Pricing surplus server capacity for mean waiting time sensitive customers

Transcription

1 Pricing urplu erver capacity for mean waiting time enitive cutomer Sudhir K. Sinha, N. Rangaraj and N. Hemachandra Indutrial Engineering and Operation Reearch, Indian Intitute of Technology Bombay, Mumbai , India September 6, 2008 Abtract Reource including variou aet of upply chain, face random demand over time and can be hared by other. We conider an operational etting where a reource i hared by two different clae of cutomer, the primary cla exiting cutomer) and the econdary cla new firm) cutomer, under a ervice level baed pricing contract with the econdary cla cutomer which pecifie the unit price of ervice a well a the quality of ervice offered. We aume that the Poion arrival rate of econdary cla of cutomer depend linearly on the unit price of ervice a well a on the ervice level offered. In our model, we ue delay dependent priority cheme for queue management and tationary mean expected time a a quality of ervice meaure. Given an exiting ervice level baed contract between the ervice provider and the primary cla of cutomer, we analyze the impact of incluion of econdary cla of cutomer ha on the ytem utilization and ervice level of the exiting cutomer. We tudy the joint problem of optimally pricing and operation of the reource with the incluion of the econdary cla of cutomer, while continuing to offer a pre-pecified quality of ervice to primary cla of cutomer. Thi non-convex contrained optimization problem ha two pricing contract) deciion variable and two operational deciion variable. While the two deciion variable, the unit price of ervice and quality of ervice level offered contitute the pricing parameter, the allowable rate of econdary cla cutomer and a parameter capturing the relative delay dependent queue priority form the two operational deciion variable. We oberve that in our model, we can firt find the optimal operating parameter and then ue them to find the optimal contract pricing) parameter. Thi follow from eparability property of linear demand function. We earch exhautively for Karuh-Kuhn-Tucker point of the optimization problem to obtain the global optimum point. We propoe an algorithm that find thee optimal parameter in cloed form expreion for variou poible value of input parameter. We alo tudy in detail the tructure and the nonlinear nature of thee optimal deciion along with their enitivity to variou input parameter. 1

2 The tudy ha implication in etting where a new firm enter into buine requiring high infratructural et up cot and i willing to ue the infratructure of an exiting firm after entering into a pricing contract. An example i the entry of private firm into the inland rail container movement buine in India. Key word: Queueing, quality of ervice, dynamic priority cheme, linear demand function, non-convex optimization 1 Introduction In addition to cot minimization, guaranteeing aured level of ervice ha been a dominant concern while operating reource. In practice, it i poible that reource remain under-utilized becaue of the random nature of demand and uage. Owner of uch reource may want to hare the exiting reource with other, including new firm, requiring uch reource. In uch a cenario, the reource will be ued by two different clae or type of cutomer; the primary cla cutomer exiting cutomer) and econdary cla cutomer cutomer of other or new firm). Queueing ytem are natural model for reource allocation to cutomer who arrive over time. In thi paper, we propoe a priority queue baed model for the optimal ue of exce capacity of a reource when cutomer of both clae arrive over time and thee cutomer will be offered pre-pecified Quality of Service QoS) level guarantee. In particular, we conider the iue of optimal pricing of exce capacity of a erver for an independent Poion tream of econdary cla cutomer whoe arrival rate i enitive to both the offered mean waiting time QoS level) a well a to the price charged per cutomer while imultaneouly enuring that the mean waiting time QoS level) of the primary cla cutomer i le than a pre-pecified level. We aume that the reource owner ha a long term agreement with the primary cla cutomer which pecifie a QoS level to the primary cla cutomer. The agreement aume a Poion demand for the primary cla cutomer. The incluion of econdary cla cutomer into the ytem increae the traffic intenity at the reource, which in turn, affect not only the ytem utilization but alo the effective ervice level offered to the primary cla cutomer. Therefore, one need to control the arrival rate of the econdary cla cutomer into the ytem. We aume that the demand of the econdary cla cutomer depend not only on the price charged but alo on the aured ervice level. Such a pricing cheme will influence the econdary cla cutomer demand and hence it can be viewed a a mechanim to control the traffic intenity at the reource. The cutomer ervice level depend on the queueing dicipline at the reource. A queue dicipline ued will therefore affect the econdary cla cutomer arrival rate whoe demand i enitive to unit admiion price a well a to the aured ervice level. That, in turn, alo affect the effective ervice level offered to the primary cla cutomer. 2

3 Thu, the queue dicipline employed can be viewed a another mechanim to control the traffic intenity at the reource. The implet queueing dicipline i firt come firt erve FCFS). But, FCFS queueing dicipline doe not provide differentiated ervice level. A differentiated ervice level between cutomer of different clae can be achieved uing priority queue management dicipline. Such a priority cheme can be either tatic or dynamic. The tatic priority cheme may caue long delay to the job of low priority cutomer. A dynamic priority cheme eliminate the diadvantage of FCFS and tatic priority dicipline a job for ervice are now elected baed on their actual waiting time a well a their priority cla. In our model for reource haring we ue a delay dependent priority queue management cheme originally propoed by Kleinrock 1964). In thi paper, we retrict ourelve to the cae when QoS level of a cla are meaured in term of the tationary expected waiting time in queue of cutomer of that cla. An aured ervice level for a cla implie that the tationary expected waiting time of cutomer in queue of that cla will be le than or equal to the aured mean waiting time. We alo aume that the potential mean arrival rate of the econdary cla cutomer i a linear function of unit admiion price and aured ervice level. The reource owner aim to elect a pair of operating parameter, a queue dicipline management parameter characterizing the dynamic priority policy and an appropriate arrival rate of the econdary cla cutomer along with a uitable pair of pricing parameter, i.e., unit admiion price and aured mean waiting time for the econdary cla cutomer, that will maximize it expected revenue from the incluion of econdary cla cutomer while enuring the prevailing mean waiting time level to the primary cla cutomer. Such a contrained reource haring problem can be viewed a a deign of a QoS level baed contract that the reource owner want to enter with the econdary cutomer. Given that econdary cutomer Poion arrival rate i linear in unit admiion price and aured ervice mean waiting time in queue) level, the reource owner would like to quote optimal value for thee two quantitie that alo enure pre-aigned QoS mean waiting time in queue) level to the primary cutomer. The econdary cutomer market will offer an additional teady Poion demand for the reource owner while availing a certain QoS mean waiting time) that i pecified in the contract by the reource owner. The reource owner will employ a dynamic priority management cheme to meet thee QoS level of both the clae of cutomer. The tationary waiting time of a cla in the queueing model can be interpreted a the ample path baed cutomer average of waiting time of member of that cla in a regenerative ytem like our Wolff 1989). Thi ugget a practical way to implement uch a long term contract that the reource owner may want to enter into with the econdary cla cutomer. The aumption on the nature of demand function and the definition of cutomer ervice level help u to reformulate the reource owner contrained problem. The deciion variable of the reformulated problem are the operating parameter, the arrival 3

