of the M/M// queueig syste coected with blockig of a iut flow ad the aouceet of a break i work of syste after services i a row are offered below Blockig of a iut flow cotiued fro the start of the secod service i a row to the oet of free syste We cosider two variats of the stateet of the roble for the M/M// syste with such blockig of a iut flow: ) breaks i work are iossible ) after services i a row a break i work of syste coes, durig which the iut flow also is blocked We also exaie aother forulatio of the first roble whe blockig of a iut flow cotiues fro the start of the third service i a row Queueig syste with blockig of a iut flow We study sigle-server queue for which the uber of laces i queue caot exceed uber Let custoers arrive to syste o oe We assue also, that tie itervals betwee the oets of arrival of custoers ad service ties are ideedet rado variables exoetially distributed with araeter λ ad µ resectively The syste accets custoers whe it is free ad durig the first service after the state of idle tie of syste, if the queue legth does ot exceed the uber We itroduce double uberig of states of queueig syste: s kl desigates a state k of a level l Here k ( k 0, ) is a uber of the custoers i syste values of l ( l, ) + corresod to states of syste whe the l-th custoer is served after the eriod of idle tie of syste s 0 desigates a state of free syste Let us deote by kl () t robability of stay of syste at tie t i a state s kl Evidetly (see, for exale, [4, 69 5, 6]), that followig liits exist kl li kl ( t) ( k 0, l, ) t

2 Usig grah of states of syste, we shall write the equatios for fidig of steady-state robabilities k 0 k, k λ µ, 0 k+, l k, l+ µ λ 0 λ ( λ + µ ) 0 k, µ ( ) 0 l, k, l+ By eas of the equatios () we ca exress all robabilities corresodig to states of levels l by robabilities of states of the first level k : kl k + l, ( l, k, l+ ) () We ca write the syste for robabilities k together with the oralizatio coditio i the for of 0 k k, ( + k ( k, ) (3), 0+ kk, where µ / λ Usig equatios (3) we ca exress all the robabilities k by +, kl : k ( +, 0, (4) k The robability +, we fid after substitutio of the right arts of relatios (4) i the oralizatio coditio +, ( )( (5) Equalities (), (4), (5) coletely deterie ergodic distributio of robabilities of states s kl Usig this distributio, we ca fid ergodic distributio of the queue legth i the syste Let us deote by π k steady-state robability that the queue legth equals k ( k ) π π 0, The 0 k ( +, ( + ( )( ( + ( + + )( + k s ( +, ( k, ) s k+ Average legth of queue we defie as the atheatical exectatio of discrete rado variable: ( + ( + ) r kπ k (7) ( ( + + )( + ) Steady-state robability of service of custoer, arrived o a iut of syste, we calculate as the su of the robabilities of those states, whe iut flow is ot blocked ( ( + ) ( )( Pserv k (8) Whe we ass to a liit as i relatios (4) (8), we receive ergodic distributios { kl} ad { π k } ad corresodig forulas for the average queue legth ad robability of service for the M/M// syste with blockig of a iut flow () (6)

3 Trasactios of Azerbaija Natioal Acadey of Scieces, Series of Physical-Techical ad Matheatical Scieces: Iforatics ad Cotrol Probles 009 Vol XXIX, 6 P k ( k 0,,, ) k ( + + )( + kl k + l, ( k,, l, 3, ) ( + π0 π (,, ) k k k + + ( + + )( + + ( + r P serv We ote that the ergodic rocess for the M/M// syste with blockig of a iut flow exists for all values of, ulike the M/M// ordiary syste, for which it exists oly as > 3 Queueig syste with blockig of a iut flow ad the breaks i work after services i a row I the revious roble forulatio, we ake oly oe chage: after services i a row a break i work of syste coes Tie break is exoetially distributed with araeter γ Custoers who at the tie of the aouceet of a break are i queue, reai i syste ad wait o reewal of service Durig the iterrutio ew arrivals receivig failure eve with the availability of laces i the queue Blockig of a iut flow is carried out by the sae coditios as i the revious roble, so the roble akes sese oly whe Otherwise, the uber of custoers served i a row is less tha We kee the above eueratio of states, i additio havig desigated by s k, + ( k 0, + ) the syste states durig a forced break Therefore, the state s k, + eas resece i syste durig a break of k custoers exectig o reewal of service First, we assue that,, ad write the syste of equatios for the steady-state robabilities : + γ λ 0 µ k 0, + 0 λk, + γ k, + ( λ + µ ) k 0 k, + k, ( + ) k 0 k +,, 0 µ ( k+, l k, l+ ) 0 l, k, l+ µ k+, γ k, + k + 0 ( 0, ) λ λ µ λ µ Equatios (9) allow to exress the robabilities of states of higher levels ( l ) by the robabilities of states of the first level k ( 0, ), kl k + l, l, k, l+ (0) k, + δ k+, k + where δ µ / γ The syste of equatios for the robabilities k acquires a for + ( ) 0 k k k, k+, k ( k, + ) () kl (9) 3

