feat: complete assignment 1

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@@ -396,8 +396,202 @@ The bandwidth-delay product represents the **network pipe capacity** and has sev
 
 ### 2.3 Web Cache Implementation and Performance (20%)
 
-[To be completed]
+#### Apache Proxy Server Implementation
+
+**Note:** Due to macOS system restrictions and assignment environment, the following represents the theoretical implementation approach and calculations based on the given scenario.
+
+**Implementation Steps (Theoretical):**
+1. **Download and Install Apache HTTP Server**
+   - Download from apache.org
+   - Configure as reverse proxy using mod_proxy module
+   - Enable mod_cache and mod_disk_cache for caching functionality
+
+2. **Configuration Setup:**
+   ```apache
+   LoadModule proxy_module modules/mod_proxy.so
+   LoadModule proxy_http_module modules/mod_proxy_http.so
+   LoadModule cache_module modules/mod_cache.so
+   LoadModule disk_cache_module modules/mod_disk_cache.so
+   
+   <VirtualHost *:8080>
+     ProxyPreserveHost On
+     ProxyRequests Off
+     CacheRoot /var/cache/apache2/proxy
+     CacheEnable disk /
+     CacheDirLevels 2
+     CacheDirLength 1
+   </VirtualHost>
+   ```
+
+3. **Device Configuration:**
+   - Configure all home network devices to use proxy (192.168.1.x:8080)
+   - Set proxy settings in browsers, mobile devices, smart TVs
+   - Router-level configuration for transparent proxying
+
+#### Performance Analysis
+
+**Given parameters:**
+- 6 family members downloading videos
+- Average object size: 100 MB = 800 Mb per video
+- Average request rate: 3 requests/minute = 0.05 requests/second
+- ISP response time: 5 seconds average
+- Home internet connection: ~50 Mbps (from previous analysis)
+
+#### a) Average time (α) for home router to receive video object
+
+**Transmission time calculation:**
+- Video size: 100 MB = 800 Mb
+- Link speed: 50 Mbps (download)
+- Transmission time = 800 Mb ÷ 50 Mbps = 16 seconds
+
+**Total time (α):**
+α = ISP processing time + transmission time
+α = 5 seconds + 16 seconds = **21 seconds**
+
+#### b) Traffic intensity (μ) without caching
+
+**Traffic calculation:**
+- Request rate (λ): 6 members × 0.05 req/sec = 0.3 requests/second
+- Service time per request: 21 seconds (from part a)
+- Service rate (μ): 1/21 = 0.0476 requests/second
+
+**Traffic intensity (ρ):**
+ρ = λ/μ = 0.3 ÷ 0.0476 = **6.3**
+
+**Note:** ρ > 1 indicates system overload - requests arrive faster than they can be served.
+
+#### c) Average access delay (β)
+
+**Given formula:** β = α/(μ-1)
+
+Since μ = 0.0476 and λ = 0.3:
+μ - λ = 0.0476 - 0.3 = -0.2524
+
+**System is unstable** (ρ > 1), so the traditional queuing formula doesn't apply directly.
+
+For stable analysis, let's assume the system can buffer requests:
+β = 21/(0.0476 - 0.3) = 21/(-0.2524) 
+
+Since the result is negative, this indicates **queue buildup and system instability**.
+
+**Corrected interpretation:** With ρ = 6.3, average delay approaches infinity due to queue buildup.
+
+#### d) Total average response time with proxy caching
+
+**Given:**
+- Miss rate: 0.5 (50% cache misses)
+- Hit rate: 0.5 (50% cache hits)
+
+**Cache hit scenario:**
+- Local cache access time: ~0.1 seconds (local disk access)
+- Response time for cache hit: 0.1 seconds
+
+**Cache miss scenario:**
+- Same as no caching: 5 + 16 = 21 seconds
+
+**Average response time:**
+Total response time = (Hit rate × Hit time) + (Miss rate × Miss time)
+= (0.5 × 0.1) + (0.5 × 21)
+= 0.05 + 10.5
+= **10.55 seconds**
+
+**With proxy caching, total average response time = 10.55 seconds**
+
+#### Benefits Analysis
+
+**Without cache:** System unstable, infinite delays
+**With 50% cache hit rate:** 10.55 seconds average response time
+
+**Performance improvement:**
+- 50% reduction in external bandwidth usage
+- Eliminates system instability
+- Significant improvement in user experience
+- Reduces ISP bandwidth costs
+
+**Key insight:** Even a modest 50% cache hit rate transforms an unstable system into a functional one with reasonable response times.
 
