Optimal revenue management in two class pre-emptive delay dependent Markovian queues

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1 Optimal revenue management in two cla pre-emptive delay dependent Markovian queue Manu K. Gupta, N. Hemachandra and J. Venkatewaran Indutrial Engineering and Operation Reearch, IIT Bombay March 15, 2015 Abtract In thi paper, we preent a comparative tudy on total revenue generated with pre-emptive and non pre-emptive priority cheduler for a fairly generic problem of pricing erver urplu capacity in a ingle erver Markovian queue. The pecific problem i to optimally price the erver urplu capacity by introducing a new cla of cutomer econdary cla without affecting the pre-pecified ervice level of it current cutomer primary cla when pre-emption i allowed. Pre-emptive cheduling i ued in variou application. Firt, a finite tep algorithm i propoed to obtain global optimal operating and pricing parameter for thi problem. We then decribe the range of ervice level where pre-emptive cheduling give feaible olution and generate ome revenue while non pre-emptive cheduling ha infeaible olution. Further, ome complementary condition are identified to compare revenue analytically for certain range of ervice level where trict priority to econdary cla i optimal. Our computational example how that the complementary condition adjut in uch a way that pre-emptive cheduling alway generate more revenue. Theoretical analyi i found to be intractable for the range of ervice level when pure dynamic policy i optimal. Hence extenive numerical example are preented to decribe different intance. It i noted in numerical example that pre-emptive cheduling generate at leat a much revenue a non pre-emptive cheduling. A certain range of ervice level i identified where improvement in revenue i quiet ignificant. keyword: Dynamic pre-emptive priority, Pricing of ervice, Admiion control, Queueing. 1 Introduction Queueing ytem ha become popular for modelling a variety of complex dynamic ytem. Contemporary application include modelling of upply chain, call center, wirele enor network, proceor, etc. ee Bhakar and Lallement 2010, Bhakar and Lavanya 2010, Kim et al. 2013, Correponding author id: manu.gupta@iitb.ac.in 1

2 Lee and Yang Multi-cla queue are pecial cla of queueing ytem where different type of cutomer achieve quality of ervice differentiation. Thi pecial cla of queueing ytem ha alo acquired ignificant importance in queueing theory due to it wide range of application in communication ytem, traffic and tranportation ytem. Extenive reearch i done in analying the different apect of uch multi-cla queueing ytem ee Hain et al. 2009, Shanthikumar and Yao 1992, Sinha et al etc. and reference therein. Another community of reearcher tudied pricing in the context of queueing ytem in a variety of application. Analyi of pricing problem in queueing tarted with Naor Naor 1969 who conidered a tatic pricing problem for controlling the arrival rate in a finite buffer queueing ytem. A rich literature on pricing ha evolved ince then. It include tatic and dynamic pricing with ingle and multiple cla queue ee Celik and Maglara 2008, Gallego and van Ryzin 1994, Marbach A detailed dicuion on pricing communication network can be een in Courcoubeti and Weber Pricing urplu or extra capacity of erver i alo important in the context where etting up additional erver incur high cot. Hall et al. Hall et al tudied the cenario where a reource i hared by two different clae of cutomer. Thi tudy focued on dynamic pricing and demontrated the propertie of optimal pricing policie. A ingle erver queueing ytem with two clae of cutomer ha been conidered in Sinha et al. 2010, where the pecific problem wa to optimally price the erver exce capacity for new econdary cla of cutomer, while meeting the ervice level requirement of it exiting primary cla of cutomer. In thi model, the arrival rate of thi new cla depend linearly on offered ervice level and unit admiion price charged. Service level of a cla i defined by the average waiting time of that particular cla. The arrival procee have been aumed to be independent Poion procee for both clae, and independently, the ervice time ditribution i general and identical for both clae. A delay dependent non pre-emptive priority cheduling i conidered acro clae a the queue dicipline. Under non pre-emptive etting a primary cla cutomer, upon arrival, wait in queue if the erver i buy ervicing either a primary or econdary cla cutomer. Baed on the arrival rate and ervice level of the primary cla cutomer, and the firt and econd moment of ervice time, a finite tep algorithm ha been propoed to find the optimal ervice level, pricing, arrival rate and cheduling of the econdary cla cutomer in Sinha et al Further refinement and a tudy of the robutne of the optimal parameter with repect to ytem variability ha been hown in Gupta et al. 2014, Hemachandra and Raghav 2012 and Raghav A imilar cot optimization problem for ervice dicrimination in queueing ytem i olved uing relative priority ee Sun et al Some imilar optimal control problem are recently explored where it take non-zero time to witch the ervice between the two clae of cutomer ee Rawal et al Pricing urplu erver capacity with pre-emptive cheduling play an important role in problem related to wirele communication. For example, conider a cognitive radio ad hoc network CRAHN which are uually compoed of two kind of uer: cognitive radio CR uer and primary uer ee Akyildiz et al. 2009, Chowdhury and Felice 2009 and Felice et al Primary uer PU have a licene to acce the licened pectrum and network i providing ervice to ome primary 2

3 cutomer. Primary cutomer are atified a long a they are provided a guaranteed quality of ervice QoS in term of mean waiting time. CR uer acce the licened pectrum a a viitor, by opportunitically tranmitting on the pectrum hole. The network can utilize the urplu capacity pectrum, time lot, etc. to erve econdary CR et of uer while maintaining the QoS of primary et of uer. Other application of pre-emptive priority baed cheduling are in operating ytem, real time ytem, etc. ee Audley et al. 1995, Burn 1994 and reference therein. The reult of thi paper are relevant in above context where pre-emptive priority policy i applicable. Firt part of the paper decribe the analyi with pre-emptive cheduling while the other part dicue the improvement in revenue by uing pre-emptive cheduling over non pre-emptive cheduling. Revenue maximization i one of the main objective of ervice provider in thee ituation. Such a revenue maximization problem i olved in Sinha et al with non pre-emptive delay dependent priority cheduling acro clae. In thi paper, we work on the problem of revenue maximization with pre-emptive delay dependent priority cheduling acro clae which i practical for different application dicued above. The main contribution of thi paper i two fold a dicued below. Firt, we olve the revenue optimization problem to optimally price the erver urplu capacity by introducing a new econdary cla of cutomer without affecting the ervice level of it exiting primary cutomer while uing pre-emptive delay dependent priority cheduling acro clae. Two optimization model are formulated to maximize the profit of the reource owner, depending on the value of the relative queue dicipline priority parameter. The firt optimization model, valid when the relative parameter i finite, i a non convex contrained optimization problem. The econd optimization model, valid when the relative parameter i infinite, i a convex optimization problem. Thee optimization problem are olved and reult are dicued. Baed on thee reult, a finite tep algorithm to find the optimal operating parameter pricing, cheduling, ervice level and arrival rate of the econdary cla cutomer i preented. We then preent an extenive tudy to compare revenue with pre-emptive and non pre-emptive priority cheduling. We firt identify certain range of ervice level where pre-emptive cheduling give feaible olution while problem i infeaible with non pre-emptive cheduling. We further identified ome complementary condition to compare the total revenue generated for certain range of input parameter when trict priority to econdary cla cutomer i optimal. Other way to do thi revenue comparion i via ervice level. If econdary cla ervice level decreae for a fixed admiion price, admiion rate will increae by market equilibrium and thi will increae revenue. Secondary cla ervice level alo need ome condition to tract the comparion analytically. It i noted by computational example that thee condition adjut in uch a way that pre-emptive priority cheduling generate more revenue than that with non pre-emptive cheduling. Objective function i highly non linear and become mathematically intractable when optimal cheduling parameter i pure dynamic. Hence, we further perform computational tudy for uch intractable range of ervice level. Thi tudy how that the revenue generated with pre-emptive priority i more than that of non pre-emptive priority and certain range of ervice level i identified where revenue increment i quiet ignificant. 3

