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Queueing Theory
An Introduction


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Motivation
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  • A Personal Experience:
    • I went to the store at 5:30
    • There was an enormous line at each cashier
  • What Caused the Lines?
    • 200 people left work at 5:00 and went to the store
    • There were 5 cashiers working
    • Each cashier can check-out 50 people per hour
    • At 5:30, only 125 people had checked out
Motivation (cont.)
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queueing_deterministic
Motivation (cont.)
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  • What Caused the Line?
    • The grocery store didn't have "enough" cashiers
  • Is This the Only Cause of Lines?
    • No! In fact, it's the unimportant case.
Motivation (cont.)
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  • Another Personal Experience:
    • The restaurant had 60 tables
    • The average group took an hour to eat
    • So, a table was coming available about every minute
    • People were arriving about every 90 seconds
    • There was a line of people waiting
  • What Caused the Lines?
    • Randomness
Motivation (cont.)
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queueing_arrivals-and-departures
A Conceptual Model
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queueing_system

\(\lambda\) denotes the arrival rate
\(\mu\) denotes the service rate

Going Further: Modeling State Transitions in Queues
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  • Visualizing Transitions:
    • queueing_transitions
  • A Steady State:
    • \(\lambda \cdot P\{Q(t) = j \} = \mu \cdot P\{Q(t) = j+1 \}\) for all \(j\)
Going Further: Modeling Steady States in Queues
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  • Simplifying the Notation:
    • We can ignore \(t\) since we are in a steady state
    • So, let \(P_j\) denote the probability that there are \(j\) people in the system when it has "settled down"
  • Stability Revisited:
    • \(P_{j+1} = \frac{\lambda}{\mu} P_j\) for all \(j\)
Going Further: Modeling Steady States in Queues
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  • Assumptions about Interarrival Times
    • :
    • Interarrival times are identically and independently distributed
    • Interarrival times are exponentially distributed (i.e., have the memoryless property)
  • Assumptions about Service Times
    • :
    • Service times are identically and independently distributed
    • Service times are exponentially distributed (i.e., have the memoryless property)
Going Further: Modeling Steady States in Queues (cont.)
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  • Some Particular Cases:
    • \(P_{1} = \frac{\lambda}{\mu} P_0\)
    • \(P_{2} = \frac{\lambda}{\mu} ( \frac{\lambda}{\mu} P_0 ) = ( \frac{\lambda}{\mu} )^2 P_0\)
    • \(P_{n} = ( \frac{\lambda}{\mu} )^n P_0\)
  • Letting \(\rho = \lambda / \mu\):
    • \(P_{n} = \rho^n P_0\)
Going Further: Modeling Steady States in Queues (cont.)
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  • What About \(P_0\)?
    • We know the probabilities must sum to 1
    • So, \(P_0 + P_1 + P_2 + P_3 + \cdots = 1\)
  • Solving:
    • \(P_0 + \rho P_0 + \rho^2 P_0 + \rho^3 P_0 + \cdots = 1\)
    • \(\Rightarrow P_0 (1 + \rho + \rho^2 + \rho^3 + \cdots) = 1 \)
    • \(\Rightarrow P_0 = \frac{1}{(1 + \rho + \rho^2 + \rho^3 + \cdots)} \)
    • We know the infinite geometric series \((1 + \rho + \rho^2 + \rho^3 + \cdots)\) equals \(1/(1-\rho)\) since \(\rho \lt 1\)
    • Hence, in such cases (i.e., when the arrival rate is less than the service rate) \(P_0 = (1 - \rho)\) and \(P_n = (1 - \rho) \cdot \rho^n\)
Going Further: Modeling Steady States in Queues (cont.)
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  • Expected Size of the System:
    • \(0 \cdot P_0 + 1 \cdot P_1 + 2 \cdot P_2 + \cdots = \rho / (1 - \rho) = \lambda / (\mu - \lambda)\)
  • Expected Time in the System:
    • The expected number of people in the system must equal the arrival rate times the expected time in the system
    • So, the expected time is the expected size divided by the arrival rate
    • So, the expected time is \(1 / (\mu - \lambda)\)
Summary of the Important Results
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  • Probability of \(n\) People in the Queue:
    • \(P_0 = (1 - \lambda/\mu)\)
    • \(P_n = (1 - \lambda/\mu) \cdot (\lambda/\mu)^n\)
  • Expected Size of the System:
    • \(\lambda / (\mu - \lambda)\)
  • Expected Time in the System:
    • \(1 / (\mu - \lambda)\)
An Example
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  • The Data:
    • Arrival rate is 30 per hour
    • Service rate is 40 per hour
  • The Expected Steady State Values:
    • Size: 30/(40-30) = 3 people
    • Time: 1/(40-30) = 1/10 hours
Designing Queueing Systems
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  • Service Mechanism:
    • Number of servers
    • Number of waiting lines (one common line, one line for each server)
    • Service rate (if it can be controlled)
  • Queueing Discipline:
    • First-come-first-served
    • Last-come-first-served
    • Priority (or preemptive) service
    • Shortest remaining processing time
There's Always More to Learn
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