Probability and Statitic Volume 5, Article ID 558, 8 page http://dxdoiorg/55/5/558 Reearch Article An (, S) Production Inventory Controlled Self-Service Queuing Sytem Anoop N Nair and M J Jacob Department of Mathematic, National Intitute of Technology Calicut, Kerala, Calicut 6736, India Correpondence hould be addreed to Anoop N Nair; nairanoopn@gmailcom Received June 5; Revied 7 September 5; Accepted 7 September 5 Academic Editor: RamónM Rodríguez-Dagnino Copyright 5 A N Nair and M J Jacob Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited We conider a multierver Markovian queuing ytem where each erver provide ervice only to one cutomer Arrival of cutomer i according to a Poion proce and whenever a cutomer leave the ytem after getting ervice, that erver i alo removed from the ytem Here the erver are conidered a a tandard (, S) production inventory Behavior of thi ytem i tudied uing a three-dimenional QBD proce The condition for checking ergodicity and the teady tate olution are obtained uing matrix analytic method Unlike claical queuing model, the number of erver varie in thi model according to an inventory policy Introduction In recent year, manufacturing commoditie or inventorie in anticipation of demand ha fallen out of favor coniderably Mot of the manufacturer produce item in repone to the actual demand The amount of time taken for the production of an item and the time at which a production order i to be placed are two important parameter in thi regard The tandard (, S) production inventory i an anwer to thee parameter which i quite different from the uual inventory policie Mot of the tudie conider inventory ytem in which a many item a needed can be replenihed all at once But in (, S) production inventory, the replenihment i done on an item-by-item bai That i, at the time at which the inventory fall to, the production proce i witched on and i witched off at the time intance at which the on-hand inventory reache S The production time of an item in the inventory i a random variable with a probability ditribution In production inventory, to know the tatu of the ytem both the inventory level and the production tatu (whether production proce i on or off) mut be known In claical inventory management problem, the ervice time of an inventory wa aumed to be negligible Inventory model with poitive ervice time have firt been invetigated by Sigman and Simchi-Levi [] Thereafter a lot of work have been carried out in thi area For further detail on inventory model with poitive ervice time we refer to a urveyarticlebykrihnamoorthyetal[]itcanbeoberved that mot of the model dicued in the literature have conidered queue in which a ingle erver ervice an inventory Inventorie with/without lead time, perihable inventory, and o forth and attached with a erver are alo examined with different ordering policie uch a (r, Q), (, S) and other general randomized policie The tudie of Parthaarathy and Vijayalakhmi [3], Deepak et al [4], Manuel et al [5], and Narayanan et al [6] are ome of the example of the above aid model In the cae of production inventory with poitive ervice time the work of Krihnamoorthy and Narayanan [7] i the firt reported work They have conidered MAP/PH/ queue attached with a production inventory The time for producing an item in the inventory follow a Markovian production cheme A mentioned above, mot of the work in the literature conider ytem where a ingle erver i providing ervice to arriving cutomer Nowaday, elf-ervice inventory facilitie are on the rie to get the cutomer atifaction It can be een that there are many ituation where the cutomer himelf ervice inventory a in retail upermarket or online hopping and o forth But tudie on uch elfervice inventorie are very le In a elf-ervice facility, there i no erver to erve the item but one or more cutomer get
