Slow MIS Distributed Algorithm
It is not actually the diameter. It is the longest chain in the partial order $P$ of the input graph $G = (V,E)$. The poset $P$ is constructed as follows. Given an input graph $G = (V,E)$ such that each vertex has a unique identifier. Build an oriented graph $G' = (V,E')$ such that an edge $(u,v) \in E'$ if $(u,v) \in E$ and $u \succ v$. Then build a partial order $P$ of $G'$ by taking the transitive closure of $G'$. Let $l$ be the length the longest chain of $P$, then the time complexity of your algorithm is exactly $l$  (assuming here that the time complexity is measured by the number of communication rounds and in each round a node can send/receive a message to/from all its neighbors). Note that $l$ is $\in O(n)$ and in $\Omega(1)$. 

What about average complexity ?
Let's $G(n,p)$ be a random graph of $n$ vertices and an edge between any pair of nodes exist with probability $p \in [0,1]$. Then, if you build a partial order $P$ of this graph (called random graph order), you will find that the longest chain $l$ is in $\Theta(n)$. This result is proven in [1]. 

[1]: Albert, Michael H., and Alan M. Frieze. "Random graph orders." Order 6.1 (1989): 19-30.

(This is for additional knowledge)
It is possible that assigning identifiers on the input graph such that the complexity of this MIS algorithm is minimized (that is, minimize $l$).  The problem is NP-Hard as it is reducible to minimum coloring. This NP-Hard problem was given in: [2] and [3]. 

[2]: Deming, R. W. "Acyclic orientations of a graph and chromatic and independence numbers." Journal of Combinatorial Theory, Series B 26.1 (1979): 101-110.

[3]: Chvátal, Vašek, and Carsten Thomassen. "Distances in orientations of 
graphs." Journal of Combinatorial Theory, Series B 24.1 (1978): 61-75.

(Ok this is just for fun, and you can skip it)
Now assume that you want to measure time with a different metric. And let's add a restriction to the definition of communication rounds (found in [4]). And let's say that a node 1) can send a message to only neighbor in a communication round, and 2) can receive a messages (more than one) from all its neighbors In such case, the order that a node follow to send a message to its neighbors will affect the time complexity. Let's consider three strategies that a node $v$ follows to visit its neighbors (with smaller identifiers) which are $\{u _1, ..., u _k\}$ such that $u _i \succ u _j$ if $i &gt; j$. Then: 


Strategy 1: $v$ sends messages in an order from $u _k$, to $u _{k-1}$ ..., $u _1$ etc .. . This strategy leads to $\Theta(n)$ time complexity. 
Strategy 2: $v$ sends messages in an order from $u _1$, to $u _{2}$ ..., $u _k$ etc .. . This strategy leads to $\Theta(n^2)$ time complexity. 
Strategy 3: $v$ sends messages to its neighbors in a uniformly random order. This strategy will also lead to $\Theta(n^2)$ in expectation. 


(Why I am interested in all that ?) - because it was found in some practical wireless networks (Bluetooth networks), that this slow MIS algorithm is the fastest although not being the theoretically fastest one. It was found that this type of networks follow the modified communication model I defined above.

[4]: Peleg, David. Distributed computing: a locality-sensitive approach. Vol. 5. Society for Industrial Mathematics, 1987.