Minimizing the Age of Information of Multiple Sources Vishrant - PowerPoint PPT Presentation

Minimizing the Age of Information of Multiple Sources Vishrant Tripathi Advisor - Prof. Sharayu Moharir Electrical Engineering Department, IIT-Bombay October 16, 2017 Vishrant (CNRG) Minimizing Age of Information October 16, 2017 1 / 37

Motivation This problem has direct applications to IoT (Internet of Things). For example, an IoT connected home or vehicle could have a large number of sensors monitoring different pieces of information, all of which needs to be sent to a central controller Having the freshest available data would be essential to making better decisions. Vishrant (CNRG) Minimizing Age of Information October 16, 2017 2 / 37

Setting n sensors communicating over m channels Time-slotted system One channel per sensor per time-slot Vishrant (CNRG) Minimizing Age of Information October 16, 2017 3 / 37

Channel Model Assumption (ON-OFF Channel Model) � 1 , if sensor i can communicate over channel j X i , j ( t ) = 0 , otherwise We have that ∀ i , j, P ( X i , j ( t ) = 1 | X i , j ( τ ) : ∀ τ < t , i , j ) ≥ p min > 0 . The processes X i , j ( t ) evolve independently across all sensor-channel pairs. Vishrant (CNRG) Minimizing Age of Information October 16, 2017 4 / 37

Channel Model Examples of such channel models include: – X i , j ( t ) is an independent Bernoulli random variable with parameter p i , j ( t ) ≥ p min . – X i , j is a Markov chain, independent across all users and channels with P ( X i , j ( t ) = 1 | X i , j ( t − 1)) ≥ p min . Vishrant (CNRG) Minimizing Age of Information October 16, 2017 5 / 37

Goal l i ( t ) is the age of the latest measurement from sensor i Our goal is to minimize the time-average cost of the age of information C ( t ) = f ( l i ( t ); 1 ≤ i ≤ n ) , where f is a non-decreasing function of the l i s Examples: n � f ( l i ( t ); 1 ≤ i ≤ n ) = g ( l i ( t )) i =1 f ( l i ( t ); 1 ≤ i ≤ n ) = max l i ( t ) i Vishrant (CNRG) Minimizing Age of Information October 16, 2017 6 / 37

A Converse Result n l ( t ) : the number of sensors with age ≥ l at time t , then n l ( t ) ≥ ( n − lm ) + . This can be proved using a simple counting argument. Vishrant (CNRG) Minimizing Age of Information October 16, 2017 7 / 37

Our Algorithm: Max-Age Matching Key Ideas - Use a locally greedy strategy to minimize the age of information increment in each time-slot Convert this problem of greedy minimization to a minimum weight perfect matching problem in bipartite graphs Vishrant (CNRG) Minimizing Age of Information October 16, 2017 8 / 37

Max-Age Matching Algorithm 1 Max-Age Matching Input: Connectivity and age information for the current time-slot Output: A valid allocation of sensors to channels 1: procedure Max-Age-Matching( X i , j ) Construct a bipartite graph G ( X , Y , E ) using connectivity and age 2: information. M = FindMaxWeightMatching( G ) 3: Use M to allocate sensors to channels 4: 5: end procedure Vishrant (CNRG) Minimizing Age of Information October 16, 2017 9 / 37

Max-Age Matching Example Figure: MAM Example with 4 sensors, 2 channels and sensor ages (2,2,1,3) Vishrant (CNRG) Minimizing Age of Information October 16, 2017 10 / 37

Optimality of Max-Age Matching Theorem n MAM ( t ) : the number of sensors with age ≥ l under MAM � n l � ( t ) = ( n − lm ) + ∀ l ≥ 0 , with probability ≥ 1 − 3 m (1 − p min ) m n MAM l m Corollary C MAM ( t ) : the cost of the age of information under MAM C OPT ( t ) : the cost under the optimal scheduling policy � n � C MAM ( t ) = C OPT ( t ) , with probability ≥ 1 − 3 m (1 − p min ) m m The probability → 1 as n , m ↑ ∞ if m grows at least as fast as Ω(log( n )) . Vishrant (CNRG) Minimizing Age of Information October 16, 2017 11 / 37

Our 2 nd Algorithm: Iterative Max-Age Scheduling Key Idea - Simply iterate through the entire set of channels, allocating sensors which can connect to a particular channel in descending order of age. The advantage here is that of reduced complexity. No other algorithm can be simpler, since this algorithm goes through all the inputs only once. Vishrant (CNRG) Minimizing Age of Information October 16, 2017 12 / 37

Iterative Max-Age Scheduling Algorithm 2 Allocate Sensors to Channels in each Time-slot Input: X i , j - the connectivity information for the current time-slot Output: A valid allocation in each time-slot 1: procedure FindAllocation() Define a priority of sensors in decreasing order of costs g ( l i ( t )). 2: For sensors with equal costs, use lexicographic ordering. 3: for every channel j do 4: Find highest priority un-allocated sensor i s.t. X i , j = 1 5: Allocate sensor i to channel j 6: end for 7: Output the Allocation for the next time-slot. 8: 9: end procedure Vishrant (CNRG) Minimizing Age of Information October 16, 2017 13 / 37

