The Profit Maximizing Cutoff in Observable Queues with State Dependent Pricing
Christian Borgs (MSR-NE) Jennifer T. Chayes (MSR-NE)
Sherwin Doroudi (CMU-Tepper) Mor Harchol-Balter (CMU-CS) Kuang Xu (MIT-LIDS)
Support provided by Microsoft Computational Thinking grant
1. Background & Motivation
4. Problem Statement & Approach
CHARGE ENTRY PRICE
OBSERVABLE
QUEUE
Hmm…
should
I join?
DELAY
SENSITIVE
CUSTOMERS
w/ LIMITED
BUDGET

INPUTS: fixed 𝜆 and 𝑉

OBJECTIVE: Find cutoff 𝑘 ∗ to maximize the
earning rate 𝑅(𝑘, 𝜆, 𝑉)

TECHNIQUE: use a “discrete derivative”
 Define Δ𝑥 𝑓 𝑥 ≡ 𝑓 𝑥 + 1 − 𝑓 𝑥
 Δ𝑥 is the forward difference operator
 Solve Δ𝑘 [𝑅 𝑘, 𝜆, 𝑉 ] = 0 for 𝑘 ∈ 𝐑
 Obtain optimal cutoff 𝑘 ∗ = ⌈𝑘⌉
CLOUD COMPUTING SERVICE
GOAL: Choose prices and admissions
policy to maximize profits
2. The Queueing Model

𝑀/𝑀/1 queue

𝜆→

𝜇=1
WLOG

5. The Analysis

ln(𝜆) 𝐺 𝜆, 𝑉 − 𝑘 𝑒 ln
All customers are identical with parameters:
𝑉
𝑣=1
Value for receiving service
Waiting cost per unit time
Solve: 𝑅 𝑘 + 1, 𝜆, 𝑉 − 𝑅 𝑘, 𝜆, 𝑉 = 0
1
Define: 𝑘 ≡ 𝑘 + 2 and 𝐺 𝜆, 𝑉 ≡ 1 − 𝜆 𝑉 +
1−𝜆
Question: How do we solve this for 𝑘?


(𝑛 + 1) + 𝑝 ≤ 𝑉
Now solve: 𝑥𝑒 𝑥 = 𝐶 for 𝑥
Answer: Lambert 𝑊 (product log) function
𝑊 𝑥 𝑒
𝑛 : # of customers in (state of) system
ln(𝜆)𝜆𝐺(𝑉,𝜆)
=
1−𝜆
Call both of
these 𝑥
𝑝: price customers must pay
I wait only if:
𝜆 𝐺 𝜆,𝑉 −𝑘
3. How to Price?
𝑊(𝑥)
=𝑥
Or equivalently

𝑊 𝑥𝑒 𝑥 = 𝑥
6. Conclusions
Charge as much as customers are willing to pay in each state:
𝑝 𝑛 = 𝑉 − (𝑛 + 1)
𝝀
0
𝝀
𝝀
1
𝝀
= ceil
𝝀
1
1
ln 𝜆 ⋅ 𝜆
1−𝜆 𝑉+
−
⋅𝑊
1 − 𝜆 ln 𝜆
1−𝜆
𝑘
𝑘−1
$𝑉 − 1 𝟏 $𝑉 − 2 𝟏
1−𝜆 𝑉+
𝑘∗
Cutoff at 𝑘
𝟏 $𝑉 − 𝑘 − 2 𝟏 $𝑉 − 𝑘 − 1𝟏
So what can the closed form do for us?
But… we use an explicit early cutoff at state 𝑘.
Why?
Asymptotic Results
For fixed 𝜆 < 1, as 𝑉 → ∞: 𝑘 ∗ ∼ 1 − 𝜆 𝑉
Answer: Cutoff  shorter queue  high prices more often
When 𝜆 = 1, as 𝑉 → ∞: 𝑘 ∗ ∼ 2𝑉
So how much do we make?
Probability:
(𝑘)
𝜋0
0
Price charged:
𝝀
(𝑘)
𝜋1
𝝀
𝝀
1
(𝑘)
𝜋𝑘−1
$𝑉 − 1 𝟏 $𝑉 − 2 𝟏
For fixed 𝜆 > 1, as 𝑉 → ∞: 𝑘 ∗ ∼ log 𝜆 ( 𝑉)
𝝀
𝑘
𝑘−1
𝟏 $𝑉 − 𝑘 − 2 𝟏
Earning rate = (arrival rate) x (average price charged)
Is the cutoff of practical use?
When 𝜆 > 1: high cutoff  huge losses!
𝑘−1
(𝑘)
𝜋𝑛 : Probability that a new arrival
sees state 𝑛, given a cutoff of 𝑘
𝜆
1
𝑅(𝑘, 𝜆, 𝑉) =
⋅
𝑘+1
1−𝜆
1−𝜆
𝑘∗ = 7
⋅ 𝑝(𝑛)}
𝑝 𝑛 = 𝑉 − (𝑛 + 1)
1 − 𝜆𝑘 1 − 𝜆 𝑉 + 𝜆𝑘 1 + 𝑘 − 𝑘𝜆 − 1
Earning Rate
𝑅 𝑘, 𝜆, 𝑉 = 𝜆 ⋅
(𝑘)
{𝜋𝑛
𝑛=0
Call this
constant 𝐶
𝜆 = 1.2
𝑉 = 50
𝑘
1
1−𝜆
−2