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What is queuing theory?

     The mathematical approach to the analysis of lines

     Useful in planning and analysis of service capacity

     Goal of queuing -- minimize total cost - costs associated with customers waiting in line
      for service and those associated with capacity

System Characteristics

  1) Population source
       o Infinite source
       o Finite source

   Number of servers (channels)

   Arrival and service patterns
             Probability distribution (exponential, Poisson, etc)

   Queue discipline (order of service)
    Four Common Variations of Queue Systems

    Single channel,
    single phase

 Single channel,
 multiple phase

 Multiple channel,
 single phase

Multiple channel,
multiple phase
Queuing Models:

  Infinite Sources
       Assumptions:
            o Poisson arrival rate
            o System operates under steady state (average arrival and service rates are

            Important note: The arrival () and service rates () must be in the same units

     Four Basic Models
           1)    Single channel, exponential service time
           2)    Single channel, constant service time
           3)    Multiple channel, exponential service time
           4)    Multiple priority service, exponential service time

  Finite Source

     (2) Appropriate for cases in which the calling population is limited to a relatively small
         number of potential calls
     (3) Example -- one person may be responsible for handling breakdown on 15 machines
     (4) The mathematics of finite-source model can be complex, analysts often use finite
         queuing tables in conjunction with simple formulas to analyze these systems
Important Note: To solve queuing problems use the Excel templates that accompany
the text

Five Typical Measures of System Performance
Operations Managers Look at

     1)       Average number of customers waiting (in line or in system)
     2)       Average time customers wait (in line or system)
     3)       System utilization (percentage of capacity used)
     4)       Implied cost of given level of capacity and its related waiting line
     5)       The probability that an arrival will have to wait for service

Infinite-source Symbols

           Customer arrival rate
           Service rate
          LQ The average number of customer waiting for service
          LS The average number of customers in the system
           The system utilization
       Wq The average time customers wait in line
       Ws The average time customers spend in the system
               Service time
          P0 The probabilit y of zero units in the system
          Pn The probabilit y of n units in the system
          M The nunber of servers (channels)
          Lmax The maximum expected number waiting in line
           Problem 1 (808)

   Repair calls are handled by one repairman at a photocopy shop. Repair time, including travel time, is
   exponentially distributed, with a mean of two hours per call. Requests for copier repairs come in at a mean
   rate of three per 8-hour day (assume Poisson).


   (a) The average number of customers awaiting repairs.

   (b) System utilization

   (c) The amount of time during an 8-hour day that the repairman is not out on call

   (d) The probability of two or more customers in the system.

Excel Solution
       Problem 2 (808)

A vending machine dispenses hot chocolate or coffee. Service time is 30 seconds per cup and is constant.
Customers arrive at a mean rate of 80 per hour, and this rate is Poisson distributed. Determine:

(a) The average number of customer waiting in line

(b) The average time customers spend in the system

(c) The average number in the system

Excel Solution
    Problem 4 (809)

A small town with one hospital has two ambulances to supply ambulance service. Requests for ambulances
during non-holiday weekends average 0.8 per hour and tend to be Poisson distributed. Travel and assistance
time averages one hour per call and follows an exponential distribution. Find:

(a) System utilization

(b) The average number of customers waiting

(c) The average time customers wait for an ambulance

(d) The probability that both ambulances will be busy when a call comes in

Excel Solution
   Problem 10 (810)

   Two operators handle adjustments for a group of 10 machines. Adjustment time is exponentially distributed
   and has a mean of 14 minutes per machine. The machines operate for an average of 86 minutes between
   adjustments. While running, each machine can turn out 50 pieces per hour. Find:

   (a) The probability that a machine will have to wait for an adjustment

   (b) The average number of machines waiting for adjustment

   (c) The average number of machines being serviced

   (d) The expected hourly output of each machine, taking adjustments into account

   (e) Machine downtime represents a cost of $70 per hour; operator cost (including salary and fringe benefits)
       is $15 per hour. What is the optimum number of operators?

Excel Solution

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