Working Set Model

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// Example 1: Loop variables exhibit extreme temporal locality
// The variable 'i' is accessed on every iteration
void loop_temporal_locality() {
    int sum = 0;        // Accessed 1001 times (initialization + all iterations)
    for (int i = 0; i < 1000; i++) {  // 'i' accessed ~3000 times
        sum += i;       // sum accessed every iteration
    }
    // During execution:
    // - Instruction at 'for' statement: accessed 1000+ times
    // - 'i' variable: accessed ~3000 times (init, compare, increment)
    // - 'sum' variable: accessed ~1001 times
    // This creates EXTREME temporal locality for these locations
}
 
// Example 2: Function calls create temporal locality in instruction space
void frequently_called_function(int x) {
    // These instructions are accessed every time function is called
    static int call_count = 0;  // Strong temporal locality
    call_count++;
    // Function code exhibits temporal locality through repeated calls
}
 
void caller() {
    for (int i = 0; i < 10000; i++) {
        frequently_called_function(i);  // Code of this function accessed 10000 times
    }
}
 
// Example 3: Working with a small active set of variables
void matrix_multiply(int n, double *A, double *B, double *C) {
    // 'n' is accessed in every loop iteration (temporal locality)
    // Loop indices i, j, k have extreme temporal locality
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            double sum = 0.0;  // sum accessed n times for each (i,j) pair
            for (int k = 0; k < n; k++) {
                sum += A[i*n + k] * B[k*n + j];
            }
            C[i*n + j] = sum;
        }
    }
    // Variables n, i, j, k, sum have MUCH stronger temporal locality
    // than the matrix elements themselves
}

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// Example 1: Excellent spatial locality - sequential access
void sum_array_good(int *arr, int n) {
    long sum = 0;
    for (int i = 0; i < n; i++) {
        sum += arr[i];  // Accesses arr[0], arr[1], arr[2], ...
    }                   // Each access is 4 bytes from the previous
    // Result: Each cache line (64 bytes) serves 16 consecutive accesses
    // Page (4KB) serves 1024 consecutive accesses
}
 
// Example 2: Terrible spatial locality - strided access
void sum_array_bad(int *arr, int n, int stride) {
    long sum = 0;
    for (int i = 0; i < n; i++) {
        sum += arr[i * stride];  // Accesses arr[0], arr[stride], arr[2*stride], ...
    }
    // If stride = 1024, each access is 4KB apart
    // Each access potentially hits a different cache line AND different page
    // Result: Catastrophic cache and TLB miss rates
}
 
// Example 3: Matrix traversal - row vs column order
#define N 1000
double matrix[N][N];  // Row-major storage in C
 
// GOOD: Row-major access matches storage layout
double sum_row_major() {
    double sum = 0;
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            sum += matrix[i][j];  // Consecutive memory access
        }
    }
    return sum;  // ~1 million accesses, ~1000 page accesses if 4KB pages
}
 
// BAD: Column-major access fights storage layout
double sum_column_major() {
    double sum = 0;
    for (int j = 0; j < N; j++) {
        for (int i = 0; i < N; i++) {
            sum += matrix[i][j];  // Stride of N elements = 8000 bytes
        }
    }
    return sum;  // ~1 million accesses, potentially 1 million page accesses!
}
 
// Example 4: Structure layout affects spatial locality
struct BadLayout {
    char flags;           // 1 byte
    double values[1000];  // 8000 bytes
    char more_flags;      // 1 byte
};
// If you iterate over array of BadLayouts accessing only flags,
// you jump 8002 bytes between each access - terrible spatial locality
 
struct GoodLayout {
    char flags;
    char more_flags;
    // padding to alignment
    double values[1000];
};
// If struct array designed for access pattern, spatial locality improves

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# Conceptual implementation of locality metrics
 
class LocalityAnalyzer:
    """Analyze memory access trace for locality characteristics."""
    
    def __init__(self, page_size=4096):
        self.page_size = page_size
        self.trace = []  # List of (timestamp, address)
        
    def record_access(self, timestamp, address):
        """Record a memory access."""
        self.trace.append((timestamp, address))
    
    def compute_working_set(self, window_size):
        """
        Compute working set size over time with given window.
        Returns list of (time, working_set_size) pairs.
        
