## On-line Pricing of Secondary Spectrum Access with Unknown Demand Function and Call Length Distribution

Citations: | 7 - 4 self |

### BibTeX

@MISC{Mutlu_on-linepricing,

author = {Huseyin Mutlu and Murat Alanyali and David Starobinski},

title = {On-line Pricing of Secondary Spectrum Access with Unknown Demand Function and Call Length Distribution},

year = {}

}

### OpenURL

### Abstract

Abstract—We consider a wireless provider who caters to two classes of customers, namely primary and secondary users. Primary users have long term contracts while secondary users are admitted and priced according to current availability of excess spectrum. Secondary users accept an advertised price with a certain probability defined by an underlying demand function. We analyze the problem of maximizing profit gained by admission of secondary users. Previous studies in the field usually assume that the demand function is known and that the call length distribution is also known and exponentially distributed. In this paper, we analyze more realistic settings where both of these quantities are unknown. Our main contribution is to derive near-optimal pricing strategies under such settings. We focus on occupancy-based pricing policies, which depend only on the total number of ongoing calls in the system. We first show that such policies are insensitive to call length distribution except through the mean. Next, we introduce a new on-line, occupancy-based pricing algorithm, called Measurement-based Threshold Pricing (MTP) that operates by measuring the reaction of secondary users to a specific price and does not require the demand function to be known. MTP optimizes a profit function that depends on price only. We prove that while the profit function can be multimodal, MTP converges to one of the local optima as fast as if the function were unimodal. Lastly, we provide numerical studies demonstrating the near-optimal performance of occupancy-based policies for diverse sets of call length distributions and demand functions and the quick convergence of MTP to near-optimal on-line profit. I.

### Citations

185 |
Blocking in a shared resource environment
- Kaufman
- 1981
(Show Context)
Citation Context ... Refs. [10] study optimal call admission policy for generally distributed call lengths whereas we study pricing policies. More information on insensitivity to call length distribution can be found in =-=[15, 16]-=-.3 III. MODEL AND PROBLEM FORMULATION In this section, we introduce our model and objective. We consider a single-cell wireless network which provides access to C channels. Calls from PUs arrive acco... |

127 | Tsitsiklis, ”Congestion-Dependent Pricing of Network Services
- Paschalidis, N
- 2000
(Show Context)
Citation Context ...insensitivity to call length distribution. We start with the well studied area of congestion-based pricing. As such, we restrict our literature review to those papers that are the most relevant. Ref. =-=[3]-=- studies pricing of network resources when arrival rate of all users can be regulated with price. They show that static pricing (a single price is advertised regardless of occupancy level) achieves go... |

89 |
Queueing Networks and Markov Chains
- BOLCH, GREINER, et al.
- 2006
(Show Context)
Citation Context ... cv (ratio of standard deviation to mean) is 1. These distributions represent divergence from exponential distribution in both directions. For hyper-exponential cv ≥ 1 and for hypo-exponential cv < 1 =-=[21]-=-. If a call length has a two-phase hyper-exponential distribution then with probability p it is exponentially distributed with mean 1/µ1 and with probability 1 − p it is exponentially distributed with... |

60 |
The stochastic knapsack problem
- Ross, Tsang
- 1989
(Show Context)
Citation Context ...lishment of a connection with coordinate convex policies in call admission control that are known to enjoy product-form equilibrium distributions and to be insensitive to the call length distribution =-=[10]-=-.2 Next, we introduce occupancy-based dynamic and threshold pricing policies. In dynamic pricing, a spectrum provider sets a different price for each occupancy level. In threshold pricing, the provid... |

54 |
Sequential Minimax Search for a Maximum
- Kiefer
- 1957
(Show Context)
Citation Context ...ch can be undefined at certain points (transition points from one RT (u) to another). Therefore, we base the MTP algorithm on the derivative-free Fibonacci search which was first introduced by Kiefer =-=[18]-=-. Fibonacci search is a sequential line search algorithm which maximizes a unimodal function. In every iteration, it makes a function evaluation. Together with the information from earlier evaluations... |

34 | Pricing in multiservice loss networks: Static pricing, asymptotic optimality and demand substitution effects
- Paschalidis, Yong
- 2002
(Show Context)
Citation Context ...with price. They show that static pricing (a single price is advertised regardless of occupancy level) achieves good performance and is optimal in some asymptotic regimes. This result was extended in =-=[4]-=- in the context of large network asymptotics. In addition to [6], which we mentioned in the previous section, the following papers consider secondary access pricing. Ref. [7] studies optimal and stati... |

32 | Primary users in cellular networks: a large-scale measurement study
- Willkomm, Machiraju, et al.
- 2008
(Show Context)
Citation Context ...engths are exponentially distributed (see again Sec. II for exceptions). A recent study based on measurement of real traces in a cellular networks shows that this assumption does not hold in practice =-=[9]-=-. In particular, [9] observes that the variance of call length is significantly higher than that of the exponential distribution and questions the validity of previous work (and [6] in particular) bas... |

27 |
Optimum Seeking Methods
- Wilde
- 1964
(Show Context)
Citation Context ...size [18]. In our case, function evaluations, i.e., measurements are conducted for discrete values of price. Therefore, we utilize a discrete version of Fibonacci search (also known as lattice search)=-=[19]-=-. While Fibonacci search might fail to converge when the function is multimodal, MTP converges to a local maximum of Rmax(u). We manage this by taking advantage of the fact that Rmax(u) is the maximum... |

