Instructions:
You are encouraged to work on the problem set in groups and turn in one problem set for the entire group. Remember to put all your names on the solution sheet. Also remember to put the name of the TA in whose discussion section you would like the problem set returned to you.

0. If the latency of a DRAM memory bank is 37 cycles, into how many banks would you interleave this memory in order to fully hide this latency when making sequential memory accesses?
1. An ISA supports an 8-bit, byte-addressable virtual address space. The corresponding physical memory has only 128 bytes. Each page contains 16 bytes. A simple, one-level translation scheme is used and the page table resides in physical memory. The initial contents of the frames of physical memory are shown below.

   
   

   Frame 0	empty
   Frame 1	Page 13
   Frame 2	Page 5
   Frame 3	Page 2
   Frame 4	empty
   Frame 5	Page 0
   Frame 6	empty
   Frame 7	Page Table

   
   A three-entry Translation Lookaside Buffer that uses LRU replacement is added to this system. Initially, this TLB contains the entries for pages 0, 2, and 13. For the following sequence of references, put a circle around those that generate a TLB hit and put a rectangle around those that generate a page fault. What is the hit rate of the TLB for this sequence of references? (Note: LRU policy is used to select pages for replacement in physical memory.)

   References (to pages): 0, 13, 5, 2, 14, 14, 13, 6, 6, 13, 15, 14, 15, 13, 4, 3.

   At the end of this sequence, what three entries are contained in the TLB? What are the contents of the 8 physical frames?

2. We have been referring to the LC-3b memory as 2^16 bytes of memory, byte-addressable. This is the memory that the user sees, and may bear no relationship to the actual physical memory. Suppose that the actual physical address space is 8K bytes, and our page size is 512 bytes. What is the size of the PFN? Suppose we have a virtual memory system similar to the VAX in which virtual memory is divided into User Space (P0) and System Space, and System Page Table remains resident in physical memory. System space includes trap vector table, interrupt vector table, operating system and supervisor stack as shown in Figure A.1 in Appendix A. The rest of the address space in Figure A.1 is user space. If each PTE contained, in addition to the PFN, a Valid bit, a modified bit, and two bits of access control, how many bits of physical memory would be required to store the System Page Table?
3. A machine with 64KB, byte addressable virtual memory and 4KB physical memory has two-level virtual address translation similar to the VAX. The page size of this machine is 256 bytes. Virtual address space is partitioned into the P0 space, P1 space, system space and reserved space. The space a virtual address belongs to is specified by the most significant two bits of the virtual address, with 00 indicating P0 space, 01 indicating P1 space, and 10 indicating system space. Assume that the PTE is 32 bits and contains only the Valid bit and the PFN in the format V0000000..000PFN.

   For a single load instruction the physical memory was accessed three times (excluding instruction fetch). The first access was at location x108 and the value read from that location (x108, x109, x10A, x10B) was x80000004. Hint: What does this value mean?

   The second access was at location x45C and the third access was at location x942.

   If SBR = x100, P0BR = x8250 and P1BR = x8350,
   

   1. What is the virtual address corresponding to physical address x45C ?
   2. What is 32 bit value read from location x45C ?
   3. What is the virtual address corresponding to physical address x942 ?


4. Consider a processor that supports a 9-bit physical address space with byte addressable memory. We would like the processor to support a virtual memory system. The features of the virtual memory system are

   
       Virtual Memory Size : 4 Kbytes (12 bit address-space)
       Page Size           : 32 bytes
       PTBR                : 0x380
       SBR                 : 0x1E0
      
   

   The virtual memory is divided into two spaces: system space and user space. System space is the first kilobyte of the virtual address space (i.e., most significant two bits of the virtual address are 00). The rest of the virtual memory is user space. The system page table remains resident in physical memory. Each PTE contains, in addition to the PFN, a Valid bit, a modified bit and 2 bits for access control. The format of the PTE is
   

   Valid

   Modified

   Access Control

   PFN

   (Valid bit is the most significant bit of the PTE and the PFN is stored in the least significant bits.)

   
   
   1. How many virtual pages does the system accommodate?
   2. What is the size of the PFN? How big is the PTE?
   3. How many bytes are required for storing the entire user space page table? How many pages does this correspond to?
      Since the user space page table can occupy a significant portion of the the physical memory, this system uses a 2 level address translation scheme, by storing the user space Page Table in virtual memory (similar to VAX).
   4. Given the virtual address 0x7AC what is the Physical address? The following table shows the contents of the physical memory that you may need to do the translation :

      Address Data x1F8 xBA x1F9 xBB x1FA xBC x1FB xBD x1FC xBE x1FD xB8 x1FE xB7 x1FF xB6

      Address Data x118 x81 x119 x72 x11A x65 x11B x34 x11C x97 x11D x83 x11E xC6 x11F xB2

   
   
5. A computer has an 8KB write-through cache. Each cache block is 64 bits, the cache is 4-way set associative and uses a victim/next-victim pair of bits in each block for its replacement policy. Assume a 24-bit address space and byte-addressable memory. How big (in bits) is the tag store?
6. An LC-3b system ships with a two-way set associative, write back cache with perfect LRU replacement. The tag store requires a total of 4352 bits of storage. What is the block size of the cache? Please show all your work.