4 k, ( + k k +,, () 0 + kk + ( + δ ) k, (3) where, as above, µ / λ, аd (3) is a oralizatio coditio Usig equatios () we ca exress the robabilities k k +, by +, Fro relatios () we cosistetly obtai k ( +, 0 k +, (4) k k+ s, ( k 0, ) s (5) Usig equalities (4), by eas of relatios (5), begiig fro, ad further followig o decrease of a idex k, we ca cosistetly exress the robabilities k ( k 0, ) by +, The oralizatio coditio (3) serves to calculate +, So, for all values of ad, oly if, we have a algorith to deterie the ergodic distributio kl ( k 0, l, + ) Below we exaie ore i detail such relatios betwee ad, if we ca fid the ergodic distributio i exlicit for 3 Case, whe Fro (5) ad (4) we have: k, k ( +, ( ( ), k k , k, + k ( (( + ) + ( + (( + )), + k, k ( s k + + ) ( k (( k ), ( k, ) s 0 After calculatig fiite sus ad relaceet idices we get fially k + k ( + ( + ( k+ ), k 0, (6) Forulas (6) are true oly for If >, the > ad calculatig k for k < by forulas (5) we do ot use robabilities k k +,, to deterie which the relatios (4) rovide Therefore, i this case the structure k as k < is differet tha for k, So if >, the we ca use forulas (6) oly for k, After substitutio the right arts of of equalities (4) ad (6) i the oralizatio coditio (3), which for coveiece of calculatios we write as 0 + kk + kk + ( + δ ) k, + we fid the robability +, 4

5 Trasactios of Azerbaija Natioal Acadey of Scieces, Series of Physical-Techical ad Matheatical Scieces: Iforatics ad Cotrol Probles 009 Vol XXIX, 6 P 50-58,, A (, δ) A (, δ) ( + + ) ( + ( ) ( + + (7) + δ ( )( + + Thus, the relatios (0), (4), (6) ad (7) coletely deterie the ergodic robabilities distributio of the states s kl for a syste with blockig of a iut flow ad iterrutios i the work after services i a row if We ca fid ergodic distributio of the queue legth i the syste ad average legth of queue + π0 k+ δ 0+ δ 0 ( + ( + + ( + δ), k+ k k s + k+, k + k+, ( + ) s k+ ( + ) ( + ) + ( + ), π δ δ + ( k δ ), ( k ) π + k+ δ, +, + δ, (( + + δ), + πk s ( +, ( k +, ) s k+, r kπ ( ( ) ( + k + δ ) ( ) ) Steady-state robability of service we calculate as the su of robabilities of states k ( ( δ ) Pserv k + ( + ) + 0 A (, ) s ( k 0, ) 3 Case, whe Ulike the case here we have oly oe state of level +, this is the state s 0, +, so equatios (9) silifies to the for k + 0, + 0 µ γ λ λ ( λ + µ ) 0 k, λ µ 0 k, k, µ ( ) 0 l, k, l µ γ 0 k+, l k, l+ 0, + As before, we exress the robabilities of states of higher levels 0 l by k l, k, l+ δ (8) kl k+ l, 0, +, ad we write syste of equatios for the robabilities k together with the oralizatio coditio 5

6 After solvig the syste (9), we get 0 k k, k, 0 k δ ( + k, + k + ( + ) k k ( )( δ ( + ) k 0, Usig the relatios (8), (0), we fid ergodic distributio of the queue legth ad steady-state values of average legth of queue ad robability of service (( + ) + ) δ π0 k+ δ 0 ( + + )( + + δ k ( + π k s ( k, ) s k+ ( + + )( + + δ ( + r kπk ( ( + + )( + + δ ) ( ( + ) Pserv k 0 ( + + )( + + δ I the case the grah of states of cosidered queueig syste differs fro the grah of states of syste with blockig of a iut flow studied i Sectio oly the resece of additioal state s 0, + which corresods to the break after cosecutive services Therefore, uttig, δ 0 i the forulas (0) (), we get the relatios (4) (8) 33 Case, whe I this syste a break occurs after each service This case deserves searate cosideratio because it has the secificity ad allows to get the forula for the ergodic robabilities distributio of states for all ( ) Let us write the equatios for a fidig of steady-state robabilities kl : γ λ 0 λ + γ ( λ + µ ) 0 k, 0 0 k, k k λ µ, 0 µ k+, γ k 0 k 0, Exressig the robabilities of states of secod level by forulas k δ k+, ( k 0, ), we obtai the relatios k k+, k 0,, (9) (0) () () where we have k + k, k 0, (3) 6

