In operating system, How MMU searches for virtual page number as key in page table

Question

1)So lets say a single level page table
3)A TLB miss happens
3)The required page table is at main memory

Question : Does MMU always fetch the page table required to a number of registers inside it so that fast hardware search like TLB can be performed? I guess no that would be costly hardware

4)MMU fetch the physical page number (I guess MMU must be saved it with a format like high n-bits as virtual page no. and low m bits as physical page frame no. Please correct and explain if I am wrong)
Question: I guess there has to be a key-value map with Virtual page no as key and physical frame no. as value. How MMU search for the key in the page table. If it is a s/w like linear search than it would be very costly.
5)With hardware it appends offset bits to page frame no.

and finally a read occurs for physical address.

So this question is bugging me a lot, how the MMU performs the search for given key(virtual page entry) in page table?

Answer 1

The use of registers for a page table is satisfactory if the page table is reasonably small(for example, 256 entries). Most contemporary computers, however, allow the page table to be very large (for example, 1 million entries). For these machines, the use of fast registers to implement the page table is not feasible. Rather, the page table is kept in main memory, and a page table base register (PTBR) points to the page table. Changing page tables requires changing only this one register, substantially reducing context-switch time.

The problem with this approach is the time required to access a user memory location. If we want to access location i, we must first index into the page table, using the value in the PTBR offset by the page number for i. This task requires a memory access. It provides us with the frame number, which is combined with the page offset to produce the actual address. We can then access the desired place in memory. With this scheme, two memory accesses are needed to access a byte (one for the page-table entry, one for the byte). Thus, memory access is slowed by a factor of 2. This delay would be intolerable under most circumstances. We might as well resort to swapping!

The standard solution to this problem is to use a special, small, fastlookup hardware cache, called a translation look-aside buffer(TLB) . The TLB is associative, high-speed memory. Each entry in the TLB consists of two parts: a key (or tag) and a value. When the associative memory is presented with an item, the item is compared with all keys simultaneously. If the item is found, the corresponding value field is returned. The search is fast; the hardware, however, is expensive. Typically, the number of entries in a TLB is small, often numbering between 64 and 1,024.

Source:Operating System Concepts by Silberschatz et al. page 333