Iterative Max-Age Scheduling Example Figure: IMAS Example with 4 sensors, 2 channels and sensor ages (2,2,1,3) Vishrant (CNRG) Minimizing Age of Information October 16, 2017 14 / 37

Optimality of IMAS Theorem n IMAS ( t ) : the number of sensors with age ≥ l, under IMAS l � n  � n − lm + O (log m ) , for 0 ≤ l < , m  � n  n IMAS � ( t ) = O (log m ) , for l = , l m � n  � 0 , for l > .  m with high probability, which goes to 1 as n , m ↑ ∞ if m grows at least as fast as Ω(log( n )) . Vishrant (CNRG) Minimizing Age of Information October 16, 2017 15 / 37

Optimality of IMAS Corollary C IMAS ( t ) : the cost of the age of information under IMAS 1 If f is defined as a sum of individual costs of sensors, we have C IMAS ( t ) � log ( m ) � C OPT ( t ) = 1 + O m 2 If f is defined as the maximum of individual costs, then C IMAS ( t ) = C OPT ( t ) + 1 with high probability, which goes to 1 as n , m ↑ ∞ if m grows at least as fast as Ω(log( n )) . Vishrant (CNRG) Minimizing Age of Information October 16, 2017 16 / 37

Complexity Comparison Now that we have compared the performance of the two proposed algorithms, we can also compare their computational costs. The complexity of Max-Age Matching (MAM) is O ( n 3 ). The complexity of Iterative Max-Age Scheduling (IMAS) is O ( mn ). Vishrant (CNRG) Minimizing Age of Information October 16, 2017 17 / 37

Energy Performance Trade-offs The above scheduling algorithms require all sensors to be ON in each time-slot Instead, we can use a batch based version of the above algorithms to save energy. Clearly, the performance of these batch based will be worse off as compared to the above algorithms. � n We use a batch size of n � k where k = so that in each batch we m have roughly the same number of active sensors as the number of channels. Vishrant (CNRG) Minimizing Age of Information October 16, 2017 18 / 37

Batched Max-Age Matching Algorithm 3 Batched Max-Age Matching Input: Connectivity and age information for the batch being served in the current time-slot (serve batches in round robin fashion) Output: A valid allocation of sensors to channels 1: procedure Max-Age-Matching( X i , j ) Construct a bipartite graph G ( X , Y , E ) as described earlier using 2: connectivity and age information of the current batch. M = FindMaxWeightMatching( G ) 3: Use M to allocate sensors to channels 4: 5: end procedure Vishrant (CNRG) Minimizing Age of Information October 16, 2017 19 / 37

B-MAM Example Vishrant (CNRG) Minimizing Age of Information October 16, 2017 20 / 37

Optimality of B-MAM Theorem n B-MAM ( t ) : the number of sensors with age ≥ l, under B-MAM l n MAM ( t ) = ( n − lm ) + l ∀ l ≥ 0 , with probability ≥ 1 − 3 n (1 − p min ) m . Corollary C B-MAM ( t ) : the cost of the age of information under B-MAM C B-MAM ( t ) = C OPT ( t ) , with probability ≥ 1 − 3 n (1 − p min ) m . Order wise identical with MAM The probability → 1 as n , m ↑ ∞ if m grows at least as fast as Ω(log( n )) . Vishrant (CNRG) Minimizing Age of Information October 16, 2017 21 / 37

B-IMAS Algorithm 4 Allocate Sensors to Channels in each Time-slot Input: Connectivity and age information for the batch being served in the current time-slot (serve batches in round robin fashion) Output: A valid allocation in each time-slot 1: procedure FindAllocation() Define a priority of sensors in decreasing order of costs g ( l i ( t )). 2: For sensors with equal costs, use lexicographic ordering. 3: for every channel j in current batch do 4: Find highest priority un-allocated sensor i s.t. X i , j = 1 5: Allocate sensor i to channel j 6: end for 7: Output the Allocation for the next time-slot. 8: 9: end procedure Vishrant (CNRG) Minimizing Age of Information October 16, 2017 22 / 37

B-IMAS Example Vishrant (CNRG) Minimizing Age of Information October 16, 2017 23 / 37

Optimality of B-IMAS Theorem n B-IMAS ( t ) : the number of sensors with age ≥ l under B-IMAS l � n  � n − l ( m − m α ) , for 0 ≤ l < , m  � n � n � n  n B-IMAS � � � ( t ) = (2 − l ) m α , for ≤ l ≤ 2 , l m m m � n  � 0 , for l > 2 .  m with probability � n ( m α (1 − p min ) m − m α +1 + ( m − m α )(1 − p min ) m α +1 ) where � ≥ 1 − m 0 < α < 1 . This probability → 1 as n , m ↑ ∞ if m grows at least as fast as Ω((log n ) 1+ a ), where 1 1+ a < α. Vishrant (CNRG) Minimizing Age of Information October 16, 2017 24 / 37