        This directly implements Denning's working set definition:
        W(t, Δ) = {pages accessed in interval [t-Δ, t]}
        """
        results = []
        for i, (t, addr) in enumerate(self.trace):
            # Find all accesses in window [t - window_size, t]
            window_start = t - window_size
            pages_in_window = set()
            
            for j in range(i, -1, -1):
                access_time, access_addr = self.trace[j]
                if access_time < window_start:
                    break
                page = access_addr // self.page_size
                pages_in_window.add(page)
            
            results.append((t, len(pages_in_window)))
        
        return results
    
    def compute_reuse_distance(self):
        """
        Compute reuse distance distribution.
        Reuse distance = number of distinct addresses accessed since last use.
        
        Lower reuse distances indicate stronger temporal locality.
        """
        last_access = {}  # address -> trace position
        distinct_since = {}  # address -> set of addresses since last access
        reuse_distances = []
        
        access_stack = []  # Stack to track unique accesses
        
        for i, (t, addr) in enumerate(self.trace):
            page = addr // self.page_size
            
            if page in last_access:
                # Find how many distinct pages accessed since last use
                # This is the reuse distance (LRU stack depth)
                if page in access_stack:
                    distance = access_stack.index(page)
                    reuse_distances.append(distance)
                    access_stack.remove(page)
            else:
                reuse_distances.append(float('inf'))  # First access
            
            access_stack.insert(0, page)  # Move to top of stack
            last_access[page] = i
        
        return reuse_distances
    
    def compute_stride_distribution(self):
        """
        Compute distribution of strides (distances between consecutive accesses).
        
        Concentration near zero indicates strong spatial locality.
        """
        strides = []
        for i in range(1, len(self.trace)):
            prev_addr = self.trace[i-1][1]
            curr_addr = self.trace[i][1]
            stride = abs(curr_addr - prev_addr)
            strides.append(stride)
        
        return strides
    
    def summarize_locality(self):
        """Generate locality summary statistics."""
        strides = self.compute_stride_distribution()
        reuse = self.compute_reuse_distance()
        
        # Spatial locality: fraction of strides within one cache line
        cache_line = 64
        spatial_score = sum(1 for s in strides if s < cache_line) / len(strides)
        
        # Temporal locality: fraction of reuses within small distance
        finite_reuse = [r for r in reuse if r != float('inf')]
        if finite_reuse:
            temporal_score = sum(1 for r in finite_reuse if r < 100) / len(finite_reuse)
        else:
            temporal_score = 0
        
        return {
            'spatial_locality_score': spatial_score,  # 0-1, higher = better
            'temporal_locality_score': temporal_score,  # 0-1, higher = better
            'average_stride': sum(strides) / len(strides),
            'median_reuse_distance': sorted(finite_reuse)[len(finite_reuse)//2] if finite_reuse else None,
            'unique_pages': len(set(addr // self.page_size for _, addr in self.trace))
        }

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# Demonstration: Locality strength determines working set effectiveness
 
import random
from collections import defaultdict
 
class WorkingSetSimulator:
    """
    Simulate working set model effectiveness under different locality conditions.
    """
    
    def __init__(self, num_pages, memory_frames, window_delta):
        self.num_pages = num_pages
        self.memory_frames = memory_frames
        self.window_delta = window_delta
        self.access_history = []  # Circular buffer of recent page accesses
        self.resident_pages = set()
        self.page_faults = 0
        self.total_accesses = 0
        