4 rate of the econdary cla cutomer and the queue dicipline management parameter, while the uitable pair of pricing contract) parameter, i.e., the unit admiion price and the aured mean waiting time for the econdary cla cutomer, are derived uing the value of the deciion variable at optimality of the reformulated problem. By chooing variou poible value of the Lagrange variable in the Karuh-Kuhn-Tucker firt order neceary condition of thi reformulated optimization problem, we exhautively earch for it Karuh-Kuhn-Tucker point. One of the deciion variable, the queue dicipline management parameter, can take the value of infinity correponding to head-of-the-line tatic high priority to the econdary cla cutomer) and hence contitute a valid deciion in our optimization problem. Therefore, we alo eparately analyze the contrained optimization problem with arrival rate of econdary cla cutomer a a ingle deciion variable, by etting the queue dicipline management parameter a infinity. We next compare the optimal olution of thee two optimization problem to obtain the global optimal olution of the original contrained reource haring problem. We identify it global optimal olution for all poible value of input parameter, except for one finite interval of S p, the prevailing QoS level of primary cla cutomer. Baed on our analyi and numerical experimentation, we conjecture an optimal olution for thi finite interval of S p alo. Thi lead to an algorithm that terminate in finite tep with cloed form expreion for optimal value of both the pair of operating parameter and pricing parameter. A conequence of the ue of the linear demand function in the context of haring a reource over time i that the joint problem of optimal pricing and operation of exce capacity can be eparated; one can find optimal operating variable firt and then ue them to find the optimal pricing parameter. Alo, it turn out that the optimal deciion variable remain inenitive to the price enitivity coefficient of the demand function. One of our finding i that there exit an interval for the ratio of co-efficient of linear demand function uch that it i beneficial for the reource owner to offer tatic high priority to the econdary cla cutomer. For value of the above ratio to the right of thi interval, it may be optimal to ue dynamic priority cheme. In uch a cae, a part of the feaible value of S p will have three interval and in thee interval exactly one of the two operating parameter, either the optimal arrival rate of the econdary cla of cutomer, λ, or the parameter correponding to optimal dynamic priority queue management, β, remain contant. But, the optimum contract pricing) parameter θ and S are different non-linear function in the different interval of S p. The enitivity analyi of optimal parameter with repect to demand coefficient how that the increae in the maximum attainable demand rate of the econdary cla cutomer mean that a delay dependent priority queue dicipline need to be ued a part of optimal policy over a wider range of S p value. The increae in ervice level enitivity coefficient of econdary cla cutomer lead to the ue of the delay dependent priority 4

5 queue dicipline for a maller range of S p value. Thi work i motivated by the recent opening up of the inland rail container movement in India to both the private and public ector player, which till recently wa olely managed by Container Corporation of India Ltd Concor), a public firm. The intereted companie have to arrange for a rail-linked inland container depot ICD). Due to the high infratructural et up cot involved, new firm may be eeking to leae ome reource like ICD from Concor preently the only one to poe a rail-linked ICD) in the initial year of operation. In a hared ICD, the exiting cutomer of Concor are the primary cla cutomer wherea cutomer of the new firm contitute the econdary cla cutomer Sinha et al. 2008). The framework we conider i optimal pricing of urplu capacity in a general commitment baed reource haring model and can be potentially relevant in many etting, e.g., a in-houe manufacturing unit of a firm utilizing it exce capacity to cater an outide firm demand, a third party logitic ervice provider erving multiple cutomer. Other type of ituation can be communication network providing ervice to different clae of cutomer. A contemporary example could be the determination of charge a mobile telephony ervice provider can ue while providing roaming ervice to cutomer of another ervice provider. Similar tudie of Palaka et al. 1998), Pekgun et al. 2008), Ray and Jewke 2004) and So and Song 1998) determine an optimum pair of price and quoted lead-time for cutomer enitive to price a well a quoted lead time. The quoted lead time i identical to aured ervice level. Thee tudie aume a ingle cla of cutomer and employ FCFS queueing dicipline. Palaka et al. 1998) and Pekgun et al. 2008) model cutomer demand a a linear function of price and quoted lead time. The linear demand nature will mean that econdary cutomer view price and ervice level a ubtitute Palaka et al. 1998). Such demand function alo exhibit nice elaticity propertie that we ummarize later. Palaka et al. 1998) model the ytem a M/M/1 queue and conider that the firm incur congetion cot a well a pay latene penaltie. They find the optimal deciion of the reulting profit maximization problem and tudy the impact of varying parameter value on thoe optimal deciion. They alo examine a ituation in which it i poible for the firm to expand capacity marginally. Pekgun et al. 2008) conider a firm where pricing and lead time deciion are made by two independent function, marketing and production, repectively. They model the firm operation a a M/M/1 queue and the equence of deciion a a Stackelberg game. They how the inefficiencie reulting from decentralized deciion making and preent a coordination cheme to overcome thoe inefficiencie. Their focu i on the deirability of coordination iue in thi etting. Ray and Jewke 2004) extend the linear cutomer demand model by auming that price itelf i function of lead time. They aume that the firm can reduce the lead time by invetment in capacity. They firt determine the profit maximizing optimal policy 5

6 and thereafter invetigate the behaviour of the optimal policy under variou change of the ytem parameter. They pecifically preent the condition under which overlooking price and lead time dependence will lead to a ub-optimal deciion. They alo extend the model by incorporating economie of cale to unit operating cot. In contrat, So and Song 1998) conider log-linear Cobb-Dougla demand function to reflect cutomer enitivity to price and lead time and model a ervice facility a a G/G/1 queue. They determine the optimal price, lead delivery) time quote and hort-term capacity expanion level which maximize the average net profit while maintaining a predetermined level of delivery reliability. We retrict ourelve to a linear demand model. We advocate the ue of dynamic queue management cheme a part of ervice level baed pricing in multi-cla queueing model. Hall et al. 2002) tudy a imilar etting where a reource i hared by two different clae of cutomer. They aume FCFS queue dicipline at reource and alo aume that the demand i enitive to jut unit price. They focu on dynamic pricing policie which depend on the production ytem queue) tatu. They demontrate the propertie of the optimal policie and how that a policy of uniform pricing up to cutoff tate i uperior according to a certain performance/complexity ratio meaure. We, in contrat, focu on tatic pricing cheme with dynamic queue dicipline management. We preent the detail of the operational etting and the optimal contrained reource haring problem in Section 2. Analyi of thi optimization problem are given in Section 3. Baed on it, we preent in Section 4 the algorithm to elect the optimal contract parameter for econdary cla cutomer and optimal parameter to operate the reource and alo illutrate the algorithm by everal numerical example that correpond to variou poible cae of the input data parameter of the model. Further, we preent detail of our undertanding of the model and enitivity analyi of it optimal olution in Section 5. A preliminary verion of the algorithm along with a numerical example are preented in Sinha et al. 2008). In the preent paper, we preent detailed argument that lead to the algorithm along with an extenive enitivity analyi. 2 A queueing model for a hared reource Let λ p and λ be independent Poion arrival rate of the cutomer of the primary and econdary clae repectively. A the ervice requirement of the primary and econdary cla cutomer are identical in nature, we aume that the ervice time, i.e., time taken by the reource to complete a job irrepective of cutomer cla, are independent and identically ditributed random variable with mean 1/µ and variance σ 2. Further, the queue dicipline employed at the reource i head-on-line non-preemptive) delay dependent priority cheme. A chematic view of the ytem i hown in Figure 1. 6