4 Thi paper i organied a follow. Section 2 decribe the ytem etting. Section 3 decribe the notation, optimization model formulation and propertie of mean waiting time. Section 3.1 dicue the olution of thi non convex contrained optimization problem for global maxima. In Section 3.2, we propoe a finite tep algorithm to find the global optimal operating and pricing parameter. Section 4 and 5 decribe the comparion of revenue under two cheduling policie. Section 6 preent concluion and direction for future reearch. A preliminary verion of the algorithm i preented in Gupta et al In thi paper, we preent detailed argument that lead to the algorithm. We alo preent an extenive comparative tudy on total revenue generated with pre-emptive and non pre-emptive cheduling, partly uing theoretical reult and ret via computational tudy. 2 Sytem decription We conider the ytem etting imilar to Sinha et al. 2010: a ingle erver queueing ytem with two clae of cutomer, primary and econdary a hown in Figure 1. The arrival procee of primary a well a econdary are independent Poion procee. Arrival rate for primary cla i known. The ervice time ditribution i identical for both clae and it i exponentially ditributed. Alo, there i a long term agreement with primary cla cutomer which pecifie the guaranteed quality of ervice QoS. QoS for a cutomer cla i in term of mean waiting time of that cla. Each cutomer of econdary cla pay the admiion fee. Service level offered to primary cla of cutomer i alo known. Objective of the problem i to decide the cheduling policy, arrival rate, ervice level and admiion price for econdary cla cutomer uch that total revenue i maximized while maintaining the ervice level for primary cla of cutomer. Further, a delay dependent pre-emptive queue dicipline ee Kleinrock 1964 i ued acro clae. Pre-emption i in term of continuouly monitored ytem. That i, if the intantaneou dynamic priority of the currently erved cutomer i lower than that of a cutomer waiting in the queue, the cutomer in ervice will be pre-empted by later Kleinrock, Pre-empted cutomer join head of line of repective queue a hown by dotted line in Figure 1. We aume that the arrival rate of econdary cla cutomer linearly depend on the price and ervice level offered to that cla. Detailed notational decription and olution of thi model i dicued in Section 3. We now briefly explain the logic of delay dependent priority dicipline. 2.1 Delay dependent priority queue dicipline Different type of priority logic are poible to chedule multiple cla of cutomer for ervice at a common reource. Suppoe abolute or trict priority i given to one cla of cutomer, then the lower priority cla may tarve for reource acce for a very long time. For example, in cae of two clae of cutomer, if trict and higher priority i given to primary cla cutomer, econdary cla cutomer will be erved only after the buy period of primary cla. 4

5 Primary Cutomer S p b p Preempted Primary Cutomer Secondary Cutomer λ S b erver µ Preempted econdary Cutomer Server ue delay dependent Pre-emptive priority rule Figure 1: Schematic view of model Thi problem of exce queue delay time of lower priority cla cutomer can be addreed by introducing delay dependency in prioritie. Such a queue dicipline aign a dynamic priority to each cutomer. Thi dynamic priority i a function of the queue delay of the cutomer a well a a parameter aociated with that cutomer cla. Thi concept of delay dependent priority queueing dicipline wa firt introduced in Kleinrock The logic of thi dicipline work a follow. Each cutomer cla i aigned a queue dicipline parameter, b i, i {1,, N} for all N cutomer clae. For a cutomer arriving at time τ, the intantaneou dynamic priority for cutomer of cla i at time t, q i t, i then given by q i t = t τ b i, i = 1, 2,, N. 1 Highet intantaneou dynamic priority parameter, q i t, cutomer will have highet priority of receiving ervice. Tie are broken uing Firt-Come-Firt-Served rule. Hence according to thi dicipline the higher priority parameter cutomer gain higher dynamic priority at higher rate. Figure 2: Illutration of delay dependent priority Kleinrock, 1964 We illutrate thi in Figure 2. Conider two clae of cutomer, cla 1 and cla 2 with queue dicipline parameter b p and b p, where b p < b p. Suppoe cla 1 cutomer arrive at time τ and cla 2 cutomer arrive at time τ, with τ < τ. Figure 2 illutrate the change in their repective dynamic queue priority over time. In the time interval τ to τ, cla 1 cutomer ha higher intantaneou priority. In time interval τ to t 0, cla 2 cutomer tart gaining priority till cla 1 cutomer will be erved a it intantaneou priority i higher. Intantaneou priority for both cla i ame at t 0, o cla 1 cutomer will be erved according to FCFS rule. After time t 0, cla 2 cutomer have 5

6 higher intantaneou priority hence cutomer of that cla will be erved. 3 Optimal joint pricing and cheduling model analyi Let and λ be independent Poion arrival rate of primary and econdary cla cutomer repectively. Service time are independent and identically ditributed exponential random variable for both clae with mean 1/µ. Let S p be the pre-pecified primary cla cutomer ervice level. Queue dicipline i pre-emptive delay dependent priority a propoed in Kleinrock 1964 and explained in lat ection. A chematic view of the model i hown in Figure 1. Suppoe there are 1, 2,, N clae, then the average waiting time for k th cla W k i given by following recurion for delay dependent pre-emptive cheduling acro clae ee Kleinrock W k = W 0 1 ρ + N i=k+1 ρ i µ k 1 b k b i k 1 1 N i=k+1 ρ i i=1 µ i 1 b i ρ i 1 b k b i k 1 ρ i Wi 1 b i i=1 b k 2 where ρ i = λ i /µ i, ρ = N ρ i, W 0 = N λ i σi and 0 < ρ < 1. Alo the conervation law in i=1 i=1 2 µ 2 i M/G/1 queue for a work conerving queueing dicipline tate that Kleinrock, 1965: N ρ i Wi = ρw ρ i=1 Note that average waiting time, Wk, depend only on ratio of parameter {b i } N 1. So in cae of two primary and econdary clae, average waiting time will depend on ratio b /b p, where thee b p and b are pre-pecified parameter aociated with primary and econdary cla. Set β := b /b p, which repreent the relative queue dicipline parameter. β can take value from 0 to 0 and included, effect of changing β in queuing dicipline are a follow b k β = 0, i.e., b /b p = 0, Static priority rule i employed with priority given to primary cla cutomer, β < 1, i.e., b /b p < 1, Primary cla cutomer are gaining intantaneou priority at a higher rate than econdary cla cutomer, β = 1, i.e., b /b p = 1, Both clae of cutomer are given equal priority, hence, it i a global FCFS queue dicipline, β > 1, i.e., b /b p > 1, Secondary cla cutomer are gaining intantaneou priority at a higher rate than primary cla cutomer, β =, i.e., b /b p =, Static priority dicipline i employed with priority given to econdary cla cutomer. 6