   Hint: 4352 = 2^12 + 2^8

7. (Based on Hamacher et al., p. 255, question 5.18) You are working with a computer that has a first level cache that we call L1 and a second level cache that we call L2. Use the following information to answer the questions.
   • The L1 hit rate is 0.95 for instruction references and 0.90 for data references.
   • The L2 hit rate is 0.85 for instruction references and 0.75 for data references.
   • 30% of all instructions are loads and stores.
   • The size of each cache block is 8 words.
   • The time needed to access a cache block in L1 is 1 cycle and the time needed to access a cache block in L2 is 6 cycles.
   • The accesses to the caches and memory are done sequentially. If there is a miss in the L1 and a hit in the L2 then the total latency is 7 cycles.
   • Memory is accessed only if there is a miss in both caches.
   • The width of the memory bus is one word.
   • It takes one clock cycle to send an address to main memory.
   • It takes one clock cycle to send one word from the memory to the processor.
   • The bus allows sending a new address to memory in the same cycle that data is sent from memory to the processor.
   • It takes 20 cycles to access the main memory.
   • Data is only accessible to the processor when the whole cache block has been brought in from the memory. However, the processor does not have to wait for the data to be written into the cache. It can access the data during the cache fill.

   1. What is the average access time per instruction?

   2. What is the average access time per instruction if the main memory is 4-way interleaved?

   3. What is the improvement obtained with interleaving?

8. Below, we have given you four different sequences of addresses generated by a program running on a processor with a data cache. Cache hit ratio for each sequence is also shown below. Assuming that the cache is initially empty at the beginning of each sequence, find out the following parameters of the processor's data cache:

   * Associativity (1, 2, or 4 ways)
   * Block size (1, 2, 4, 8, 16, or 32 bytes)
   * Total cache size (256B, or 512B)
   * Replacement policy (LRU or FIFO)
   

   Assumptions:All memory accesses are one byte accesses. All addresses are byte addresses.
   Address traces
   sequence 1:  0, 2, 4, 8, 16, 32                                                    hit ratio: 0.33
   sequence 2:  0, 512, 1024, 1536, 2048, 1536, 1024, 512, 0      hit ratio: 0.33
   sequence 3:  0, 64, 128, 256, 512, 256, 128, 64, 0                    hit ratio: 0.33
   sequence 4:  0, 512, 1024, 0, 1536, 0, 2048, 512                      hit ratio: 0.25
   

The following problems are meant to help you study for the test. These problems do NOT need to be turned in.



1. You will be given a cache simulator (just the executable) with a hard-coded configuration. Your job is to determine the configuration of the cache. The simulator takes a trace of memory addresses as input and provides a hit ratio as output. Find the following:

   * Associativity (1, 2, 4, or 8 ways)
   * Block size (1, 2, 4, 8, 16, or 32 bytes)
   * Total cache size (256B, 512B, or 1024B)
   * Replacement policy (LRU or Pseudo-LRU)
   

   Show the traces you used to determine each parameter of the cache. Assumptions:All memory accesses are one byte accesses. All addresses are byte addresses.
   Simulator
   The syntax for running the program is:
   %./cachesim   <trace.txt>
   The traces are just text files with one integer memory address per line. For example, the following trace would cause conflict misses in a direct-mapped, 256B cache:
   0
   256
   512
   0
   256
   Simulator for Linux
   Simulator for Solaris
   After downloading the file, please do "chmod 700 cachesim.linux" or "chmod 700 cachesim.solaris".

 
2. The virtual address of variable X is x3456789A. Find the physical address of X.

   Assume a Virtual Memory model similar to VAX.

   Remember that in VAX each Virtual Address consists of 
   
   2  bits to specify the Address Space 
   21 bits to specify Virtual Page Number
   9  bits to specify the byte on the page 
   
   

   You will need to know the contents of P0BR: x8AC40000 and SBR: x000C8000.

   You will also need to know the contents of the following physical memory locations:

   x1EBA6EF0:    x80000A72
   x0022D958:    x800F5D37

   Some intermediate questions to help you:
   • What virtual page of P0 Space is X on?
   • What is VA of the PTE of the page containing X?
   • What virtual page of System Space is this PTE on?
   • What is the PA of the PTE of this page of System Space?
   • What is the PA of the PTE of the page containing X?
3. Let's say we added a virtual memory system to the LC-3b. Which instructions can possibly generate a page fault? What is the maximum number of page faults an instruction can possibly generate while it is being processed? Which instructions can possibly generate that maximum number of page faults?

   Assume that the virtual memory system added uses a one-level translation scheme and the page table is always resident in physical memory.

   An instruction is said to generate a page fault if a page fault occurs at any time during the processing of that instruction.

4. (Hamacher, pg.255, question 5.13)  A byte-addressable computer has a small data cache capable of holding eight 32-bit words. Each cache block consists of one 32-bit word. When a given program is executed, the processor reads data from the following sequence of hex addresses:

        200, 204, 208, 20C, 2F4, 2F0, 200, 204, 218, 21C, 24C, 2F4

   This pattern is repeated four times.

   a. Show the contents of the cache at the end of each pass throughout this loop if a direct-mapped cache is used. Compute the hit rate for this example. Assume that the cache is initially empty.

   b. Repeat part (a) for a fully-associative cache that uses the LRU-replacement algorithm.

   c. Repeat part (a) for a four-way set-associative cache that uses the LRU replacement algorithm.