Probability and Statitic ervice imultaneouly depending on the availability of the inventory That i, cutomer themelve erve the inventory and the time for erving the inventory can be conidered a ervice time In particular, we can think about a elferviceable retail outlet which ell a particular type of inventory or an online hopping ite excluively for a ingle type of inventory Such a ytem ha the characteritic of a multierver queuing ytem A elf-ervice queuing ytem with poitive ervice time wa firt invetigated by Anoop et al [8] They have conidered a tandard (, S) inventory without lead time a erver and obtained the teady tate probabilitie, conditional ditribution on the ytem ize and inventory level, ditribution of the inventory cycle time, and an optimization problem which optimize the reorder quantity Nair and Jacob [9] have analyzed a queuing ytem with retrial of demand by conidering the (, S) inventory a erver Production inventory i very relevant when the manufacturer himelf i meeting the demand It could be a treet vendor preparing nack baed on the demand for the item Alo it can be the manufacturer of cutomized item which will be produced depending on the actual demand In the preent work, we try to addre a elf-ervice inventory where the inventory i the (, S) production inventory We conider the (, S) production inventory a erver of the queuing ytem Unlike in claical queuing model, the number of erver varie in thi model according to an inventory policy The aim of thi paper i to tudy implet Markovian queuing ytem with varying number of erver which replenihed according to an (, S) productioninventoryitcanbeobervedthat there i a trong dependence between the ytem ize and the inventory level Therefore, a product form olution i not anticipated a decribed in Krihnamoorthy and Viwanath [] We ue matrix analytic method to find the teady tate probabilitie Model Decription We conider a queuing ytem where arrival of cutomer (demand for an item) follow a Poion proce with rate λ Wehaveatandard(, S) production inventory a erver and the ervice time follow an exponential ditribution with parameter μ Therefore, when there are n cutomer and i itemintheinventory,theeffectiveervicerateofthe ytem i min(nμ, iμ) After ervice completionan item in the inventory a well a a cutomer leave from the ytem which reult in the decrement of the ytem ize and the number of erver (inventory) When the inventory level reache, the production proce i witched on and it i witched off when the inventory level reache S The time required to produce an item in the inventory i aumed to be exponentially ditributed with parameter α An arrivingcutomer join the ytem with probability if he find a free erver (inventory) But whenever the number of cutomer exceed or i equal to the available inventory, an arriving cutomer join the ytem with probability p and he leave the ytem without waiting with probability p We analyze the ytem by conidering a threedimenional QBD proce Ω t = {N(t), I(t), C(t); t }, Table : Tranition rate of A (n) From To Rate Condition (n, i, ) (n, i, ) min(n, i)μ For n>, i> (n, +, ) (n,, ) min(n, + )μ For n> (n, i, ) (n, i, ) min(n, i)μ For n>, +<i S where N(t) repreent the number of cutomer in the ytem at time t, I(t) repreent the inventory level at time t, and C(t) repreent tatu of the production proce The tatepaceoftheabovecontinuoutimemarkovchaini given by {(n,i,); i } {(n,i,c);+ i S ;c=, } {(n, S, )} for n Furtherc = indicate that the production proce i off and c = indicate that the production proce i on Propoition The infiniteimal generator