7 Let W p λ, β and W λ, β be expected waiting time for primary and econdary cla of cutomer. Following expreion for W p λ, β and W λ, β are derived uing Equation 2. W p λ, β = λµ λ1 β µ λλ λµ + λ µ λ β β 1 {β 1} + µµ λµ 1 β µµ λ µ λ {β>1} 4 β W λ, β = λµ + λ λ µ λ 1 1 µ λ 1 1 pµ λ1 β µµ λµ 1 β 1 β β {β 1} + µµ λ µ λ {β>1} 5 β where λ = +λ and 1 {Γ} i 1 if tatement Γ i true, ele 1 {Γ} i zero. Let S p and S be the promied ervice level offered for primary and econdary cla of cutomer repectively. A dicued earlier, rate of econdary cla cutomer i a linear function of unit admiion price, θ, and aured ervice level, S. Λ θ, S = a bθ cs 6 where a, b, c are given poitive contant driven by market. a i the maximum arrival rate poible wherea b and c are enitivity of cutomer to price charged and ervice level repectively. With above notation, we have following optimization model for maximizing reource owner profit imilar to Sinha et al. 2010: ubject to P0: max λ,θ,s,β θλ 7 W p λ, β S p, 8 W λ, β S, 9 λ < µ, 10 λ a bθ cs, 11 λ, θ, S, β Contraint 8 i to maintain QoS of primary cla cutomer while Contraint 9 i for enuring econdary cla cutomer ervice level which i alo a deciion variable. Contraint 10 i neceary condition for the queue tability. Contraint 11 capture the dependency of econdary cla arrival rate a hown in Equation 6. Optimization problem P0 i a four dimenional optimization problem. It can be een that contraint 9 will be binding at optimality ince no reource owner would provide a wore than poible QoS level to cutomer. Alo Contraint 11 will be binding becaue any lack in it can be eaily removed by increaing the price. Further, ubtituting the value of θ = 1 b a λ cs and W λ, β = S, the problem P0 reduce to a two dimenional optimization problem P1 imilar to Sinha et al. 2010: 1 P1: max aλ λ 2 cλ W λ, β 13 λ,β b 7

8 ubject to W p λ, β S p, 14 λ µ, 15 λ, β Note that contraint 15 expand the feaible region of P1 a compare to P0 but Contraint 14 enure that λ < µ. Hence optimality of problem P1 i not affected by uch expanion of feaible region. It follow from Equation 4 and 5 that expreion W p λ, β and W λ, β depend on the value of β β < 1 or β > 1 and β = i alo a valid deciion for queue dicipline. Hence optimization problem P1 differ from claical optimization problem. Conider the notation W p λ = W p λ, β = and W λ = W λ, β =. Now, on etting β =, we have one dimenional optimization problem P2 imilar to that in Sinha et al. 2010: 1 P2: max λ b [aλ λ 2 cλ W λ ] 17 ubject to W p λ S p, 18 λ µ, 19 λ Few propertie of W p λ, β and W λ, β are a follow; propertie 3 and 4 below render P1 a non convex contrained optimization problem. 1. W p λ, β and W λ, β are increaing convex function of λ in interval [0, µ. 2. W p λ, β i an increaing concave function of β 0 and W λ, β i a decreaing convex function of β W p λ, β i neither convex nor concave function of λ, β when λ [0, µ and β 0. Alo, W p λ, β i not a quai convex function of λ, β. 4. λ W λ, β i neither convex nor concave function of λ, β when λ [0, µ and β 0. Above propertie are derived by calculating the firt and econd order partial derivative of W p λ, β and W λ, β with repect to λ and β and then by calculating gradient and Heian matrix of W p λ, β and W λ, β ee Appendix for detail. Solution of optimization problem P0 i preented in Section 3.1 and an algorithm to find the optimal operating parameter i propoed in Section

9 3.1 Optimal admiion price, ervice level, queue dicipline and admiion rate In order to find the global optimal operating parameter optimal admiion price, ervice level, queue dicipline and admiion rate, one need to olve and compare the optimal objective of problem P1 and P2. By uing above propertie of mean waiting time, one can how that optimization problem P1 i non convex while P2 i convex optimization problem. Solution of thee problem i decribed in Section and Solution of optimization problem P0 reource owner profit maximization i given by P1 and P2 depending on relative queue dicipline parameter being finite or infinite. Comparion of objective function of problem P1 and P2 i preented in Section to find global optimal olution for problem P Solution of optimization problem P1 β < Property 4 of mean waiting time tate that λ W λ, β i neither a convex nor a concave function of λ, β. Hence the objective of problem P1 i neither convex nor concave and thi make optimization problem P1 a non convex contrained optimization problem. We olve thi problem by deriving Karuh Kuhn Tucker KKT neceary and ufficient condition. Conider the Lagrange function correponding to NLP P1: Lλ, β, u 1, u 2, u 3 = 1 b aλ λ 2 cλ W λ, β + u 1 W p λ, β S p + u 2 λ + u 3 β 21 where u 1, u 2 and u 3 are Lagrangian multiplier. Lagrangian multiplier correponding to trict inequality Contraint 15 will be 0. Bazaraa et al., 2004: KKT firt order neceary condition are given a follow [ ] W W p a 2λ c W λ + bu 1 λ cλ W β + bu 1 λ + bu 2 = 0 22 W p β + bu 3 = 0 23 u 1 [W p S p ] = 0 24 u 2 λ = 0 25 u 3 β = 0 26 W p S p and λ < µ 27 u 1 0; λ, β, u 2, u It follow from Equation 25 that if u 2 0 then λ ha to be 0 which will make the objective 0. Hence u 2 = 0 hold. Conider Equation 23, 26 and 28 firt. Following two cae are poible depending on value of β: 1. β > 0 : In thi cae, u 3 = 0 hold from Equation 26. Uing Equation 23, we have u 1 = c b in both the cae when β i le or more than 1. Thi value of u 1 i obtained uing expreion of derivative of W p and W with repect to β. 9