    def generate_high_locality_trace(self, length, hot_pages=10):
        """
        Generate trace with strong temporal locality.
        Most accesses go to a small set of 'hot' pages.
        """
        hot_set = list(range(hot_pages))
        trace = []
        for _ in range(length):
            if random.random() < 0.9:  # 90% accesses to hot pages
                trace.append(random.choice(hot_set))
            else:  # 10% accesses to random pages
                trace.append(random.randint(0, self.num_pages - 1))
        return trace
    
    def generate_low_locality_trace(self, length):
        """
        Generate trace with poor temporal locality.
        Accesses uniformly distributed across all pages.
        """
        return [random.randint(0, self.num_pages - 1) for _ in range(length)]
    
    def compute_working_set(self, current_time):
        """Compute working set from recent history."""
        working_set = set()
        window_start = max(0, current_time - self.window_delta)
        
        for t in range(window_start, current_time + 1):
            if t < len(self.access_history):
                working_set.add(self.access_history[t])
        
        return working_set
    
    def access_page(self, page, current_time):
        """Process a page access using working set policy."""
        self.total_accesses += 1
        self.access_history.append(page)
        
        # Check if page is resident
        if page not in self.resident_pages:
            self.page_faults += 1
            
            # Compute current working set
            ws = self.compute_working_set(current_time)
            
            # Make room if needed
            if len(self.resident_pages) >= self.memory_frames:
                # Evict pages not in working set
                pages_to_evict = self.resident_pages - ws
                if pages_to_evict:
                    # Remove one page not in working set
                    self.resident_pages.remove(pages_to_evict.pop())
                else:
                    # All pages are in working set - need more memory or evict LRU
                    # (In true working set model, this indicates insufficient memory)
                    oldest = min(self.resident_pages, 
                                key=lambda p: self.access_history[::-1].index(p) 
                                if p in self.access_history else float('inf'))
                    self.resident_pages.remove(oldest)
            
            self.resident_pages.add(page)
    
    def simulate(self, trace):
        """Run simulation on given trace."""
        for t, page in enumerate(trace):
            self.access_page(page, t)
        
        return {
            'page_faults': self.page_faults,
            'total_accesses': self.total_accesses,
            'fault_rate': self.page_faults / self.total_accesses,
            'final_working_set_size': len(self.compute_working_set(len(trace) - 1))
        }
 
# Compare high vs low locality workloads
def compare_locality_impact():
    """
    Demonstrate that working set model works well BECAUSE of locality.
    """
    print("Comparing Working Set Effectiveness vs Locality Strength")
    print("=" * 60)
    print(f"Pages: 1000, Memory Frames: 50, Window: 100 accesses")
    print(f"Trace length: 10000 accesses")
    print()
    
    # High locality workload
    sim_high = WorkingSetSimulator(num_pages=1000, memory_frames=50, window_delta=100)
    trace_high = sim_high.generate_high_locality_trace(10000, hot_pages=20)
    result_high = sim_high.simulate(trace_high)
    
    # Low locality workload
    sim_low = WorkingSetSimulator(num_pages=1000, memory_frames=50, window_delta=100)
    trace_low = sim_low.generate_low_locality_trace(10000)
    result_low = sim_low.simulate(trace_low)
    
    print("HIGH LOCALITY (90% to 20 hot pages):")
    print(f"  Page faults: {result_high['page_faults']}")
    print(f"  Fault rate: {result_high['fault_rate']:.4f}")
    print()
    
    print("LOW LOCALITY (uniform random):")
    print(f"  Page faults: {result_low['page_faults']}")
    print(f"  Fault rate: {result_low['fault_rate']:.4f}")
    print()
    
    print("CONCLUSION:")
    print(f"  High locality fault rate is {result_low['fault_rate']/result_high['fault_rate']:.1f}x better")
    print(f"  Working set model effectiveness DEPENDS on locality presence!")
 
# Output example:
# HIGH LOCALITY: Fault rate ~0.02 (2%)
# LOW LOCALITY: Fault rate ~0.95 (95%)  
# Working set works because locality exists!