7 Primary Cutomer λ p, S p, b p Secondary Cutomer λ, S, b Server µ Service Rule Head on Line Delay Dependent Priority Figure 1: Schematic view of the hared reource The delay dependent priority cheme i an example of dynamic priority cheme. A queueing ytem with delay dependent priority cheme wa firt tudied by Kleinrock 1964). It conit of P priority clae aociated with a et of variable parameter {b p } P 1, where 0 b 1 b 2... b P. The intantaneou priority at time t of a cla p job that arrived at time T p i given by q p t) = t T p )b p. After a ervice completion, the erver chooe the next job with highet intantaneou priority q p ) from all available job. If there i a tie for the highet intantaneou priority, then it i broken by uing FCFS rule. Here, a higher priority job gain priority at fater rate than lower priority job. The teady tate expected waiting time in queue for a cla p job in M/G/1 head-on-line delay dependent queue i given by the following recurion Kleinrock 1964, Kanet 1982): where ρ i = λ i µ i, ρ = p 1 W 0 1 ρ ρ i W i 1 b ) i b i=1 p W p = P 1 ρ i 1 b ), p {1, 2,..., P } 1) p b i i=p+1 P ρ i, W 0 = p=1 P p=1 λ p 2 σp ) µ 2 p and 0 ρ < 1. We note that the queue parameter {b p } P 1 only appear a ratio b p /b p+1 in the expreion for W p. Alo, the conervation law for M/G/1 ytem with non-preemptive work-conerving queueing dicipline a ytem in which work i neither created nor detroyed within the ytem) Kleinrock 1976) tate that P ρw 0 ρ < 1 ρ p W p = 1 ρ ρ 1 p=1 Let b p and b be the aociated parameter of the primary and econdary cla cutomer repectively in our ytem. Alternatively, we define relative priority queue dicipline management parameter β a a ratio of the aociated queue parameter to the primary and the econdary cla cutomer, i.e., β := b /b p. The election of the relative priority control parameter β define different regime of the delay dependent priority 7

8 queue. β < 1 b < b p ). Thi implie that the primary cla cutomer get higher priority than the econdary cla cutomer. When β approache zero, the queuing ytem become equivalent to a tatic priority queue with high priority to primary cla cutomer. β = 1 b = b p ). Thi implie that both clae of cutomer get equal prioritie. Thu, the queueing dicipline i FCFS. β > 1 b > b p ). Thi implie that the econdary cla cutomer get higher priority than the primary cla cutomer. When β approache infinity the queuing ytem become equivalent to a tatic priority queue with high priority to econdary cla cutomer. Figure 2 ummarize the relationhip between the control parameter β and prioritie of the primary and econdary cla cutomer. Delay dependent, high priority to primary cutomer b p >b Delay dependent, high priority to econdary cutomer b p <b β = 0 β = 1 β = Static high priority to primary cutomer FCFS β Static high priority to econdary cutomer Figure 2: Effect of queue dicipline management parameter on queue dicipline Recall that we aume that the prevailing agreement between the reource owner and the primary cla cutomer reult in a Poion arrival rate λ p of the primary cla cutomer. In the abence of the econdary cla cutomer, it i aumed that the reource owner i able to fulfill the aured mean waiting time requirement of the primary cla cutomer, given by S p, and till the facility i under-utilized. Thi aumption hold if and only if S p Ŝp, where Ŝp i the expected waiting time in queue for the primary cla cutomer when the reource i dedicated to the primary cla cutomer. In the M/G/1 etting, Ŝp = λ pψ where ψ = 1 + µµ λ σ2 p) µ 2 /2. Let Λ θ, S ) be the correponding potential Poion arrival rate of the econdary cla cutomer with a charged unit admiion price of θ per completed job and aured ervice level of S. We aume that thi rate i a linear function of the unit admiion price and aured ervice level, i.e., Λ θ, S ) = a bθ cs 2) 8

9 for ome given contant a, b, c > 0. The coefficient a, b and c repreent the maximum attainable mean arrival rate market potential), price enitivity and ervice level enitivity repectively. We have aumed linear demand function becaue it i convenient and it exhibit the following deirable propertie Palaka et al. 1998). 1. The demand function i eparable in price and aured ervice level and thu make price and ervice level a ubtitute. 2. The price elaticity of demand, bθ/a bθ cs ), increae with increae in θ and S. Thi reult in higher price elaticity of demand at higher unit price θ and higher ervice level S. Thu, cutomer are more enitive to high price when they wait for longer period of time in the queue. A imilar property i valid regarding ervice level elaticity. A the arrival rate of the primary cla cutomer remain fixed at λ p, the expected waiting time of the cutomer depend only on the mean arrival rate of the econdary cla cutomer λ and relative queue dicipline management parameter β. Let W p λ, β) and W λ, β) be the expected waiting time in the queue for the primary and the econdary cla cutomer repectively. The reource owner aim to elect an appropriate arrival rate of the econdary cla cutomer λ, a uitable pair of pricing parameter θ and S for the econdary cla cutomer and a queue dicipline management parameter β that will maximize it expected revenue from the incluion of econdary cla cutomer while enuring the prevailing mean waiting time to the primary cla cutomer. The reulting contrained reource haring problem of the reource owner i a follow P0: max λ,θ,s,β θλ 3) Subject to: W p λ, β) S p 4) S W λ, β) 5) λ µ λ p 6) λ a bθ cs 7) λ, θ, S, β 0. 8) Here, contraint 4) enure that the reource owner doe not violate the prevailing ervice level commitment to the primary cla cutomer while haring the reource. Contraint 5) retrict the reource owner to offer a ervice level commitment to econdary cla cutomer within ytem capability. Contraint 6) et a retriction on maximum permiible mean arrival rate of econdary cla cutomer baed on proceing capability of the ytem, i.e., it avoid intability of the multi-cla queue. Later we how that thi contraint indeed remain non-binding at the optimum and enure tability of the 9