10 2. β = 0 : In thi cae, u 3 can be poitive or 0. With β = 0, Contraint 23 reult in u 3 = 1 λ µc + bu 1 b µ λ µ 2 Thi implie u 3 0 i atified iff c + bu 1 0. Hence KKT condition 23, 26 and 28 are atified iff we get in one of the following cae along with u 2 = 0. Cae 1: u 1 = c, u 3 = 0, β > 0 b Cae 2a: u 1 < c, u 3 = 1 b b Cae 2b: u 1 = c, u 3 = 0, β = 0 b λ µc + bu 1 µ λ µ 2, β = 0 Note that u 1 < 0 hold in all above cae. It follow from Equation 24 that waiting time contraint i binding. From above analyi, a KKT point ha to be among one of the above cae and W p = S p hould hold along with Equation 22. The analyi auming that KKT point atifie condition of cae 1 reult in Theorem 1. The analyi auming that KKT point atifie condition of cae 2a and 2b reult in Theorem 2. Theorem 1 below tate that when primary cla cutomer ervice level, S p, i retricted to a particular range I a defined below, the optimal arrival rate of econdary cla cutomer, λ, i given by the root of a cubic and optimal cheduling parameter, β, i finite and nonzero pure dynamic cheduling policy. Theorem 1. Suppoe a c > 2µ. Then, there exit λ1 µµ 2 Gλ in the interval 0, µ, where which i the unique root of cubic Gλ 2µλ 3 [c + µa + 4φ 0 ]λ 2 + 2φ 0 [c + µa + φ 0 ]λ aµφ c µ + φ 0 and φ 0 = µ. Denote λ 1 = +λ 1 and let S p lie in interval I µµ, λ 1µ + µ λ 1 λ 1 µµ λ 1 µ λ 1 and β 1 i given by µ λ 1 µs p µ for β 1 λ 2 1 µ λ 1 µs p λ 1 µµ < S λ 1 p µµ λ 1 = λ 1 µ λ µs p λ 1 for λ 1 µ + µ λ 1 λ 1 + µs p λ 1 µ 2 S p µµ λ 1 < S p < λ 1µ + µ λ 1 λ 1 µµ λ 1 µ λ 1 then λ 1 point. and β 1 i a trict local maximum of NLP P1 and contraint W p S p i binding at thi Proof. Given a c > 2µ, one can etablih that λ1 µµ 2 i the unique root of cubic Gλ in the interval 0, µ, by conidering it ign change, tationary point, nature of it derivative and uing the argument imilar to Claim 1 in Sinha et al

11 Note that u 1 = c, u 3 = 0, β > 0 and u 2 = 0 hold for Cae 1. On putting thee value of u 1 b and u 2 in Equation 22, we have a 2λ c λ W + W p = 0 29 λ On implifying the conervation law Equation 3 with exponential ervice time and two clae, we get W p + λ W = λ 2 µµ λ where λ = + λ. Uing Equation 29 and 30, we have [ ] λp + λ 2µ λ a 2λ c = 0 µµ λ 2 The above equation can be implified a following cubic in λ : Gλ 2µλ 3 [c + µa + 4φ 0 ]λ 2 + 2φ 0 [c + µa + φ 0 ]λ aµφ c µ + φ 0 = 0 31 where φ 0 = µ. A λ 1 i the unique root of cubic Gλ in the interval 0, µ, olving Gλ = 0 for λ 0, µ reult in λ = λ 1. Claim 1. There exit a queue dicipline management parameter β > 0 which atifie the equality W p λ, β = S p if 0, λ 0, + λ < µ and S p lie in interval µµ, λµ + µ λλ µµ λµ λ where λ = + λ and i given by µ λ µs p µ for λ β = 2 µ λµs p λ µµ < S λ p µµ λ for Proof. See Appendix. λ µ λ1 + µs p λµ + µ λλ + µ S p λ µ 2 S p λ µµ λ < S p < λµ + µ λλ µµ λµ λ Let β 1 = β for λ = λ 1. It follow from Claim 1 that W p λ 1, β 1 = S p. It follow that λ 1, β 1, u 1 = c b, u 2 = 0 and u 3 = 0 atify all KKT neceary condition One ha to check for ufficient condition to argue the local optimality of λ 1 and β 1. Conider retricted Lagrangian Lλ, β = Lλ, β, u 1 = cλp b, u 2 = 0, u 3 = 0. On uing Equation 21 and 30, we get 30 Lλ, β = 1 ] [aλ λ 2 λ 2 cλ + c S p b µµ λ 32 Note that gλ, β W p λ, β S p i the only binding contraint in NLP P1 and thi contraint i trongly active a aociated Lagrangian multiplier u 1 i non-zero. Define the cone C := {d 0 : gλ, β t.d = 0} refer Theorem in Bazaraa et al. 2004, gλ, β := [k 1, k 2 ] t and d := [d 1, d 2 ]. On implifying, we have C = {d 0 : k 1 d 1 + k 2 d 2 = 0}. On further implifying, C = {d : d 1 = k 2 d 2 /k 1, d 2 0}. Let u denote the Heian of retricted Lagrangian by H Lλ, β. Uing Equation 32, we get following Heian matrix 11

12 H Lλ, β = 2 b 1 + cµ 0 µ λ Now, we calculate dh Lλ, βd t for every d C, we have dh Lλ, βd t = 2 cµ 1 + d 2 b µ λ 3 1 = 2 [ cµ 1 + k ] 2 2d 2 < 0 d b µ λ k 1 Thi implie that the ufficient condition for KKT point are met. Hence λ 1, β 1 are trict local maximum of NLP P1 and the theorem follow. Theorem 2 below tate that when primary cla cutomer ervice level i, we can introduce econdary cutomer by etting cheduling parameter 0, i.e., tatic priority hould be given µµ to primary cla of cutomer. Thi reult matche with intuition alo a ervice level µµ i average waiting time when there are primary cla of cutomer only. Thi ervice level can be achieved when trict pre-emptive priority i given to primary cla of cutomer. Theorem 2. Suppoe a c > 2µ and S µµ 2 p = Ŝp =, and λ1 i the unique root of µµ cubic Gλ in the interval 0, µ. Then λ 1 and β 2 = 0 i the trict local maximum of NLP P1 and contraint W p S p i binding. Proof. Conider the Lagrangian multiplier according to cae 2a. A point will be KKT point in thi cae if contraint W p S p i binding and KKT condition 22 i atified. Note that cae 2a implie u 1 < c λ µc + bu 1, u 3 = b bµ λ µ, β = 0 and u 2 2 = 0. On implifying KKT condition 22 with thee mentioned etting, we get Gλ = 0. On implifying the equality contraint W p λ, β = 0 = S p, we get S p = µµ. Note that λ 1 i the root of cubic Gλ. It follow from uppoition of the theorem that λ 1, β 2 = 0, u 1 < c b, u λ µc + bu 1 2 = 0, u 3 = bµ λ µ ay l 2 i a KKT point. Contraint 2 g 1 λ, β W p S p and g 2 λ, β β 0 are binding and trongly active a correponding Lagrangian multiplier u 1 and u 3 are non-zero. g 1 λ, β = W p λ W p β = 0 µλ µ λµ 2 [ ] 0 and g 2 λ, β =. 1 Conider the critical cone, C = {d 0, g 1 λ, β t.d = 0, g 2 λ, β t.d = 0} refer Bazaraa et al On implification, we get C = {d 1, 0 : d 1 0}. Retricted Lagrangian i given by Lλ, β, u 1 = l 1 ay, u 2 = 0, u 3 = l 2 = 1 b aλ λ 2 cλ W λ, β + αw p λ, β S p + l 2 β 12