21 |
Dynamic pricing without knowing the demand function: Risk bounds and near optimal pricing algorithms. Oper. Res., forthcoming
- Besbes, Zeevi
- 2008
(Show Context)
Citation Context ...line algorithm for static pricing of calls with exponentially distributed call lengths. It considers a parametric demand function while we consider a more general non-parametric demand function. Ref. =-=[13]-=- studies a different model than ours where the pricing problem is finite horizon and there is a single product with a finite inventory. They provide on-line learning algorithms for parametric and non-... |

19 |
Secondary pricing of spectrum in cellular CDMA networks
- Daoud, Alanyali, et al.
(Show Context)
Citation Context ...t optimization problem with two types of customers, which are akin to our SUs and PUs. Ref. [11] provides a game theoretic analysis of revenue maximization problem for secondary spectrum access. Ref. =-=[8]-=- studies secondary spectrum access pricing strategies capturing the effects of network-wide interferences. When applicable, the previous work mentioned above assume a known demand function and exponen... |

16 | Simplification of network dynamics in large systems
- Lin, Shroff
- 2005
(Show Context)
Citation Context ...th a finite inventory. They provide on-line learning algorithms for parametric and non-parametric demand functions. Finally, we provide related work on insensitivity to call length distribution. Ref. =-=[5]-=- shows that static pricing policy is insensitive to call length distribution and is still asymptotically optimal. Ref. [14] also studies static pricing for generally distributed call lengths and shows... |

10 |
Pricing and capacity rationing for rentals with uncertain durations
- Gans, Savin
(Show Context)
Citation Context ...s shapes depending on the parameter β: λs(u) = α (umax − u) β, u ∈ [0, umax]. (8) umax For β = 1 it is linear, for β < 1 it is convex and for β > 1 it is concave. This demand function is also used in =-=[7]-=- and satisfies Assumptions 3.1 and 3.2. We study different shapes of this demand function for α = 10 and umax = 10. In Fig. 2, we compare the performances of the optimal occupancy-based, the optimal t... |

10 | A Note on optimal pricing for finite capacity queueing systems with multiple customer classes
- Ziya, Ayhan, et al.
- 2008
(Show Context)
Citation Context ...y, we provide related work on insensitivity to call length distribution. Ref. [5] shows that static pricing policy is insensitive to call length distribution and is still asymptotically optimal. Ref. =-=[14]-=- also studies static pricing for generally distributed call lengths and shows that the profit function of static pricing is unimodal. It assumes that the demand function is known and that all users ar... |

8 | On-line tuning of prices for network services
- Campos-Náñez, Patek
- 2003
(Show Context)
Citation Context ...ion and exponentially distributed call lengths, on the contrary to the model presented in this paper. Next, we present related work on less studied field of pricing with unknown demand function. Ref. =-=[12]-=- introduces an online algorithm for static pricing of calls with exponentially distributed call lengths. It considers a parametric demand function while we consider a more general non-parametric deman... |

8 |
Relationships among three assumptions in revenue management
- Ziya, Ayhan, et al.
- 2004
(Show Context)
Citation Context ...u ≤ umax. Then λsu(λs) is concave with respect to λs. Assumption 3.2 implies that the marginal instantaneous profit is decreasing with respect to user demand. It ensures a wellbehaved demand function =-=[17]-=-. This assumption is widely made in the literature [3, 13, 17] and is satisfied by variety of demand functions such as functions with exponential, linear and polynomial decay. We assume that PU and SU... |

5 | A revenue enhancing Stackelberg game for owners in opportunistic spectrum access - Ercan, Lee, et al. - 2008 |

5 | Spot pricing of secondary spectrum access in wireless cellular networks
- Mutlu, Alanyali, et al.
- 2009
(Show Context)
Citation Context ...t hold in practice [9]. In particular, [9] observes that the variance of call length is significantly higher than that of the exponential distribution and questions the validity of previous work (and =-=[6]-=- in particular) based on the exponential distribution assumption. In this paper, we develop an on-line, measurement-based pricing framework for secondary access applicable to settings where both the d... |

3 | Optimizing a 2d function satisfying unimodality properties
- Demaine, Langerman
- 2005
(Show Context)
Citation Context ...rements. In conclusion to this section, we note that the number of measurements required by MTP is the same as in the Fibonacci search which is logφ(|U|)+O(1) where φ = (1+ √ 5)/2 is the golden ratio =-=[20]-=-. MTP can easily be adapted to converge to the global maximum of Rmax(u). To do so, the threshold should be kept fixed throughout the MTP algorithm. This should be repeated for all possible thresholds... |

1 |
Insensitivity of blocking probabilites in a circuit-switching network
- Burman, Lehoczky, et al.
- 1984
(Show Context)
Citation Context ... Refs. [10] study optimal call admission policy for generally distributed call lengths whereas we study pricing policies. More information on insensitivity to call length distribution can be found in =-=[15, 16]-=-.3 III. MODEL AND PROBLEM FORMULATION In this section, we introduce our model and objective. We consider a single-cell wireless network which provides access to C channels. Calls from PUs arrive acco... |