10 multi-cla queue. Contraint 7) enure that the mean arrival rate of econdary cla cutomer hould not exceed the demand generated by charged price θ and offered ervice level S. The lat contraint capture the non-negativity of the mean arrival rate of econdary cla cutomer λ, price θ, aured ervice level S and queue dicipline management parameter β. Contraint 7) will hold a an equality in an optimal olution given that the demand i a eparable function of both price and aured ervice level Palaka et al. 1998). Next, we claim that contraint 5) will alo hold a an equality in an optimal olution. For, uppoe that the optimal olution of the optimization problem i given by mean arrival rate of econdary cla cutomer λ, price θ, aured ervice level S and queue dicipline management parameter β that atify S > W λ, β ). We already know that contraint 7) will hold a an equality in an optimal olution. Therefore, the objective function can be rewritten a 1 b aλ λ 2 cλ S. Since the objective function i a decreaing function of S, an aured ervice level S uch that S > S W λ, β ) will increae the earned revenue of the reource owner. Therefore, the contraint 5) mut hold a an equality in the optimal olution. In view of the fact that the above contraint are binding at optimality, the reource haring problem of the facility owner, P0) can be rewritten a P1: max λ,β 1 aλ λ 2 cλ W λ, β) 9) b Subject to: W p λ, β) S p 10) λ µ λ p 11) λ, β 0. 12) Once the optimal mean arrival rate of econdary cla cutomer λ and queue dicipline management parameter β i known, the optimal price θ and aured ervice level S i obtained uing equalitie λ = a bθ cs and S = W λ, β ). The objective function 9) indicate that the optimal choice of λ and β remain inenitive to price enitivity co-efficient b of the econdary cla cutomer. Further, we claim that the contraint 11) hould remain non-binding at the optimal point, i.e., the contraint 11) will hold with trict inequality at optimality, λ < µ λ p. To arrive at a contradiction, let u aume that the contraint 11) i binding at the optimal point, i.e., λ = µ λ p. We note that a S p i finite, W p λ, β ) will be finite at optimality. Suppoe S > a bθ ; then, the demand function 2) reult in λ c < 0. Thu, λ 0 if and only if S Ŝθ) where Ŝθ) = a bθ. A Ŝθ) i a finite number, the optimum S c will remain finite. Alo, at optimality S = W λ, β ). Therefore, W λ, β ) hould take a finite value at optimality. When λ p + λ = µ, it i not poible to have finite W p λ, β) and W λ, β) imultaneouly for any feaible β. Thi contradict the initial aumption 10

11 that the contraint 11) i binding at optimality, i.e., λ = µ λ p. Hence, the contraint 11) will remain non-binding at the optimal olution. 3 Optimal pricing and operation of the reource haring model Uing recurion 1), the expected waiting time in queue for the primary and the econdary cla cutomer are given by W p λ, β) = W λ, β) = λψ µ λ 1 β µ µ λ µ λ p 1 β 1 {β 1} + λψ µ λ µ λ p 1 β 1 {β 1} + λψ 1 {β>1} 13) µ λ µ λ 1 1 β λψ µ λ 1 1 β 1 {β>1} 14) µ µ λ µ λ 1 1 β where λ = λ p + λ, ψ = 1 + σ 2 µ 2 /2 and 1 {.} denote the indicator function which i equal to 1 if the tatement between brace i true and 0 otherwie. We note that the objective function and the contraint 10) of the reource haring problem P1 are defined differently in the region correponding to β 1 and β > 1. Thi apect ditinguihe the optimization problem P1 from a claical optimization problem. Firt, we note ome ueful propertie of W p λ, β) and W λ, β) whoe detail are given in Appendix. Next, we how that the above optimization problem i a non-convex problem. 1. W p λ, β) and W λ, β) are increaing convex function of λ in the interval 0, µ λ p ). 2. W p λ, β) i an increaing concave function of β 0 wherea W λ, β) i a decreaing convex function of β W p λ, β) i neither a convex nor a concave function of λ, β) where λ 0, µ λ p ) and β 0. example i given below. Alo, W p λ, β) i not quai-convex function of λ, β); a numerical 4. λ W λ, β) i neither a convex nor a concave function of λ, β) where λ 0, µ λ p ) and β 0. We demontrate below by a numerical example that the W p λ, β) i alo not a quaiconvex function of λ, β) where λ 0, µ λ p ) and β 0. Let u aume that λ p = 8, µ = 10 and σ = 0.1. We note that W p 1.5, 0) = 0.475, W p 0.5, 1) = 0.567, W p 1, 0.5) = and W p , 0) , 1)) = W p 1, 0.5) > max {W p 1.5, 0), W p 0.5, 1)}. 11

12 The above inequality violate the neceary condition for a function to be a quai-convex function Bazaraa et al. 1993); hence, W p λ, β) i not a quai-convex function over λ 0, µ λ p ) and β 0. Thi mean that, becaue contraint 10), the feaible region will be non-convex. Alo, the Heian matrix of λ W λ, β) at λ = 0.1, β = 0.5) i ) The eigenvalue of thi Heian matrix are and Thi implie that λ W λ, β),i.e., third term of the objective function, i neither a convex nor a concave function of λ, β). Hence, optimization problem P1 i a non-convex contrained optimization problem. The Lagrangian function correponding to the non-linear programming NLP) problem P1 can be expreed a L 1 λ, β, u 1, u 2, u 3 ) = 1 aλ λ 2 cλ W λ, β) +u 1 W p λ, β) S p +u 2 λ +u 3 β 15) b where u 1, u 2 and u 3 are the Lagrangian multiplier. The optimum value of the vector λ, β, u 1, u 2, u 3 ) hould atify the Karuh-Kuhn-Tucker firt order neceary condition. Thee are given a follow Bazaraa et al. 1993): W W p a 2λ c W + λ + bu 1 λ cλ W β + bu 1 + bu 2 = 0 16) λ W p β + bu 3 = 0 17) u 1 W p S p = 0 18) u 2 λ = 0 19) u 3 β = 0 20) W p S p and λ < µ λ p 21) u 1 0; λ, β, u 2, u ) A Karuh-Kuhn-Tucker KKT) point i defined by a pecific vector λ, β, u 1, u 2, u 3 ) that atifie the condition 16)-22). If the KKT point alo atifie the econd order ufficient condition then it can be either a local or a global optimum point of the NLP P1. We note that if the Lagrangian multiplier u 2 i uch that u 2 > 0, then the KKT condition 19) i atified if and only if λ = 0, in which cae objective function value i zero. A the objective of the reource owner i to earn a trict poitive revenue, we ignore value of u 2 > 0 in further analyi and aume throughout that u 2 = 0. The analyi below exhautively earche for all poible KKT point of the optimization problem P1 where u 2 = 0, i.e., aign pecific value to the remaining four unknown element of a poible KKT point. 12