of the QBD proce Ω i given by Δ= ( ( A () A () A () A () A () A () A () A (), () ) ) where the dimenion of each block matrix i dim =S and the block matrice give the tranition rate which are decribed a follow: (a) A (n), n, repreent the tranition rate matrice due to ervice completion at level n (ee Table ) (b) A (n), n, repreent the tranition rate matrice which leave the firt coordinate fixed at level n (ee Table ) (c) A (n), n, repreent the tranition rate matrice due to arrival of cutomer at level n (ee Table 3) The proof of Propoition conit of finding the tranition rate for the CTMC Ω t and then arranging in the matrix formweomittheproofherebecaueitiratherlongand trivial Propoition CTMC Ω t i table if and only if S j=+ λ< μ p [ jθ j + j(θ j, +θ j, )+Sθ S, ], () j= [ ] where θ j are the olution of the equation ΘA = with A= + + and Θ=(θ,,θ,θ +,,θ +,,,θ S,, θ S,,θ S, ) Proof Let n Sand p>definea= +A(S) +A(S) Let Θ=(θ,θ,,θ,θ +,,θ +,,,θ S,,θ S,,θ S, ) (3)
Probability and Statitic 3 Table : Tranition rate of A (n) From To Rate Condition (n, i, ) (n, i +, ) α For n, i<s (n,s,) (n,s,) α For n (n,i,) (n,i,) (λ+α+min(n, i)μ) For n<i (n, i, ) (n, i, ) (pλ + α + min(n, i)μ) For i n (n, i, ) (n, i, ) (pλ + min(n, i)μ) For + i S, i n (n, i, ) (n, i, ) (λ + min(n, i)μ) For + i S, n i Table 3: Tranition rate of A (n) From To Rate Condition (n, i, c) (n +, i, c) λ For n<i, i,andc=, (n, i, c) (n +, i, c) pλ For i n, c=, be the teady tate probability vector of the Markov proce with generator matrix A Therefore Θ can be obtained from the following equation: ΘA =, (4) Θe =, where e i a column vector of order dim with all the element being The following et of linear equation are obtained from (4): αθ +μθ =, αθ j (α+(j )μ)θ j +jμθ j = jμθ j, + (j + ) μθ j+, = for j=,3,,, for j=+,+,,s Equation (5) and (6) are implified a follow: θ j = αj j!μ j θ ; θ j, = + θ j +, ; for j=,,,, for j=+,+3,,s Alo we have αθ (α+() μ) θ, + (+) μ[θ +, +θ +, ]=, αθ (α+(+) μ) θ +, + (+) μθ +, =, αθ j, (α+(j )μ)θ j, +jμθ j, = along with the normalizing condition j= θ j + for j=+3,+4,,s, αθ S, (α+(s ) μ) θ S, =, S j=+ αθ S, Sμθ S, = (5) (6) (7) (8) (θ j,o +θ j, )+θ S, = (9) Equation (7) to (9) can eaily be olved and thereby the teady tate probability Θ can be obtained Neut[]haprovedthatQBDprocewithgenerator matrix Δ i table if and only if Θ e<θa(s) e () Subtituting Θ, and in (), we get λ< μ p [ [ j= jθ j + S j=+ 3 Steady State Probabilitie j(θ j, +θ j, )+Sθ S, ] () ] When p = (lo ytem), the QBD proce reduce to a Markov proce with finite tate pace and the analyi i rather eay So we omit the dicuion of the lo ytem Aume that p > ItieaytoeethattheQBDproce Ω t = (N(t), I(t), C(t)) i irreducible Denote the tationary probabilitie of the proce by π (n, i, c) = lim t > P (N (t) =n,i(t) =i,c(t) =c) ; n, i S, c =, () Define the infinite dimenional teady tate probability vector Π = (π,π,π,),whereeachπ j i a dim dimenional vector decribed by π j = (π (j,, ), π (j,, ) π (j,, ), π (j, +, ), π (j, +, ),, π (j, S, ), π (j, S, ), π(j,s,)) (3) The tructure of the infiniteimal generator in thi model i imilar to the model decribed by Anoop et al [8] The algorithm for calculating the teady tate ytem probabilitie i decribed below The proce i level independent for n S Therefore the olution i of the form π +j =π R j for j=,,, (4) where matrix R i the olution of the quadratic equation R +R + = (5)
