3.2 Single-Server and Multi-Server Models

Queuing theory helps us understand how lines form and move. Single-server models, like a lone cashier, are simpler than multi-server setups with parallel workers. These models use math to predict wait times and line lengths.

The key difference is how they handle multiple customers. Single-server models focus on one worker's efficiency, while multi-server models balance workload across staff. Understanding both helps businesses optimize their operations and keep customers happy.

Single vs Multi-Server Queuing Models

Fundamental Differences and Applications

• Single-server queuing models involve one service facility with a single server, while multi-server models have multiple servers working in parallel
• M/M/1 model represents the basic single-server queuing system, where M denotes Markovian (exponential) interarrival and service times, and 1 represents a single server
• M/M/c model embodies the fundamental multi-server queuing system, where c represents the number of parallel servers
• Single-server models typically apply to simple systems (single checkout counter), while multi-server models represent more complex systems (call centers, multi-lane toll booths)
• Mathematical formulations for performance measures differ between single-server and multi-server models, particularly in terms of waiting times and queue lengths
• Utilization factor calculations vary:
  • Single-server: $ρ = λ/μ$
  • Multi-server: $ρ = λ/(cμ)$
  • λ represents arrival rate, μ denotes service rate, and c signifies the number of servers

Performance Measure Comparisons

• Average number in the system (L) calculation differs:
  • M/M/1: $L = λ/(μ-λ)$
  • M/M/c: More complex formula involving Erlang C function
• Average waiting time in queue (Wq) calculation varies:
  • M/M/1: $Wq = ρ/(μ-λ)$
  • M/M/c: Involves probability of waiting and Erlang C function
• System stability conditions differ:
  • M/M/1: Stable when $ρ < 1$
  • M/M/c: Stable when $ρ < c$ (utilization per server less than 1)
• Probability of zero customers in the system (P0) formulas are distinct:
  • M/M/1: $P0 = 1 - ρ$
  • M/M/c: More complex expression involving summation and factorial terms

Applying M/M/1 and M/M/c Models

M/M/1 Model Application

• M/M/1 model requires knowledge of arrival rate (λ) and service rate (μ) to calculate key performance measures
• Essential formulas for the M/M/1 model include:
  • Utilization factor: $ρ = λ/μ$
  • Average number in the system: $L = λ/(μ-λ)$
  • Average number in the queue: $Lq = ρ²/(1-ρ)$
  • Average time in the system: $W = 1/(μ-λ)$
  • Average waiting time in the queue: $Wq = ρ/(μ-λ)$
• Probability of n customers in the system: $Pn = (1 - ρ)ρ^n$
• Probability of waiting: $Pw = ρ$ (same as utilization factor in M/M/1)
• Application example: Analyzing a single-server coffee shop with customer arrivals every 5 minutes (λ = 0.2/min) and service time of 4 minutes (μ = 0.25/min)

M/M/c Model Application

• M/M/c model requires knowledge of arrival rate (λ), service rate (μ), and number of servers (c)
• Key formulas for the M/M/c model include more complex expressions involving the Erlang C formula for the probability of waiting
• Erlang C formula: $C(c,ρ) = \frac{(cρ)^c}{c!(1-ρ)} / [\sum_{n=0}^{c-1} \frac{(cρ)^n}{n!} + \frac{(cρ)^c}{c!(1-ρ)}]$
• Probability of waiting: $Pw = C(c,ρ)$
• Average number in the queue: $Lq = \frac{C(c,ρ)ρ}{c(1-ρ)}$
• Average waiting time in the queue: $Wq = \frac{C(c,ρ)}{cμ(1-ρ)}$
• Application example: Analyzing a call center with 5 agents, calls arriving every 2 minutes (λ = 0.5/min), and average call duration of 8 minutes (μ = 0.125/min)

Model Assumptions and Limitations

• Both models assume Poisson arrival processes, exponential service times, and first-come-first-served queue discipline
• Limitations include:
  • Assumption of unlimited queue capacity
  • No consideration of customer balking or reneging
  • Assumes steady-state conditions
• Real-world applications may require adjustments or more complex models to account for these limitations
• Sensitivity analysis helps assess model robustness to violations of assumptions