13 Firt, we invetigate the KKT condition 17), 20) and 22) in detail. Let u aume that β > 0 at optimum. If β > 0, then the KKT condition 20) i atified iff u 3 = 0. Given u 3 = 0, the KKT condition 17), uing work conervation relation, reult in u 1 = cλ W β b W p β = cλ p b. 23) Next, let u conider that β = 0 at optimum. The implification of the KKT condition 17) at β = 0 reult in u 3 = λ λψcλ p + bu 1 bµ λµ λ p 2. 24) We note that u 3 0 iff u 1 cλ p given that 0 < λ b < µ λ p and λ p 0. In particular, u 3 = 0 at u 1 = cλp. Thu, the KKT condition 17), 20) and 22) are atified if and b only if one of the following hold true: C1: u 1 = cλ p b, u 3 = 0 and β 0. C2: u 1 < cλ p, u b 3 = λ λψcλ p + bu 1 and β = 0. bµ λµ λ p 2 The above invetigation explicitly aign pecific value to two unknown element of a poible KKT point. The remaining two unknown element of a poible KKT point i obtained by olving the Equation 16) and 18) note that u 2 = 0 at optimum). Next, we invetigate the KKT condition 18) conidering the fact that the contraint 10) can be either binding or non-binding at optimum. If the contraint 10) i binding at the optimum, then the KKT condition 18) get automatically atified irrepective of the value of u 1. On the other hand, if the contraint 10) i non-binding at optimum, then the KKT condition 18) i atified if and only if u 1 = 0. But, we note from condition C1-C2 that u 1 hould be le than or equal to cλ p cλ p b b to atify 17), 20) and 22). A 0, the KKT condition 17), 18), 20) and 22) are not atified imultaneouly at u 1 = 0. Thi implie that it i not poible to have a KKT point with the contraint 10) non-binding. Therefore, the contraint 10) will alway be binding at optimum. Further, we note that the claical optimization theory and thereby the KKT firt order neceary condition inherently aume finite value of the deciion variable. The above analyi which reult in a particular value of Lagrangian multiplier alo conider that both λ and β are finite. But, β = i a valid deciion in our optimization problem. Therefore, we eparately analyze the reulting one-dimenional optimization problem by etting β = in the NLP P1) in Section 3.2. In Section 3.3, we aim to identify global optimal point uing both thee olution. 13

14 3.1 Relative queue dicipline management parameter β < The above analyi etablihe that a KKT point hould atify either condition C1 or condition C2. Alo, the contraint 10) i binding at the optimum. In the analyi below, we conider C1 and C2 individually and olve the equality relationhip W p λ, β) = S p and Equation 16) for unknown element of the KKT point. The analyi auming that KKT point atifie condition C1 reult in Theorem 1 below. Theorem 1. Suppoe a c > λ p 2µ λ p ) 2 ψ. Then, there exit λ1) which i the unique µ µ λ p ) root of the cubic Gλ ) in the interval 0, µ λ p ): Gλ ) = 2µλ 3 cψ + µa + 4φ 0 )λ 2 + 2φ 0 cψ + µa + φ 0 )λ aµφ cψλ p µ + φ 0 ) 25) where φ 0 = µ λ p. Denote λ 1 = λ ) p + λ 1) and further aume that S p lie in the interval ψλ 1 I µµ λ p, ψλ 1 and β 1) i given by µ λ 1) µ λ 1 β 1) = Then, λ 1) binding at thi point. µ λ 1 µs p µ λ p ψλ 1 ψλ 2 1 µs p λ p µ λ 1 S p λ 1) µ λ 1 ψλ 1 S p µ λ 1) µ λ 1 for for ψλ 1 µµ λ p S p ψλ 1 µµ λ 1, ψλ 1 µµ λ 1 < S p < ψλ 1 µ λ 1) µ λ 1. 26) and β 1) i a trict local maximum of the NLP P1 and the contraint 10) i Proof. Firt, we how that λ 1) 0, µ λ p ). i the unique root of the cubic Gλ ) in the interval Claim 1. If a c > λ p 2µ λ p ) µ µ λ p ) 2 ψ, then, the cubic Gλ ) ha an unique root in the interval 0, µ λ p ). Proof. See Appendix. The work conervation law applied to our etting reult in λ W + λ p W p = λ2 ψ µ µ λ. 27) Let u aume that the Lagrangian multiplier are u 1 = cλp b, u 2 = 0 and u 3 = 0. Note that the contraint 10) i binding at the optimum; therefore, thee value of the Lagrangian multiplier atify KKT condition 17)-20). When the Lagrangian multiplier 14

15 are u 1 = cλp b, u 2 = 0 and u 3 = 0, then the KKT condition 16) can be rewritten a a 2λ c λ λ W + λ p W p = 0. 28) Uing equation 27) in equation 28) reult in a cubic equation given a Gλ ) 2µλ 3 cψ + µa + 4φ 0 )λ 2 + 2φ 0 cψ + µa + φ 0 )λ aµφ cψλ p µ + φ 0 ) = 0 where φ 0 = µ λ p. A λ 1) i the unique root of the cubic Gλ ) in the interval 0, µ λ p ), olving Gλ ) = 0 for λ 0, µ λ p ) reult in λ = λ 1). Claim 2. There exit a queue dicipline management parameter β 0 which atifie the equality W p λ, β) = S ) p if λ p 0, λ 0, λ p + λ < µ and S p lie in the interval ψλ µµ λ p, ψλ, where λ = λ p + λ. The value of µ λ µ λ β i β = µ λµs p µ λ p ψλ ψλ 2 µs p λ p µ λ S p λ µ λ ψλ S p µ λ µ λ for for ψλ µµ λ p S p ψλ µµ λ ψλ µµ λ < S p < ψλ µ λ µ λ 29) Proof. See Appendix. Let β 1) = β for λ = λ 1) and S p I. From Claim 2, we know that W p λ 1), β 1) ) = S p. The point given by λ 1), β 1), u 1 = cλ p, u b 2 = 0 and u 3 = 0 atifie KKT condition 16)-22) and i a KKT point of the problem P1. The retricted Lagrangian function L 1 λ, β), page no. 168, ) Bazaraa et al. 1993)), at thi KKT point i given by L 1 λ, β; u 1 = cλ p, u b 2 = 0, u 3 = 0. Uing Equation 27) and 15), we get L 1 λ, β) = 1 aλ λ 2 λ 2 ψ c b µ µ λ + cλ ps p. 30) We note that the retricted Lagrangian function i independent of β. The Heian of the retricted Lagrangian function H L1 λ, β) i given by 2 b ) 1 + cµψ µ λ) The above matrix i negative emi-definite a µ λ > 0. It i evident that at S p = ψλ 1 µµ λ p, the contraint g 1 λ, β) W p λ, β) S p and g 2 λ, β) β 0 are binding for thi KKT 15.