4 Probability and Statitic R canbecalculateduingthefollowingiterativeprocedure (refer Neut []): R n+ = ( +R n A(S) )(A(S) ) ; R = (6) Now to find the tationary vector π j for j=,,,swe proceed a follow From the equation ΠΔ = we get Thu, π S A (S ) +π S +π S+ = (7) π S =π S T S, where T S = A (S ) ( +R ) (8) Proceeding like thi, we get π j =π j T j, where T j = A (j ) (A (j) +T j+a (j+) ), for j=s,s,, (9) Now we can olve for π from the following ytem of equation: where π =π U, π Ve =, U=( T A () )(A() )( ), S V=I+ i= k i T k +( S k= T k ) (I R) () () Equation (4), (8), (9), and () give the teady tate olution of the entire ytem After calculating the teady tate olution, one can find the variou characteritic of the ytem performance Remark 3 The expected number of cutomer in ytem L i L = n= + i= ( S c= i=+ nπ (n, i, ) nπ (n, i, c) +nπ(n, S, )) () Remark 4 The expected number of cutomer waiting in the queue L q i L q = i= n=i+ + + S i=+ n=s+ (n i) π (n, i, ) n=i+ c= (n i) π (n, i, c) (n S) π (n, S, ) (3) Remark 5 Expected inventory level L inv i L inv = + n= i= ( S c= i=+ iπ (n, i, ) iπ (n, i, c) +Sπ(n, S, )) (4) Remark 6 Fraction of the time the production proce i on P on i S P on = π (n, i, ) (5) n= i= Remark 7 We ay that the ytem i buy when the inventory levelilethanorequaltothenumberofcutomerinthe ytem An arriving cutomer to a buy ytem may leave without waiting with probability pa decribed in the model If the ytem i not buy, an arriving cutomer join the ytem for ervice with probability LetP buy be the probability that the ytem i buy Then i P buy = π (n, i, ) + S + n= i= n= π (n, S, ) S Remark 8 Effective arrival rate λ eff i i π (n, i, c) i=+ n= c= (6) λ eff =λ( ( p)p buy ) (7) Remark 9 Expected waiting time of a cutomer in the ytem W i W = L λ eff (8) Remark Expected waiting time of a cutomer in the queue W q i W q = L q λ eff (9) 4 Optimality of (, S) Production Inventory with Poion Demand and Poitive Service Time Supply chain management i an emerging area which deal with production and ditribution of commodity Cot optimization i inevitable in upply chain management Many reearcher have invetigated the cot optimization problem With regard to the model decribed here, producing more commoditie will reduce the waiting time of a cutomer a wellathehortagecot,butitincreaetheholdingcotby reducing the production order point, the holding cot can be minimized but reult in the diatifaction of cutomer
Probability and Statitic 5 and thereby lo of demand In thi ituation, a cot function can be defined which optimize the expected total cot with repect to variou inventory parameter In thi model, cutomer are being elf-erved with a production inventory Aume that the hortage cot incurred by the lo of cutomer i c per cutomer and the holding cot of the inventory be c h per inventory There can be a cot aociated with waiting of cutomer called waiting cot Denote the waiting cot of a cutomer by c w Sincewe conider a ingle tage upply chain model, the other cot like tranportation, warehoue, and o forth can be neglected Thu the total cot of the ytem conit of the following: Average hortage cot: T =c (λ λ eff ) Average holding cot: T h =c h L inv Average waiting cot: T w =c w L q Thuwehavetheexpectedtotalcot,TC, TC =T +T h +T w =c (λ λ eff )+c h L inv +c w L q (3) Here TC i a function of the production order point when the maximum inventory level S i fixed Thu an optimum cot TC i guaranteed for an optimum production order point Numerical reult are included to illutrate thi 5 Numerical Illutration Now we preent the numerical reult by aigning particular value for the variable to illutrate the behavior of the ytem Table 4 give the tationary probabilitie of the ytem with =and S=5for different value of p Hereλ=, μ = 3,andα = 3Variouperformancemeaureofthe (, 5) production inventory are lited in Table 5 and 6 Table 5 give the effect of ervice rate on thee characteritic while Table 3 give the effect of production rate The meaure are calculated for the value λ=and α=3in