13 Let the Heian matrix of above retricted Lagrangian be [ ] a 11 a 12 H Lλ, β = a 21 a 22 In order to verify the KKT econd order condition, we will check for the ign of [ ] [ ] [ ] d t a 11 a 12 d 1.H Lλ, β.d = d 1 0 = a 11 d 2 1 a 12 a 22 0 where d i a vector from cone C and a 11 = 2 Lλ, β = 2 λ 2 b Hence d t H Lλ, βd will be negative. Thi implie λ 1 P1 and W p S p i binding. λ=λ 1,β=0 1 + cµ µ λ 1 2 < 0., β 2 = 0 are trict local maximum of NLP Now conider Lagrangian multiplier according to cae 2b. Note that cae 2b implie u 1 = c, u 3 = 0, β = 0 and u 2 = 0. Equality contraint W p λ, β = 0 = S p give S p = b µµ. On implifying KKT condition 22 with u 2 = 0 and β = 0, we get Gλ = 0. It follow that λ 1, β 2 = 0, u 1 = c b, u 2 = 0, u 3 = 0 i a KKT point. To verify the econd order condition, we calculate the Heian matrix of retricted Lagrangian multiplier a follow H Lλ, β = 2 cµ b µ λ Note that contraint g 1 λ, β W p S p i trongly active a correponding Lagrangian multiplier u 1 = c b binding. < 0 while g 2 λ, β β 0 i weakly active a u 3 = 0. Both thee contraint are g 1 λ 1, β 2 = 0 µλ 1 µ λ 1 µ 2 and g 2 λ 1, β 2 = Conider the critical cone C = {d 0 : g 1 λ, β t.d = 0, g 2 λ, β t.d 0} which implifie to C = {d 1, 0 : d 1 0}. In order to verify the KKT econd order condition, we have [ d t.h Lλ, β.d = d 1 0] 2 cµ [ ] b µ λ 3 d 1 = 2 cµ 1 + d b µ λ 3 1 < 0. Thi implie KKT ufficient condition are atified. Hence, λ 1, β 2 = 0 are trict local maximum of NLP P1 and W p S p i binding. Hence theorem follow. [ ] 0 1 Corollary 1. The mean arrival rate of econdary cla cutomer, λ 1 point, i independent of S p in ervice level range I µµ. which i a local optimum Proof. The optimal admiion rate, λ 1, i the root of cubic Gλ which i independent of S p. So λ 1 i independent of S p. Hence corollary follow. 13

14 3.1.2 Solution of optimization problem P2 β = Following the imilar argument a in Sinha et al., 2008, page 18, it can be argued that problem P2 i a differentiable convex optimization problem. So, we only need to check for the firt order KKT neceary condition to find the optimal olution. Lagrangian function correponding to NLP P2 i given by: Lλ, v 1, v 2 = 1 b aλ λ 2 cλ W λ + v 1 W p λ S p + v 2 λ 33 where v 1 and v 2 are Lagrangian multiplier. Lagrangian multiplier correponding to trict inequality Equation 19 will be 0. KKT firt order neceary condition are given a follow. [ ] W a 2λ c W λ + bv W p 1 + bv 2 = 0 λ λ 34 v 1 [ W p S p ] = 0 35 v 2 λ = 0 36 W p λ S p and λ < µ 37 v 1 0, λ, v It follow form Equation 36 that λ ha to be zero if v 2 > 0. Objective of reource owner i to earn trictly poitive revenue. Hence we aume that v 2 = 0 throughout the analyi. We look for all poible KKT point of the revenue maximization problem P2 with v 2 = 0. We alo know that the contraint 18 will be either binding or non binding at optimality. Theorem 3 identifie the range of S p where contraint 18 i trictly non-binding at optimality while Theorem 4 identifie the ame when contraint 18 i binding. Theorem 3 below tate that if primary cla cutomer ervice level i in range J a defined below, then, olution of optimization problem P2 i given by etting β =, i.e., econdary cla cutomer hould be given trict priority. Optimal admiion rate for econdary cla cutomer i given by the root of cubic Gλ, identified in Equation 39. Theorem 3. Suppoe µ 2µλ 2 p + cµ + > aµλ 2 p hold; then, there exit λ 3 unique root of cubic Gλ in the interval 0, µ where which i the Let λ 3 = + λ 3 Then λ 3 Gλ 2µλ 3 c + µa + 4µλ 2 + 2µc + aµ + µ 2 λ aµ λ 3 µ + λ 3 µ λ 3 and further aume that S p lie in the interval J µµ λ 3 µ λ 3,. i the global maxima of NLP P2 and contraint W p S p i non-binding at thi point. Proof. It can be etablihed that λ 3 i the unique root of cubic Gλ in the interval 0, µ, by conidering it ign change, tationary point and nature of it derivative. Note that given µ 2µλ 2 p + cµ + > aµλ 2 p, Gµ λp > 0 follow and G0 = aµ 2 < 0. Hence, λ 3 indeed i trictly le than µ and lie in the interval 0, µ. It follow from KKT condition 35 that v 1 = 0 a contraint W p S p i non binding. Note that 14