16 point. Alo, the contraint g 1 λ, β) i trongly active i.e., the aociated Lagrangian multiplier i non-zero) wherea the contraint g 2 λ, β) i weakly active i.e., the aociated Lagrangian multiplier i zero). The gradient of thee binding contraint are given by g 1 λ, β) = κ 1 κ 2 and g 2 λ 2), 0 ) = where κ 1 = W p λ and κ 2 = W p. We know that W p β λ, W p > 0 for λ β 0, µ λ p ) and β 0. A non-zero vector d d 1, d 2 ) that atifie d. g 1 λ 1), β 1) ) = 0 and d. g 2 λ 1), β 1) ) 0 imultaneouly i given by d 1 = κ 2d 2 κ uch that d 2 > 0. We note that dh L1 λ 1), β 1)) d T = cµψ ) κ ) 2 2d 2 < 0 for all d b µ λ 1 ) 3 2 > 0. κ 1 Next, if S p > ψλ 1, then only contraint g µµ λ p 1λ, β) W p λ, β) S p i binding at that KKT point and alo it i trongly active. A non-zero vector d d 1, d 2 ) that atifie d. g 1 λ 1), β 1) ) = 0 i given by d 1 = κ 2d 2 κ 1 uch that d 2 0. Again, we note that dh L1 λ 1), β 1)) d T = cµψ ) κ ) 2 2d 2 < 0 for all d b µ λ 1 ) κ 1 Hence, the KKT point λ 1), β 1), u 1 = cλ p, u b 2 = 0 and u 3 = 0 i trict local maximum of the NLP P1 if S p lie in the interval I. Corollary 1. The mean arrival rate of the econdary cutomer λ 1) point, i independent of S p in the interval I. which i a local optima Proof. The optimal λ 1) i the root of cubic Gλ ) which i independent of S p. Therefore, λ 1) i independent of S p in the interval I. We now find KKT point which atify condition C2. Thi reult in a trict local maximum of the NLP P1 for S p lying left to the interval I. Theorem 2 below tate thi reult. Theorem 2. Suppoe a c > λ p 2µ λ p ) ) µ µ λ p ) 2 ψ and S p lie in the interval I ψλp, ψλ 1 µµ λ p µµ λ p where λ 1 = λ p + λ 1) and λ 1) i the unique root of the cubic Gλ ) of Equation 25) in the interval 0, µ λ p ). Then, λ 2) = µµ λ ps p λ p and β 2) = 0 i a trict local maximum ψ of the NLP P1 and the contraint 10) i binding at thi point. Proof. Let u firt aume that the queue dicipline management parameter β = 0 and the Lagrangian multiplier u 2 = 0 at the optimum. Note that the contraint 10) i binding 16

17 at the optimum. Given β = 0, the equality relationhip W p λ, β) = S p reult in λ 2) λ = µµ λ ps p ψ λ p. 31) We note that λ 2) i an increaing function of S p a λ2) and ψ > 0. Alo, λ 2) = λ 1) at S p = ψλ 1 µµ λ p u 2 = 0 and β = 0, the KKT condition 16) reult in = µµ λp. Therefore, 0 < λ2) < λ 1) u 2) 1 u 1 = a 2λ 2) cψ µλ 2 + λ 2) ) λ 2 2 µ λ p )µ λ 2 ) 2 where λ 2 = λ p + λ 2). Alo, we note that if 0 < λ < λ 1), then cλ p b a 2λ cψ µ λ + λ ) λ 2 µ λ p ) µ λ) 2 µ µ λ p ) bψ > 0 for 0 < λ ψ p < µ for S p I. A µµ λ p ) bψ = µ λ p)gλ ) bψ µ λ) 2 < 0. Thi inequality follow from the proof of Claim 1 which etablihe that Gλ ) < 0 when 0 < λ < λ 1) and a > λp2µ λp) ψ. Thi implie that u 2) c µµ λ p ) 2 1 < cλp a 0 < λ 2) b < λ 1). Take u 2) 3 = u 3, obtained uing λ = λ 2) and u 1 = u 2) 1 in Equation 24). We note that u 2) 3 > 0 a u 2) 1 < cλ p. The point λ2) b, β = 0, u 2) 1, u 2 = 0 and u 2) 3 atifie the KKT condition 16)-22). Thu, it i a KKT point. Given S p I, we note that the contraint g 1 λ, β) W p λ, β) S p and g 2 λ, β) β 0 are binding for thi KKT point. Alo, thee contraint are trongly active. The gradient of thee binding contraint at thi KKT point are g 1 λ 2), 0 ) = ψ µ λ p λ 2) λ 2 ψ µ λ p)µ λ 2 ) 2 and g 2 λ 2), 0 ) = where λ 2 = λ p + λ 2). We oberve the both term of g 1 λ 2), 0) are trictly non zero a λ p, λ 2) > 0 and λ p + λ 2) < µ. Hence, the gradient of thee binding contraint, g 1 λ 2), 0) and g 2 λ 2), 0), at the KKT point are linearly independent. Therefore, thi KKT point i a trict local maximum Corollary of Theorem 4.4.2, Bazaraa et al. 1993)). Corollary 2. λ 2) i a linearly increaing function of S p in the interval I ) Proof. We have λ2) = µµ λ p > 0 a µ > λ ψ p > 0 and ψ > 0. Alo, 2 λ 2) S 2 p = 0. 17

18 3.2 Relative queue dicipline management parameter β = In thi ection, we analyze the reulting one-dimenional optimization problem by etting β = in P1. Let W λ ) = W λ, β = ) and W p λ ) = W p λ, β = ). The reulting optimization problem, P2, i given a 1 P2: max aλ λ 2 cλ W λ ) λ b 33) Subject to: Wp λ ) S p 34) λ µ λ p 35) λ 0 36) Let u define f 1 λ ) = aλ λ 2 and f 2 λ ) = λ W λ ). f 1 λ ) i concave function of λ a 2 f 1 = 2 < 0. We oberve that λ λ 2 0 and W λ ) 0 for λ 0, µ λ p ). We note that λ i linear and W λ ) convex increaing function of λ in the interval 0, µ λ p ). We know that product of two poitively valued, increaing convex function of the real variable defined on the ame interval i an increaing convex function. Thi implie that f 2 λ ) i an increaing convex function of λ in the interval 0, µ λ p ). A objective function i 1 b f 1λ ) cf 2 λ ), it i a concave function of λ in the 0, µ λ p ). Alo, we know that W p λ ) i a convex function of λ in the interval 0, µ λ p ). So, if the objective function and contraint of the optimization problem P2 atify the KKT ufficiency condition, the KKT point, if it exit, will be a global optimum. Further, we note that the firt term of the objective function i increaing wherea the lat two term are decreaing function of λ in the interval 0, µ). Therefore, the objective function i a unimodal function of λ in the interval 0, µ). Following the earlier argument, we oberve that the contraint 35) will remain nonbinding at the optimum. The Lagrangian function correponding to the NLP P2 can be expreed a L 2 λ, v 1, v 2 ) = 1 b aλ λ 2 cλ W λ ) + v 1 Wp λ ) S p + v 2 λ 37) where v 1 and v 2 are the Lagrangian multiplier. The KKT firt order neceary condition for the NLP P2 are given a follow: a 2λ c W + λ W λ + bv 1 W p λ + bv 2 = 0 38) v 1 Wp S p = 0 39) v 2 λ = 0 40) W p S p and λ < µ λ p 41) v 1 0; λ, v ) 18