System Parameters Impact on Queuing Performance

Utilization Factor and System Stability

• Utilization factor (ρ) critically affects all performance measures in both M/M/1 and M/M/c models
• As ρ approaches 1, queue lengths and waiting times increase exponentially, indicating system instability
• Impact of ρ on key performance measures:
  • Average queue length (Lq) grows non-linearly as ρ increases
  • Probability of waiting (Pw) approaches 1 as ρ nears 1
  • Average waiting time (Wq) becomes very large as ρ approaches 1
• Example: In an M/M/1 system, as ρ increases from 0.5 to 0.9, Lq increases from 0.5 to 8.1 customers

Arrival and Service Rates

• Arrival rate (λ) and service rate (μ) have inverse effects on system performance
• Increasing λ degrades performance:
  • Longer queue lengths
  • Increased waiting times
  • Higher system utilization
• Increasing μ improves performance:
  • Shorter queue lengths
  • Reduced waiting times
  • Lower system utilization
• Trade-off between service speed and quality must be considered when adjusting μ
• Example: In an M/M/c system with 3 servers, doubling λ from 10 to 20 customers/hour while keeping μ constant at 8 customers/hour/server increases Wq from 0.05 to 0.33 hours

Number of Servers and System Variability

• In M/M/c models, increasing the number of servers (c) generally improves system performance, but with diminishing returns
• Impact of adding servers:
  • Reduces average waiting time (Wq)
  • Decreases probability of waiting (Pw)
  • Lowers overall system utilization (ρ)
• Coefficient of variation of interarrival and service times affects model accuracy
• Higher variability leads to poorer performance than predicted by M/M/1 and M/M/c models
• Example: In an M/M/c system with λ = 20 customers/hour and μ = 8 customers/hour/server, increasing c from 3 to 4 reduces Wq from 0.33 to 0.08 hours

Optimal Server Number in Multi-Server Systems

Economic Analysis and Cost Functions

• Optimal number of servers balances the cost of providing service with the cost of customer waiting time or lost business
• Economic analysis involves calculating the total cost function, typically including:
  • Server costs (e.g., wages, equipment)
  • Waiting costs (e.g., customer dissatisfaction, lost sales)
• Total cost function: $TC(c) = csCs + λWq(c)Cw$
  • c: number of servers
  • Cs: cost per server per unit time
  • Cw: waiting cost per customer per unit time
  • λ: arrival rate
  • Wq(c): average waiting time in queue as a function of c
• Decision variable in optimization: number of servers (c), usually constrained to be a positive integer
• Example: A retail store with Cs = $20/hour, Cw = $15/hour, λ = 30 customers/hour, μ = 10 customers/hour/server

Optimization Techniques

• Marginal analysis finds the optimal number of servers by comparing the marginal benefit of adding a server to its marginal cost
• Steps in marginal analysis:
  1. Calculate total cost for c and c+1 servers
  2. If TC(c+1) < TC(c), increase c
  3. Repeat until TC(c+1) > TC(c)
• Queuing cost models often exhibit a convex total cost curve, with the optimal number of servers at the minimum point
• Graphical method: Plot total cost against number of servers to visually identify the minimum point
• Integer programming techniques may be employed for more complex scenarios with additional constraints

Sensitivity Analysis and Practical Considerations

• Sensitivity analysis assesses how the optimal solution changes with variations in:
  • Cost parameters (Cs and Cw)
  • Arrival rates (λ)
  • Service rates (μ)
• Techniques for sensitivity analysis:
  • One-way analysis: Vary one parameter while holding others constant
  • Two-way analysis: Examine interactions between two changing parameters
• Practical factors influencing the final decision on server numbers:
  • Space constraints in physical queuing systems
  • Labor regulations and shift scheduling
  • Service level agreements and customer satisfaction targets
• Example: Analyzing how the optimal number of servers changes when Cw increases from $15/hour to $25/hour, reflecting higher customer value