Table 5 and λ= and μ=3in Table 6 Itcanbeobervedthatatheervicerateincreae the fraction of the time the production proce i on alo increae Probability that the ytem i buy varie with repect to an increae in ervice rate or production rate and it depend on the value of p In other word, the lo of cutomer ( < p < ) ha an impact on P buy Figure 6 detail the effect of production order point on variou performance meaure like L inv, L q, W, P on, P buy, and the lo rate of cutomer from the ytem The lo rate of cutomer from the ytem i defined by the difference between the actual arrival rate and the effective arrival rate That i, lo rate =λ λ eff The expected total cot i hown in Figure 7 for two different value of p(ie, p = 8 and p = 5) when c h =, c w =,andc = It i clear from the figure that for p = 8, the expected total cot i minimum when the production order point i 5 Similarly the expected total cot i minimum when =3for p = 5 Average number of inventorie (L inv ) 6 55 5 45 4 35 3 5 p = p = 5 p= 3 4 5 6 7 Figure : Effect of production order point on average number of inventorie in the ytem for λ=3, μ=5, α=4,ands=8 Average number of cutomer in queue (L q ) 4 8 6 4 p = p= p = 5 3 4 5 6 7 Figure : Effect of production order point on average number of cutomer in the queue for λ=3, μ=5, α=4,ands=8 Average waiting time in the ytem (W ) 65 6 55 5 45 4 35 3 5 p= p = 5 p = 3 4 5 6 7 Figure 3: Effect of production order point on average waiting time in the ytem for λ=3, μ=5, α=4,ands=8
6 Probability and Statitic Table 4: Stationary ditribution of M/M/(, 5) p production inventory queuing ytem n Inventory level (3, ) (3, ) (4, ) (4, ) 5 (When p=) 357878 478545 777679 57666 69765 5648738 378597 5577886 48355 3984797 56838 3859435 4345977 384 5577468 376585 969684 5656673 47735 9347 64565 8887 93758 8479 3 45889 67983 65553 95339 536437 973767 454857 3448 4 976743 466935 674735 67579 3944 555475 699878 5795 5 668 683979 3645597 747 694 7 349 94 6 44488 458396 3737 5583 7735 447 7759 44656 7 9835 3679 59 5384 4969 663 774 466 8 99355 3 787 456 3666 3964 733 563 9 33645 33575 67733 949 6958 98 46559 (When p = 5) 345775 593656 958634 6683685 7348486 665799 448949 663438 888 45677 6467979 4463355 53889 445579 988876 443866 7684857 368543 4885 4936 76378 4956 44354 493883 3 884 39743 456 374683 453 338799 34744 335655 4 9795 8669 945875 434945 5769 63375 4599 5655 5 33337 36854 357 49467 4935 5373 44775 7985 6 4 838 6564 565 4864 4977 746 634 7 37588 3859 76 666 75 548 788 73 8 558 79 6484 9 9 88 5 43 9 49 45 7 6 79 65 4 (When p = ) 439896 65775 984585 753 778343 7696 466865 7938 4839 458 6636 467396 54434 4676748 35679 467473 44533 3597 56965 554737 7438959 5558697 433476 5599336 3 36 9495 693969 4334 33743 343534 3434 343779 4 555 555 7589 63755 6543 358 555848 5 45 55 66 65 56 934 48 7397 6 9 3 9 6 6 6 985 Probability that the production i on (P on ) 65 6 55 5 45 p = p = 5 p= Lo rate of cutomer 5 5 5 p = p = 5 p = 8 4 3 4 5 6 7 3 4 5 6 7 Figure 4: Effect of production order point on the fraction of the timetheproductionionforλ=3, μ=5, α=4,ands=8 Figure 5: Effect of production order point on the lo rate of cutomer from the ytem for λ=3, μ=5, α=4,ands=8