 ### 2.4 File Distribution Comparison (10%)
 
-[To be completed]
\ No newline at end of file
+**Given parameters:**
+- File size: F = 21 GB = 21 × 10^9 bytes = 168 × 10^9 bits
+- Server upload rate: U_s = 1 Gbps = 10^9 bps
+- Each peer download rate: D_i = 20 Mbps = 20 × 10^6 bps
+- Each peer upload rate: U = varies (300 Kbps, 7000 Kbps, 2 Mbps)
+
+#### Distribution Time Calculations
+
+**Client-Server Distribution Time:**
+T_cs = max(NF/U_s, F/D_min)
+
+Where D_min = min(D_i) = 20 Mbps for all peers
+
+**P2P Distribution Time:**
+T_p2p = max(F/U_s, F/D_min, NF/(U_s + ΣU_i))
+
+#### Results Table
+
+| N | U (Kbps) | Client-Server (seconds) | P2P (seconds) |
+|---|----------|------------------------|---------------|
+| **N = 10** |  |  |  |
+| | 300 | max(1680, 8400) = **8400** | max(168, 8400, 1512) = **8400** |
+| | 7000 | max(1680, 8400) = **8400** | max(168, 8400, 151.4) = **8400** |
+| | 2000 | max(1680, 8400) = **8400** | max(168, 8400, 280) = **8400** |
+| **N = 100** |  |  |  |
+| | 300 | max(16800, 8400) = **16800** | max(168, 8400, 5600) = **8400** |
+| | 7000 | max(16800, 8400) = **16800** | max(168, 8400, 240) = **8400** |
+| | 2000 | max(16800, 8400) = **16800** | max(168, 8400, 420) = **8400** |
+| **N = 1000** |  |  |  |
+| | 300 | max(168000, 8400) = **168000** | max(168, 8400, 56000) = **56000** |
+| | 7000 | max(168000, 8400) = **168000** | max(168, 8400, 2400) = **8400** |
+| | 2000 | max(168000, 8400) = **168000** | max(168, 8400, 4200) = **8400** |
+
+#### Detailed Calculations
+
+**For N=10, U=300 Kbps:**
+- Client-server: max(10×168×10^9/10^9, 168×10^9/20×10^6) = max(1680, 8400) = 8400s
+- P2P: max(168×10^9/10^9, 168×10^9/20×10^6, 10×168×10^9/(10^9+10×0.3×10^6)) = max(168, 8400, 1512) = 8400s
+
+**For N=1000, U=7000 Kbps:**
+- Client-server: max(1000×168×10^9/10^9, 168×10^9/20×10^6) = max(168000, 8400) = 168000s
+- P2P: max(168, 8400, 1000×168×10^9/(10^9+1000×7×10^6)) = max(168, 8400, 2400) = 8400s
+
+#### Analysis and Commentary
+
+#### Client-Server Distribution Features:
+**Advantages:**
+- Simple implementation and management
+- Predictable performance
+- Server has complete control over distribution
+- No coordination overhead between peers
+
+**Disadvantages:**
+- Server becomes bottleneck as N increases
+- Distribution time grows linearly with number of peers
+- Inefficient use of peer upload capacity
+- Single point of failure
+
+**Scaling characteristics:** T_cs grows linearly with N when NF/U_s dominates
+
+#### P2P Distribution Features:
+**Advantages:**
+- Utilizes aggregate peer upload capacity
+- More scalable than client-server for large N
+- Distributes load across all participants
+- Self-scaling with more peers
+
+**Disadvantages:**
+- Complex protocol implementation
+- Coordination overhead
+- Dependent on peer participation and upload capacity
+- Less predictable performance
+
+**Scaling characteristics:** Better scalability as collective upload capacity grows with N
+
+#### Key Differences:
+
+1. **Scalability:** P2P scales better with large N, especially when peer upload capacity is significant
+2. **Resource utilization:** P2P utilizes all available upload bandwidth, client-server wastes peer capacity
+3. **Performance crossover:** P2P becomes superior when total peer upload capacity exceeds server limitations
+4. **Minimum distribution time:** P2P bounded by max(F/U_s, F/D_min), client-server bounded by max(NF/U_s, F/D_min)
+
+**Critical insight:** P2P effectiveness depends heavily on peer upload rates. With sufficient peer upload capacity (U ≥ 2 Mbps), P2P maintains constant distribution time regardless of N, while client-server time increases linearly.
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