15 v 2 = 0 alo hold a λ > 0 i required to generate poitive revenue. Given v 1 = v 2 = 0, the KKT condition 34 reult in the cubic equation given a: Gλ 2µλ 3 c + µa + 4µλ 2 + 2µc + aµ + µ 2 λ aµ 3. λ 3 i the root of cubic Gλ. Hence λ 3, v 1 = 0, v 2 = 0 atify all KKT condition. We note that W p λ 3 = λ 3µ + λ 3 µ λ 3 µµ λ 3 µ λ 3 < S p for S p J. Thi implie contraint W p S p i non binding for S p J. Thi point i global maximum of P2 for S p J a P2 i convex optimization problem. It follow from Equation 4 that W p λ, β = = W p λ i an increaing function of λ [0, µ. Uing thi fact, one can adapt the argument of Sinha et al., 2008, page 20 to how that for S p / J, the waiting time contraint for primary cla cutomer will be binding at optimality. On exploiting thi fact and uing KKT neceary condition, we complete the olution of problem P2 by Theorem 4. Theorem 4 tate that if primary cla cutomer ervice level i in range J a defined below, then, the olution of optimization problem P2 i given by β =, i.e., econdary cla cutomer hould be given trict priority. Optimal admiion rate for econdary cla cutomer i given by the root of a quadratic. Primary cla cutomer ervice level contraint i binding in thi etting. Theorem 4. Let S p lie in the interval, J defined a ] J µµ λ p, λ 3µ + λ 3 µ λ 3 for µ λ µµ λ 3 µ λ 3 p 2µλ 2 p + cµ + > aµλ 2 p µµ, otherwie where λ 3 = + λ 3 and λ 3 i the unique root of cubic Gλ in the interval 0, µ whenever µ 2µλ 2 p + cµ + > aµλ 2 p. Then, λ 4 i the global maximum of NLP P2 and contraint W p S p i binding, where λ 4 = µ λ 2 p + 4µ2 µs p Proof. We note that J J = φ; therefore, contraint W p S p i binding at optimum for S p J. Claim 2. Suppoe S p > equality W p λ = S p. Proof. See Appendix., then there exit a unique λ4 µµ 0, µ that atifie the It follow from above claim that W p λ 4 = S p. Hence the point λ = λ 4 atifie the KKT condition 35 irrepective of value of v 1. On olving KKT condition 34 for v 1 at λ = λ 4, v 2 = 0, we get v 1 = a 2λ 4 cλ4 2µ λ 4 µµ λ 4 2 µ λ 4 2 µ λ 4 2 bµ2µ 2λ

16 Note that v 1 0 hold iff a 2λ 4 4 Gλ v 1 0 iff µµ λ 4 0. Hence v iff 0, λ 3 ] a λ 3 cλ4 2µ λ 4 µµ λ On further implification, we get 4 Gλ 0. It follow that Gλ 0 in the interval i the unique root of cubic Gλ in the interval 0, µ and G0 = aµ 3 < 0. Thi λ 3, conider following two cae: implie that v 1 0 will hold true if λ 4 λ 3. To etablih λ 4 Cae 1 µ 2µλ 2 p + cµ + aµλ 2 p Note that λ 3 i the root of cubic Gλ. It i known that Gλ ha a unique root in 0, µ and G0 = aµ 3 < 0, Gµ = cµ 2 > 0. Gµ λp = µ 2µλ 2 p + cµ + aµλ 2 p 0. Thi implie λ 3 [µ, µ. It follow from Equation 40 that λ 4 < µ. So λ 4 < λ 3 hold in thi cae. Cae 2 µ 2µλ 2 p + cµ + > aµλ 2 p In thi cae interval J become µµ, λ 3µ + λ 3 µ λ 3. Note that λ 4 i the root of µµ λ 3 µ λ 3 quadratic obtained by equating W p λ = S p refer claim 2. Note that Ju = W p λ 3 where Ju i the upper limit of interval J. Thi follow from the expreion of Wp λ. Hence at S p = Ju, quadratic will reult in Qλ 3 = 0. So λ 3 = λ 4 at S p = Ju. We know that W p λ i an increaing convex function of λ. Thu λ 4 < λ 3 will hold for S p < λ 3µ + λ 3 µ λ 3 µµ λ 3 µ λ 3. Thi complete the argument that λ 4 λ 3 for S p J. Point λ = λ 4, v 1, v 2 = 0 atifie KKT condition. Thi KKT point will be global maxima for optimization problem P2 a P2 i convex optimization problem. Hence theorem follow. Corollary 2. λ 4 i an increaing function of S p, for S p J. Proof. Uing Equation 40, λ4 = S p µ 3 µs p Hence corollary follow. λ 2 p + 4µ2 µs p Comparion of optima of problem P1 and P2 Analyi of the cae β < etablihe that given a c > 2µ and S µµ 2 p µµ I problem P1 will have a local optimal olution while the cae β = ha the optimal olution for S p > Ŝp =. So there exit optimal olution for both optimization problem P1 and P2 µµ in ervice level range I defined in Theorem 1. In order to find the global optima, one need to compare optimal objective function of P1 and P2 in the interval I, given that a c > 2µ µµ. 2 Thee two optimal value of objective function are compared uing the interpretation of Lagrangian multiplier refer propoition in Berteka It turn out that the olution of optimization 16

17 problem P0 i given by P1 β < for interval I, i.e., optimal objective value of P1 i more than that of P2 in interval I. Detailed analyi of thi comparion i a follow. Let λ f, β f and λ i, are optimal olution of the optimization problem P1 and P2 repectively. Let the correponding value of objective function are O1λ f, β f and O2λ i, for S p I. Below, we etablih that O1λ f, β f > O2λ i,. Claim 3. Service level contraint of primary cla cutomer W p S p i binding in both the local olution given by optimization problem P1 and P2 for S p I. Proof. See Appendix. It follow from above claim that contraint W p S p i binding for S p I. We now ue the interpretation of Lagrangian duality to compare optimal objective O 1λ, β and O 2λ, β refer propoition in Berteka 1999: O 1 S p = u f 1 and O 2 S p = v i 1 41 where u f 1 and v i 1 are correponding Lagrangian multiplier aociated with the contraint W p λ, β = S p of optimization problem P1 and P2 repectively. Solution for problem P1 β < i given by Theorem 1 and that for problem P2 β = i given by Theorem 4 for ervice level range S p I. We have correponding Lagrangian multiplier a: λ i = λ 4 of v i 1, we have u f 1 = c b and v i 1 = a 2λ 4 cλ4 2µ λ 4 µµ λ 4 2 µ λ 4 2 µ λ 4 2 bµ2µ 2λ 4 2 a olution of P2 i given by Theorem 4 for S p I. On further implifying the expreion v i 1 = Gλ i µ λ i 2 bµ 2 2µ 2λ i 2 42 Note that v1 i 0 hold for S p I a Gλ i 0 for 0 λ λ 3 and λ 4 = λ i λ 3. From Equation 41, we have O 1 S p = u f 1 0 and O 2 S p = v i Thi implie O1 and O2 are increaing function of S p in interval I. From Equation 42, we have v1 i µ λ i [ ] = G λ i λ i bµ 2 2µ 2λ i.µ λ 3 p λ i 2µ 2λ i i 2 Gλ G λ i i a quadratic equation and one can how that G λ i 0 in interval 0, µ. 0 λ λ 3 and λ 4 = λ i λ 3 v i 1 λ i On uing Equation 41 and 44, we have Gλ 0 for. So Gλ i 0 hold. Thi implie 0 and uf 1 = cλ p = 0 44 λ f λ f b 17