19 A KKT point i defined by a pecific λ, v 1, v 2 ) that atifie the condition 38)-42). Again, note that if the Lagrangian multiplier v 2 > 0 then the KKT condition 40) i atified if and only if λ = 0, in which cae objective function value i zero. A the objective of the reource owner i to earn a trict poitive revenue, we ignore value of v 2 > 0 in further analyi and aume throughout that v 2 = 0. The analyi below look for all poible KKT point of the revenue maximization problem P2 with v 2 = 0, i.e., aign pecific value to the remaining two unknown element of a poible KKT point. We alo know that the contraint 34) will be either trictly binding or non-binding at the optimum. Theorem 3 identifie an interval of S p where the contraint 34) i trictly non-binding at optimality. Theorem 3. Suppoe a c > λ p µ 2 ψ and µ λ p > aλ p cψ. Then, there exit 2µλ 2 p + cψµ + λ p ) λ 3) which i the unique root of the cubic Gλ ) in the interval 0, µ λ p ): Gλ ) = 2µλ 3 aµ + cψ + 4µ 2 λ 2 + 2µ aµ + cψ + µ 2 λ µ aµ 2 cψλ p Denote λ 3 = λ p +λ 3) Then, λ 3) at thi point. ) ψλ 3 and further aume that S p lie in the interval J µ λ 3) µ λ 3,. i the global maximum of the NLP P2 and the contraint 34) i non-binding 43) Proof. Firt, we how that λ 3) 0, µ). i the the unique root of the cubic Gλ ) in the interval Claim 3. If a c > λ p µ 2 ψ, then, cubic Gλ ) ha unique root in in the interval 0, µ). Proof. See Appendix. Oberve that given µ λ p aλ > p cψ, the unique root of cubic Gλ 2µλ 2 p+cψµ+λ p) ) in the interval 0, µ) indeed i trictly le than µ λ p. Thi follow from the fact that Gµ λ p ) = µ λ p )2µλ 2 p + cψµ + λ p )) aλ p cψ) > 0 a µ λ p > aλ p cψ 2µλ 2 p +cψµ+λ p). Let u aume that the Lagrangian multiplier v 1 = 0 at optimum. Given v 1 = v 2 = 0, the KKT condition 38) reult in a cubic equation given a Gλ ) 2µλ 3 aµ + cψ + 4µ 2 λ 2 + 2µ aµ + cψ + µ 2 λ µ aµ 2 cψλ p = 0 Note that under aumption of a > λp ψ and µ λp c µ 2 > root of the cubic Gλ ) in the interval 0, µ λ p ). λ 0, µ λ p ) reult in λ = λ 3). Further, Wp λ 3) ) = We note that Wp λ 3) ) < S p for S p J. The λ 3) aλ p cψ, 2µλ 2 p+cψµ+λ p ) λ3) i the unique Therefore, olving Gλ ) = 0 for ψλ 3 where λ µ λ 3) 3 = λ p +λ 3). µ λ 3, v 1 = 0 and v 2 = 0 atifie KKT 19

20 condition 38)-42) and therefore it i a KKT point. Thi point i global maximum of P2 for S p J a P2 i a convex optimization problem. Let u aume that the input parameter atify the aumption of Theorem 3. Thi implie that λ 3) < µ λ p and interval J i defined. Earlier, we demontrated that the objective function of the revenue maximization problem P2, ay O 2 λ ), i unimodal function of λ in the interval 0, µ). Given S p J, each contraint of P2 i trictly nonbinding at λ = λ 3). Therefore λ 3) hould alo correpond to the unimodal point of the objective function. Thi implie that the objective function i increaing in the interval 0, λ 3). Now, we claim that the contraint 34) i binding at optimum for S p / J. Thi i proved uing contradiction. Note that S p / J lie left to the interval J. Let u aume that λ i an optimum point for S p / J and the contraint 34) i trictly non-binding at the optimum, i.e., Wp λ ) < S p. Note that, if λ > λ 3), then W p λ 3) ) < S p alo hold a W p λ ) i an increaing function of λ 0, µ λ p ). Thi contradict the aumption that λ i optimum becaue λ 3) i unimodal point of the objective function. Therefore, λ < λ 3). Aume that λ = ˆλ atifie W p ˆλ ) = S p for the ame S p / J. A W p λ ) i an increaing function of λ 0, µ λ p ), the inequality λ < ˆλ λ 3) will hold. Thi reult in O 2 λ ) < O 2 ˆλ ) a the the objective function i increaing in the interval 0, λ 3). Thi contradict the initial aumption that λ i optimum and the contraint 34) trictly non-binding at the optimum. Hence, the contraint 34) hould be binding at optimum for S p / J. A graphical illutration of thi argument i given in Figure 3. O 2 ) or W p ) W p O 2 S p 4) 3) Figure 3: Illutration of the optimization problem P2 Alo, we note that if µ λp aλ p cψ 2µλ 2 p+cψµ+λ p, then the unimodal point of the objective ) function lie outide the interval µ λ p ). Again, the imilar argument etablih that the contraint 34) hould be binding at optimum for S p > Ŝp. The above dicuion aert that the contraint 34) i binding at optimum for S p / J. Theorem 4 define uch interval of S p where the contraint 34) i binding. 20

21 Theorem 4. Suppoe a c > λ p µ 2 ψ and S p lie in the interval J that i defined a: J = ψλ p µ µ λ p, ψλ 3 µ λ 3) µ λ 3 ) ψλp µ µ λ p, if µ λ p > otherwie aλ p cψ 2µλ 2 p + cψµ + λ p ), 44) where λ 3 = λ p + λ 3) and λ 3) i the unique root of the cubic Gλ ) of Equation 43) in aλ p cψ the interval 0, µ λ p ) whenever µ λp >. 2µλ 2 p+cψµ+λ p ) Then, λ 4) = 1 S p 2µ λ p + ψ S p λ p + ψ 2 + 4µψS p 2S p the NLP P2 and the contraint 34) i binding at thi point. i the global maximum of Proof. We note that J J = ; therefore, the contraint 34) will be binding at optimum for S p J. We firt how that there exit unique λ 0, µ λ p ) that atify the equality W p λ ) = S p for S p > λ pψ µµ λ p. Claim 4. Given S p > λ pψ, there exit unique λ µµ λ p in the interval 0, µ λ p ) that atify the equality W p λ ) = S p. It i given by Proof. See Appendix. λ = 1 S p 2µ λ p + ψ S p λ p + ψ 2 + 4µψS p 2S p Take λ 4) = λ. A W p λ 4) ) = S p, the point λ = λ 4) atifie the KKT condition 39) irrepective of the value of the Lagrangian multiplier v 1. Given λ = λ 4) and v 2 = 0, the KKT condition 38) reult in 1 v 1 = a 2λ 4) v 4) µ cψ λ p + 2λ 4) µ µ λ 4) 2 λ 4) where λ 4 = λ p + λ 4). We note that v 4) 1 0 if and only if a 2λ4) 2 2 µ λ 4 2 µ λ 4) bψ µ µ + λ p ) λ 2 4 cψ µ h λ p > 0 and 0 < λ 4) < µ λ p. The rearrangement of thi inequality reult in i i λ p +2λ 4) hλ 4) 2 i µ hµ λ 4) 2 45) a Gλ 4) ) 0. Given a > λp ψ, the Claim 3 ha already etablihed that Gλ c µ 2 ) 0 in the interval 0, λ 3) where λ 3) i the unique root of the cubic Gλ ) in the interval 0, µ). Thi implie that if λ 4) λ 3) then the inequality v 4) 1 0 will hold true. To etablih that λ 4) λ 3), firt conider the cae when µ λp ) thereby the interval J = λp ψ, µµ λ p. Given µ λ p aλ p cψ 2µλ 2 p+cψµ+λ p ) and aλ p cψ 2µλ 2 p+cψµ+λ p), we note from the 21