Probability and Statitic 7 Table 5: Effect of the ervice rate on the performance meaure μ L L q L inv P on P buy p= 3 6883 594666 5653367 4375868 6664 4 39567 539567 586 4793339 97679 6 8683875 493554 494654 584376 3434 6646 4646 46489 545573 7694 p = 5 3 695987 86633 8448398 535689 9563 4 5389554 776769 8838887 54784 939734 6 3854 7376 775554 577435 97967 59999 6598 7444 59763 78435 p = 3 59946945 9344 9434997 539764 577 4 45636599 88538 955334 55953 57359 6 394557 747367 86776 58444965 6478359 89533 6974 8365769 666953 799 Table 6: Effect of the production rate on the performance meaure α L L q L inv P on P buy p= 3 6883 594666 5653367 4375868 6664 5 79599 6496 33833 346497 443986 67834 36753 375348654 69889 963944 5 6735435 683769 3846786 3464 56999 p = 5 3 695987 86633 8448398 535689 9563 5 6638 7985 3476889 33973 3354 66853 4543 37599946 739488 5697 5 6663988 954 384933937 39649 9736 p = 3 59946945 9344 9434997 539764 577 5 63558435 5334 345359 336566 8564 65875 775 37646767 735 475 5 6554966 5887 385354 43779 945 6 Concluion We have conidered a queuing model where the erver are regarded a an inventory with (, S) production inventory policy The behavior of thi ytem i decribed by a QBD proce where the firt S level are the boundary condition Stability condition, teady tate ditribution, important performance meaure, and an optimization problem have been invetigated uing matrix analytic method Detailed analye of the numerical reult are alo preented Analyi of a elf-ervice inventory uing the method of queuing theory by regarding the inventory a erver i a new framework The future work include the generalization of thi idea by conidering heterogeneou inventorie and multitage upply chain Probability that the ytem i buy (P buy ) 4 35 3 5 5 5 p= p = 5 p = 3 4 5 6 7 Figure 6: Effect of production order point on the fraction of the time the ytem i buy for λ=3, μ=5, α=4,ands=8 Expected total cot (TC) 95 9 85 8 75 7 65 6 55 p = 8 p = 5 3 4 5 6 7 Figure 7: Effect of production order level on the expected total cot when λ=3, μ=5, α=4, c h =, c w =, c =,ands=8 Conflict of Interet The author declare that there i no conflict of interet regarding the publication of thi paper Acknowledgment The author thank the two anonymou reviewer and the editor for the helpful comment that improved the earlier verion of the paper Reference [] K Sigman and D Simchi-Levi, Light traffic heuritic for an M/G/ queue with limited inventory, Annal of Operation Reearch,vol4,no,pp37 38,99 [] A Krihnamoorthy, B Lakhmy, and R Manikandan, A urvey on inventory model with poitive ervice time, OPSEARCH, vol48,no,pp53 69, [3] P R Parthaarathy and V Vijayalakhmi, Tranient analyi of an inventory model: a numerical approach, International Computer Mathematic,vol59,pp77 85,996
8 Probability and Statitic [4] T G Deepak, A Krihnamoorthy, V C Narayanan, and K Vineetha, Inventory with ervice time and tranfer of cutomer and/inventory, Annal of Operation Reearch,vol6, no, pp 9 3, 8 [5] P Manuel, B Sivakumar, and G Arivarignan, A perihable inventory ytem with ervice facilitie, MAP arrival and PH ervice time, Sytem Science and Sytem Engineering,vol6,no,pp6 73,7 [6] V C Narayanan, T G Deepak, A Krihnamoorthy, and B Krihnakumar, On an (,S) inventory ytem with ervice time vacation to the erver and correlated lead time, Quality Technology and Quantitative Management,vol5,no,pp9 43, 8 [7] A Krihnamoorthy and V C Narayanan, Production inventory with ervice time and vacation to the erver, IMA Journal of Management Mathematic,vol,no,pp33 45, [8] N N Anoop, M J Jacob, and A Krihnamoorthy, The multi erver M/M/(,S) queuing inventory ytem, Annal of Operation Reearch, vol 33, no, pp 3 333, 3 [9] A N Nair and M J Jacob, Inventory with poitive ervice time and retrial of demand: an approach through multierver queue, ISRN Operation Reearch,vol4,ArticleID5963, 6page,4 [] A Krihnamoorthy and N C Viwanath, Stochatic decompoition in production inventory with ervice time, European JournalofOperationalReearch,vol8,no,pp358 366, 3 [] M F Neut, Matrix-Geometric Solution in Stochatic Model An Algorithmic Approach, John Hopkin Univerity Pre, Baltimore, Md, USA, 98
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