  Consider an online facility assignment problem where a set of facilities $F =
\{ f_1, f_2, f_3, \cdots, f_{|F|} \}$ of equal capacity $l$ is situated on a
metric space and customers arrive one by one in an online manner on that space.
We assign a customer $c_i$ to a facility $f_j$ before a new customer $c_{i+1}$
arrives. The cost of this assignment is the distance between $c_i$ and $f_j$.
The objective of this problem is to minimize the sum of all assignment costs.
Recently Ahmed et al. (TCS, 806, pp. 455-467, 2020) studied the problem where
the facilities are situated on a line and computed competitive ratio of
"Algorithm Greedy" which assigns the customer to the nearest available
facility. They computed competitive ratio of algorithm named "Algorithm
Optimal-Fill" which assigns the new customer considering optimal assignment of
all previous customers. They also studied the problem where the facilities are
situated on a connected unweighted graph.
  In this paper we first consider that $F$ is situated on the vertices of a
connected unweighted grid graph $G$ of size $r \times c$ and customers arrive
one by one having positions on the vertices of $G$. We show that Algorithm
Greedy has competitive ratio $r \times c + r + c$ and Algorithm Optimal-Fill
has competitive ratio $O(r \times c)$. We later show that the competitive ratio
of Algorithm Optimal-Fill is $2|F|$ for any arbitrary graph. Our bound is tight
and better than the previous result. We also consider the facilities are
distributed arbitrarily on a plane and provide an algorithm for the scenario.
We also provide an algorithm that has competitive ratio $(2n-1)$. Finally, we
consider a straight line metric space and show that no algorithm for the online
facility assignment problem has competitive ratio less than $9.001$.