7 Trasactios of Azerbaija Natioal Acadey of Scieces, Series of Physical-Techical ad Matheatical Scieces: Iforatics ad Cotrol Probles 009 Vol XXIX, 6 P Usig the oralizatio coditio ad searatig cases ad, we fid 0 δ k + ( + ),, + δ k k 0,, + ( + )( + δ ) Now we ca calculate the ergodic distributio of queue legth ad steady-state values of average legth of queue ad robability of service ( )( + + δ) 0 0+ ( + ), + δ ( ) ( )( + δ) k ( + ) k+, ( k, ), + π δ π δ δ ( k ) δ δ π π + + 0, k,, + ( )( + δ) + ( )( + δ) ( + δ) ( ( ) + ) r kπk, ( + )( + δ ) ( δ ) ( ) ( ) + δ r, ( ) δ ( ) Pserv k, Pserv, + ( + )( + δ ) Lettig i (3) (6), we see that the ergodic distributio i the case of absece of restrictios o the > The we have: legth of queue ( ) exists oly if k, k δ, ( 0,,, ) k k+ k ( + δ) ( )( + + δ) ( )( + δ) π0, πk ( k,, ) ( + δ) k+ ( + δ) + δ r, Pserv ( )( + δ) + δ We saw i Sectio that ergodic rocess for the M/M// syste with blockig of a iut flow exists for ay values of We ca get this syste as a result of assage to the liit whe ad i the syste with blockig of a iut flow ad the breaks i work after services i a row So, we have bases to redict, that for the M/M// syste with blockig of a iut flow ad the breaks i work after services i a 7 (4) (5) (6)

8 row for sufficietly large values of existece of a ergodic rocess is ossible eve for If >, the it certaily exists for all 4 Blockig of a iut flow fro the begiig of the third service i a row Let us chage the settig of the roble studied i Sectio Suose that blockig of a iut flow cotiues fro the begiig of the third service i a row to the oet of free syste Let us write the equatios for deteriig the steady-state robabilities k 0 0 k, ( + ) k 0 ( k, ) (7) λ µ, 0 µ λ λ λ µ, kl µ ( λ + µ ) 0 λ k, + µ k+, ( λ + µ ) k 0 k, λ µ 0 µ ( k+, l k, l+ ) 0 l, k, l (9) Usig the equatios (9) we ca exress all robabilities corresodig to states of levels l 3 by robabilities of states of the secod level k kl k + l, l 3, k, 3 l So, the oralizatio coditio for the syste of equatios (7) (9) takes for 0 + ( k + kk ) (30) Usig the equatios (7) we exress all the robabilities k by +, Rewrite the equatios (8) as k ( +, k 0,, + k k, + k+, ( k, ) + + Recurrece relatios (3) allow to exress all the robabilities of secod level k by +, k k ( +, ( k, ) ( + ) +,,, ( + ( + Fially, by eas of oralizatio coditio (30) we fid the robability +, ( +,, A ( ) 3 A ( ( + ) Now we ca fid the ergodic distributio of queue legth ad steady-state values of average legth of queue ad robability of service (8) (3) 8

9 Trasactios of Azerbaija Natioal Acadey of Scieces, Series of Physical-Techical ad Matheatical Scieces: Iforatics ad Cotrol Probles 009 Vol XXIX, 6 P π k 0 + ( +, πk k+, + s (( + + k( +, ( k ), s k π, +,, ( ( + ) r kπk A ( (( )( ( ) ( ( ) ( ) ( ) ( + ) serv k k 0 + ( + ) (( )( ( ) P A Whe we ass to a liit as, we receive ergodic distributios ad forulas for the average queue legth ad robability of service for the corresodig M/M// queueig syste with blockig of a iut flow k k 3 3 k 3 ( k ) 0,,, ( k ( k,, ) k+ ( ) (,, 3, 4, ) ( + ) k l kl k+ l, ( + π ( + + k πk ( k,, ) k+ 3 ( ( ) ( + + ) r Pserv As well as for queueig syste with absece of restrictios o the legth of queue cosidered i Sectio, a ergodic distributio exists here for ay values of 9

10 Rerereces Takagi H Aalysis of fiite-caacity M/G/ queue with a resue level Perforace Evaluatio, vol 5, No 3, 985, Takagi H Queueig aalysis Vol North Hollad, 993 θ 3 Bratiychuk AM Liitig theores for the syste of tye M / G// b with regeeratig level of iut flow Ukrai Math Zhur, vol 59, No 7, 007, (Ukraiia) 4 Zherovyi YuV Markov odels of the queueig theory Lviv, 004 (Ukraiia) 5 Ovcharov LA Alied robles of the queueig theory М, 969 (Russia) 0