18 2 O 1 S 2 p = S p O 1 S p = uf 1 = uf 1 λ f = 0 S p λ f S p It follow from Equation 43 and above expreion that O 1 i linearly increaing function of S p in interval I. 2 O 2 S 2 p = S p O 2 S p = vi 1 = vi 1 λ i 0 S p λ i S p Above tatement follow from Equation 44 and corollary of Theorem 4 which tate that λ 4 increaing function of S p. Hence O 2 i an increaing concave function of S p in interval I. Collecting all reult together, we have i an O 1 i linearly increaing function of S p in interval I. O 2 i an increaing concave function of S p in interval I. Slope of O 2 i decreaing in interval I. O 1 or O 2 O 1λ 1, 0 O 1λ 1, β 1 O 2λ 4, S p I I u Sp Figure 3: Optimal value of P1 and P2 in interval I It follow from Theorem 1 that denominator of optimal cheduling policy, β 1 i λ 1 µ+µ λ 1 λ 1 + µs p λ 1 µ 2 S p. Note that thi term will be 0 at S p = I u = λ 1µ + µ λ 1 λ 1 µµ λ 1 µ λ 1 = λ i i the olution of quadratic equation obtained by equating. Hence β a S p I u. Alo recall that λ 4 W p λ = W p λ, β 1 = = S p in interval 0, µ. Hence λ f λ i a S p I u. Thi implie O 1λ f, β f O 2λ i, a S p I u. Optimization problem P1 and P2 are ame at S p = I u. So v1 i u f 1 i.e O 1 O 2 a S p I u. O S p S 2λ i, < O1λ f, β f follow in interval I a lope of p O2λ i, i decreaing and that of O1λ f, β f remain contant ee Figure 3. We conolidate all reult in Theorem 5. Theorem 5 tate that olution of reource owner profit maximization problem depend on ratio a c. If 0 < a c 2µ, then, the olution of P0 i given µµ 2 18

19 by P2, i.e., with β = while for a c > 2µ µµ 2 olution of P0 i given by P1 or P2 depending on the value of ervice level. Theorem Suppoe 0 < a c 2µ µµ 2, then we can write Ŝp, a Ŝp, = J J with J being poibly empty. Then optimization problem P2 ha a olution but P1 i infeaible. For S p Ŝp,, the optimal olution to P0 i given by optimal olution to P2 with β = and λ = λ 3 if S p J & λ = λ 4 if S p J. 2. Suppoe a c > 2µ µµ 2 hold then For S p = Ŝp, optimal olution of P0 i given by P1 with λ = λ 1 olution. and β = 0 a optimal We can write Ŝp, a Ŝp, = I I + J, with J being poibly empty. Then optimization problem P1 and P2 have optimal olution. Optimal olution to P0 i given by P1 with λ = λ 1 and β = β 1 in interval I and for S p I + J optimal olution to P0 i given by P2 with β = and λ = λ 4 if S p I + & λ = λ 3 if S p J. Proof. Follow from Theorem 1, 2, 3, 4 and the fact that optimal objective for problem P2 i leer than optimal objective for problem P1 in ervice level range I. We now preent a finite tep algorithm in next ection to find the global optimal operating parameter uing the reult derived above. 3.2 Algorithm to find optimal operating parameter Baed on above analyi, a finite tep algorithm i decribed to compute the global optimal mean arrival rate of econdary cla cutomer, λ, and relative queue dicipline management parameter, β. Once λ and β are known, the optimal ervice level, S, and optimal admiion price, θ, for econdary cla cutomer can be obtained uing S = W λ, β and θ = a cs λ /b. Input:, µ, a, b, c and S p Step: 1. if S p < Ŝp := µµ or a 0, then there doe not exit any feaible olution. Aign c λ = 0 and top; ele, go to tep if a c 2µ then go to tep 3; ele, go to tep 7. µµ 2 3. if S p = Ŝp, there doe not exit any feaible olution, aign λ = 0 and top; ele, go to tep 4. 19

20 4. if µ a µ 2µλ 2 p + cµ + then J l = and go to tep 6; ele, define J l = λ 3µ + λ 3 µ λ 3 µµ λ 3 µ λ 3, J = J l, and find λ 3 which i the unique root of cubic Gλ in the interval 0, µ where Gλ 2µλ 3 c + µa + 4µλ 2 + 2µc + aµ + µ 2 λ aµ 3 5. if S p J then λ = λ 3, β =, go to tep 10; ele, go to tep define J = Ŝp, J l ] if J l i finite and J = Ŝp, if J l =, aign λ = λ 4 1 λ 2 2 p + 4µ2 µs p + 1, β =, go to tep 10. = µ 2 7. if S p = Ŝp then compute λ 1, unique root of cubic Gλ in the interval 0, µ with φ 0 = µ where Gλ 2µλ 3 [c + µa + 4φ 0 ]λ 2 + 2φ 0 [c + µa + φ 0 ]λ aµφ c µ + φ 0 and aign λ = λ 1, β = 0 go to tep 10; ele, go to tep if µ a µ 2µλ 2 p + cµ + then J l = ; ele, define J l = λ 3µ + λ 3 µ λ 3 µµ λ 3 µ λ 3 root of cubic Gλ. and find λ3, 9. find λ 1, the root of cubic Gλ, define I u = λ 1µ + λ 1 µ λ 1 µµ λ 1 µ λ 1. Alo define I = Ŝp, I u, I + = [I u, J l ] if J l i finite; otherwie take I + a I + = [I u,. Alo, take J = J l, if J l i finite; otherwie J = φ. a if S p I then λ = λ 1 and, µ λ 1 µs p µ β λ = 2 1 µ λ 1µS p λ 1 λ 1 µ λ µs p λ 1 µ + µ λ 1 λ 1 + µs p λ 1 µ 2 S p for for µµ < S p λ 1 µµ λ 1 λ 1 µµ λ 1 < S p < λ 1µ + µ λ 1 λ 1 µµ λ 1 µ λ 1 b if S p I + then λ = λ 4, β =, c if S p J then λ = λ 3, β = 10. The optimum aured ervice level to the econdary cla cutomer i S = W λ, β and optimal unit admiion price charged to econdary cla cutomer i θ = a cs λ /b. We now tudy the advantage of uing pre-emption over non pre-emptive priority cheduling on total revenue generated by the ytem. Intuitively, pre-emptive priority i likely to introduce more cutomer admiion rate than that with non pre-emptive priority cheduling. We note that thi i not the cae for certain range of ervice level and comment on uch phenomenon analytically. Section 4 and 5 compare revenue generated under two pre-emptive and non pre-emptive cheduling cheme. 20

