Service Operations and Waiting Lines Dr. Everette S. Gardner, Jr.

Waiting Lines2 Case study: Single-server model Reference Vogel, M. A., “Queuing Theory Applied to Machine Manning,” Interfaces, Aug. 79. Company Becton - Dickinson, mfg. of hypodermic needles and syringes Bottom line Cash savings = $575K / yr. Also increased production by 80%. Problem High-speed machines jammed frequently. Attendants cleared jams. How many machines should each attendant monitor? Model Basic single-server: Server—Attendant Customer—Jammed machine

Waiting Lines3 Case study (cont.) Solution procedure Each machine jammed at rate of λ = 60/hr. With M machines, arrival rate to each attendant is λ = 60M Service rate is μ = 450/hr. Utilization ratio = 60M/450 Experimenting with different values of M produced an arrival rate that minimized costs (wages + lost production) M = 5 was optimal, compared to M = 1 before queuing study

Waiting Lines4 Case study: Multiple-server model Reference Deutch, H. and Mabert, V. A., “Queuing Theory Applied to Teller Staffing,” Interfaces, Oct., Company Bankers Trust Co. of New York Bottom line Annual cash savings of $1,000,000 in reduced wages. Cost to develop model of $110,000. Problem Determine number of tellers to be on duty per hour of day to meet goals for waiting time. Staffing decisions needed at 100 branch banks. Model Straightforward application of multi-channel model in text.

Waiting Lines5 Case study (cont.) Analysis Development of arrival and service distributions by hour and day of week at each bank. Arrival and service shown to be Poisson / Exponential. Experimentation with number of servers in model showed that full-time tellers were idle much of the day. Result Elimination of 100 full-time tellers. Increased use of part-time tellers. Today, the multi-channel model is a standard tool for staffing decisions in banking.

Waiting Lines6 Queuing model structures Single-server model Pop. Arrival Queue Service time can be ratecapacity can usually exp., finite or must be be finite but can be infinite Poisson or infinite anything Source pop. Service facility

Waiting Lines7 Queuing model structures (cont.) Multiple-server model Pop. Arrival Queue must be rate capacity infinite must be must be Poisson infinite Service time for each Note: There is only one queue server must regardless of nbr. of servers have same mean and be exp. Source pop. Service facility #1 Service facility #2

Waiting Lines8 Applying the single-server model 1. Analyze service times. - plot actual vs. exponential distribution - if exponential good fit, use it - otherwise compute σ of times 2. Analyze arrival rates. - plot actual vs. Poisson Distribution - if Poisson good fit, use it - if not, stop—only alternative is simulation 3. Determine queue capacity. - infinite or finite? - if uncertain, compare results from alternative models

Waiting Lines9 Applying the single-server model (cont.) 4. Determine size of source population. - infinite or finite? - if uncertain, compare results from alternative models 5. Choose model from SINGLEQ worksheet. SINGLEQ.xls

Waiting Lines10 Applying the multiple-server model 1. Analyze service times. - Must be exponential 2. Analyze arrival rates. - Must be Poisson 3. Queue capacity must be infinite. 4. Source population must be infinite. 5. Apply MULTIQ worksheet. MULTIQ.xls

Waiting Lines11 Single-server equations Arrival rate= λ Service rate= μ Mean number in queue= λ 2 /(μ(μ-λ)) Mean number in system = λ /(μ-λ) Mean time in queue = λ /(μ(μ-λ)) Mean time in system= 1/(μ-λ) Utilization ratio= λ /μ (Prob. server is busy) SINGLEQ.xls

Waiting Lines12 Utilization ratio vs. queue length λ μ λ/μ Queue length people ,  SINGLEQ.xls

Waiting Lines13 Single-server queuing identities A. Number units in system = arrival rate * mean time in system B. Number units in queue = arrival rate * mean time in queue C. Mean time in system = mean time in queue + mean service time Note: Mean service time = 1/ mean service rate If we can determine only one of the following, all other values can be found by substitution: Number units in system or queue Mean time in system or queue

Waiting Lines14 State diagram: single-server model A A A # in system S S S ●# in system also called state. ●To get from one state to another, an arrival (a) must occur or a service completion (s) must occur. ●In long-run, for each state: Rate in = Rate out Mean # A = Mean # S

Waiting Lines15 Balance equations for each state State Rate in= Rate out 0 SP 1 AP 0 Probability in Probability in state 1 state 0 The only way The only way into state 0 out of state 0 is service is to have completion from 1 an arrival

Waiting Lines16 Balance equations for each state (cont.) State Rate in=Rate out 1 AP 0 + SP 2 =AP 1 + SP 1 Can arrive Two ways state 1 by out of state 1, arrival from 0 arrival or or service service completion completion from 2 2AP 1 + SP 3 =AP 2 + SP 2 3AP 2 + SP 4 =AP 3 + SP 3 etc.

Waiting Lines17 Solution of balance equations Expected number in system = ΣnP n Solve equations simultaneously to get each probability. Given number in system, all other values are found by substitution in queuing identities.