22 proof of Theorem 3 that λ 3), the root of the cubic Gλ ), will alway be greater than or equal to µ λ p. We have etablihed in the Claim 4 that λ 4) < µ λ p and therefore λ 4) < λ 3) under thi aumption. Next, conider the cae when µ λp aλ > p cψ 2µλ 2 p+cψµ+λ p ) and thereby the interval J = λp ψ, ψ λ µµ λ p. A the contraint 34) i binding at µ λ 3) optimum, we note that λ 4) = λ 3) at S p = µ λ ψ λ µ λ 3). We alo know that W p λ ) i an µ λ ψ λ increaing convex function of λ in the interval 0, µ λ p ). Therefore, S p < µ λ 3) µ λ will alway reult in λ 4) < λ 3). Thi complete the argument that λ 4) λ 3) for S p J. The λ = λ 4), v 1 = v 4) 1, v 2 = 0 atifie KKT condition 38)-42). Therefore, it i a KKT point and thereby a global maximum of the optimization problem P2. Corollary 3. λ 4) i an increaing function of S p for S p J. Proof. Given λ 4), a defined in Theorem 4, we note that λ 4) = ψ 1 2Sp 2 S p 2µ + λ p ) + ψ S p λ p + ψ 2 + 4ψµS p. We note that λ4) 0 a which implie that S p 2µ + λ p ) + ψ 2 = S p λ p + ψ 2 + 4ψµS p + 4µµ + λ p )Sp 2 S p λ p + ψ 2 + 4ψµS p S p 2µ+λ p )+ψ S p λ p +ψ 2 +4ψµS p Search for global optima The analyi in Section 3.1 etablihe that if a > λp2µ λp) ψ, then the optimization c µµ λ p) 2 problem P1 will have a local optimal olution with β < provided that S p I I. The analyi in Section 3.2 etablihe that if a > λ p ψ, then the optimization problem c µ 2 P2 will have a local optimal olution for S p > Ŝp =. Note that the local optimal λpψ µµ λ p olution of P2 alo correpond to the local optimal olution of the optimization problem P1 with β =. Further, we oberve that a c > λ p 2µ λ p ) µ µ λ p ) 2 ψ = 2 µ λ p + λ p λp µ λ p ) 2 µ ψ > λ p µµ λ p ) ψ > λ p µ ψ. 2 The above inequalitie follow a 0 < λ p < µ. The above inequality implie that if a > λ p2µ λ p ) ψ then a > λ p ψ automatically hold. Alo, if a λ p ψ, then the root c µµ λ p ) 2 c µ 2 c µ 2 of cubic Gλ ) and Gλ ) are negative and doen t contitute feaible point for the 22

23 optimization problem P1 and P2. Given S p > Ŝp, the relationhip among the input parameter reult in the following poibilitie: D1: a λ p ψ: There doe not exit optimum olution to the optimization problem P1 c µ 2 and P2. D2: λp ψ < a λp2µ λp) ψ: There exit an optimum olution to the optimization problem P2, but there doe exit any optimum olution to the optimization µ 2 c µµ λ p ) 2 problem P1. D3: a c > λ p2µ λ p ) µµ λ p) 2 ψ: There exit optimum olution to both the optimization problem P1 and P2 for S p I I. Equivalently, the original optimization problem P1 ha two local optimal olution; one with β < and another with β =. But, for S p > I u where I u = ψλ 1 µ λ 1) µ λ 1 optimal olution to the optimization problem P2 only. i the upper limit of the interval I, there exit an Given a > λ p2µ λ p ) ψ, let λ f c µµ λ p), β f) and λ i, ) are optimal olution of the optimization 2 problem P1 and P2 repectively for given S p I I. Alo, let the correponding value of objective function are O1 λ f, β f) and O2 λ i, ). Below, we eek to etablih that O1 λ f, β f) O2 λ i, ). Firt, oberve from Theorem 3-4 that the feaible region of S p, i.e., Ŝp, ), i divided into interval J and J. Note that each of the contraint of the optimization problem P2 i non-binding at optimum for S p J, while, contraint 34) of the optimization problem P2 i binding at optimum for S p J. Further, if µ λ p aλ > p cψ, then 2µλ 2 p +cψµ+λ p) interval J ψλ = Ŝp, J l and J = J l, ) where J l = 3. On the other hand, if µ λ 3) µ λ 3 µ λ p aλ p cψ, then J 2µλ 2 p+cψµ+λ p = Ŝp, ) and J =. Given µ λp aλ ) > p cψ 2µλ 2 p+cψµ+λ p, we will ) now etablih that J l > I u, i.e., J l lie to the right of I u. Thi mean that for S p I I, in both the local olution given by optimization problem P1 and P2 the ervice level contraint correponding to primary cla cutomer i binding. We note that I u = ξλ 1) and J l = ξλ 3) ψλ ) where ξλ ) = µ λ and λ = λ µ λ p + λ. A ξλ ) λ = µµ+λp) λ2 ψ µ λ > 0 2 µ λ 2 for λ p > 0, λ > 0 and λ p + λ < µ, the inequality J l > I u will hold if λ 3) > λ 1). We argue below that λ 3) > λ 1). We oberve from Claim 1 that λ 1) i the unique root of the cubic Gλ ) of Equation 25) in the interval 0, µ λ p ) whenever a > λ p2µ λ p ) ψ. That i, λ 1) c µµ λ p) 2 0, µ λ p ) for a a l, ) where a l = λp2µ λp) cψ. We further oberve from Claim 3 and proof of µµ λ p ) 2 Theorem 3 that λ 3) i the unique root of the cubic Gλ ) of Equation 43) in the interval 0, µ λ p ) whenever µ λ p > aλ p cψ and a > λ p 2µλ 2 p+cψµ+λ p) ψ. That i, λ 3) c µ 2 a ã l, ã u ) where ã l = λp cψ and ã µ 2 u = 2µ λ p ) + cψ µ λ p λ p aλ p cψ, i.e., a ã 2µλ 2 p+cψµ+λ p ) u, then µ λ p λ 3) ) 0, µ λ p ) for 1 + µ λp)2. We alo note that if < µ. From λp µ 2 ψ < λp2µ λp) µµ λ p ) 2 ψ, we have that ã l < a l. Note that λ 1) = 0 at a = a l, λ 3) = 0 at a = ã l and λ 3) a = ã u. Baed on relative value of a l, ã l and ã u, we oberve following: 23 = µ λ p at