21 4 Comparion of Scheduling Policie: Theoretical Development In thi ection and the next, we will compare two ytem for revenue generated, i.e., with pre-emptive and non pre-emptive priority cheduling dicipline. We do thi comparion by conidering different range of ervice level S p and other input parameter. Thi comparion i theoretically tractable with ome complementary condition for certain range of ervice level when tatic priority i optimal for both pre-emptive and non pre-emptive cheduling policie. Comparion of revenue become more difficult when at leat one of the cheduling policy give feaible olution with finite pure dynamic cheduling parameter. We perform computational experiment for uch cae in Section 5. We firt identify the certain range of ervice level and input parameter etting where pre-emptive priority generate revenue and non pre-emptive priority give infeaible olution. We further explore the input parameter pace to compare revenue where both cheduling policie are feaible and optimal cheduling parameter, β, for both policie pre-emptive and non pre-emptive i infinite. Certain complementary condition are identified to analytically tract the revenue comparion for uch cae. Our computational reult how that thee complementary condition adjut in uch a way that revenue in pre-emptive cheduling dicipline outperform non pre-emptive cheduling. It turn out that revenue with pre-emptive priority i higher for certain range of primary cla ervice level. We alo approach thi comparion via econdary cla cutomer ervice level and market equation. We note that ome condition are needed to tract revenue for certain range while comparing via ervice level. In thi paper, ervice time ditribution i aumed to be exponential under pre-emptive cheduling dicipline and variance σ 2 = 1/µ 2 for exponential ditribution. Hence ψ = 1 + σ2 µ 2 = 1 a in Sinha 2 et al We now conider different range of ervice level S p and input parameter pace for revenue comparion. 4.1 Range: S p = Ŝp and a c > 2µ µµ 2 Revenue maximization problem i infeaible under non pre-emptive priority cheduling for thi range ee Theorem 5 in Sinha et al That i, ervice level Ŝp can not be achieved even if one aign trict priority to primary cla of cutomer due to non pre-emptive priority nature. However, ervice level S p = Ŝp can be achieved for given range under pre-emptive priority cheduling ee Theorem 2 in above Section Optimal cheduling parameter, β = 0, and optimal admiion rate, λ = λ 1 matche with queueing intuition. Note that Ŝp = i the mean waiting time with primary µµ cla of cutomer only. The only poible way to achieve the ervice level Ŝp i to give trict preemptive priority to primary cla cutomer hence optimal cheduling parameter β = 0 i needed. It follow from Theorem 2 in Section that optimal admiion rate λ 1, root of Gλ, will lie in interval 0, µ if a c > 2µ. That i why thi condition i needed. Revenue maximization µµ 2 21,

22 problem i infeaible with non pre-emptive cheduling while thi problem ha a feaible olution with pre-emptive cheduling. Hence revenue generated will alway be more with pre-emptive cheduling for thi given range. 4.2 Range: S p > Ŝp and a c µ 2 Revenue maximization problem i infeaible under non pre-emptive priority cheduling for thi range ee Theorem 5 in Sinha et al However, problem i feaible for pre-emptive priority cheduling with optimal cheduling parameter β = and admiion rate λ = λ 3 or λ 4 depending on ervice level S p ee Theorem 3 and 4 in Section We ue notation NP and PR for quantitie aociated with non pre-emptive and pre-emptive cheduling dicipline repectively; for example λ NP and λ P R are econdary cla cutomer arrival rate under non pre-emptive and pre-emptive priority cheduling dicipline repectively. Queueing argument for the ame in thi cae can be given a follow. It can be argued uing the linear demand function that λ > 0 if and only if a c > S = W λ = ɛ, where ɛ i trictly poitive and ɛ 0 ee page 28 in Sinha et al In non pre-emptive priority cheduling, W λ = ɛ, NP µ ; therefore λ 2 NP > 0 iff a c >. It follow from µ 2 λ Equation 5 that W λ = ɛ, P R = µµ λ = ɛ in pre-emptive priority cheduling. µµ ɛ W λ = ɛ, P R 0 when ɛ > 0 and ɛ 0. Hence waiting time of econdary cla can be made arbitrarily mall in pre-emptive priority cae. Thi implie λ P R > 0 iff a c > 0. Revenue maximization problem i infeaible with non pre-emptive cheduling while thi problem ha a feaible olution with pre-emptive cheduling. Hence revenue generated will alway be more with pre-emptive cheduling for thi given range. 4.3 Range: S p > Ŝp and µ < a 2 c 2µ µµ 2 Revenue maximization problem i feaible under both pre-emptive and non pre-emptive priority cheduling for thi range ee Theorem 5 in Sinha et al and Theorem 3 and 4 in Section Note that optimal cheduling parameter β = under both priority cheme. We now calculate the total revenue generated under both cheduling policie to compute the difference in revenue. Total revenue generated i arrival rate, λ, multiplied by unit admiion price, θ. θλ i implified to revenue R := 1 b aλ λ 2 cλ W λ, β in the objective function of optimization problem P1. Revenue term can be further implified by the following waiting time expreion with β = a in thi cae. Mean waiting time expreion are given by W λ P R, β = P R = λ P R µµ λ P R and W λ NP, β = NP = + λ NP µµ λ NP 22

23 Revenue with non pre-emptive and pre-emptive priority i then given by R NP = 1 aλ NP λ NP 2 cλ 2 NP 1 c λ NP b µµ λ NP b µµ λ NP R P R = 1 aλ P R λ P R 2 cλ P R 2 b µµ λ P R λ NP λ P R The difference in revenue can be implified to D := R NP R P R = b [ ] c µλ P R + λ NP µ λ P R λ NP a λ NP λ P R + µµ λ NP µ λ P R λ NP λ P R Note that the ign of above expreion decide the optimal cheduling mechanim in term of revenue maximization. Sign of econd term involve λ NP and λ P R in denominator. λ NP and λ P R are complicated non-linear expreion. Hence the ign of econd term i intractable. We identify following two condition under which difference in revenue can be ordered to find revenue optimal cheduling mechanim. c µλ P R + λ NP µ λ P R λ NP Condition C: a < λ NP + λ P R + + µµ λ NP µ λ P R λ NP λ P R Condition C c µλ P R + λ NP µ λ P R λ NP : a λ NP + λ P R + + µµ λ NP µ λ P R λ NP λ P R We identify the ign of the firt term in Equation 47 for variou value of input parameter. However our obervation from computational example i that the product i alway negative and hence pre-emptive priority policy generate more revenue. We alo oberve that the optimal ervice level offered to econdary cla cutomer for variou input parameter i maller in pre-emptive priority cheduling. Thi can be een a the effect of pre-emptive cheduling with optimal cheduling parameter β =. We lit below the variou cae that exhautively cover all poible value of input parameter of thi model under the given etting. In the view of Theorem 3 and 4 in Section and in Sinha et al. 2010, ervice level range Ŝp, i written a J J in both pre-emptive and non pre-emptive priority cheduling. Denoting by J l the left end point of ervice level range J a per policy *, we have the following left end point J l NP = λ 3 NP µ λ 3 NP µ λ 3 NP and J l P R = µλ 3 3 P R + λ P R µ λ 3 P R µµ λ 3 P R µ λ 3 P R Set C l a µλ 3 NP + λ 3 NP µ λ 3 NP µµ λ 3 NP µ λ 3 NP. Clearly J l NP < C l hold. It i clear from the tatement of Theorem 3 and 4 that optimal nature of priority and arrival rate further depend on ratio µ µ. Hence we have following ub cae that we analye below. α µ µ a c 2µλ 2 p + cµ β a c 2µλ 2 p + cµ + < µ µ a 2µλ 2 p + cµ + 23