OPSM 405 Service Management Class 19: Managing waiting time: Queuing Theory Koç University Zeynep Aksin

Telemarketing: deterministic analysis  it takes 8 minutes to serve a customer  6 customers call per hour –one customer every 10 minutes  Flow Time = 8 min Flow Time Distribution Flow Time (minutes) Probability

Telemarketing with variability in arrival times + activity times  In reality service times –exhibit variability  In reality arrival times –exhibit variability

Why do queues form?  utilization: –throughput/capacity  variability: –arrival times –service times –processor availability

A measure of variability  Needs to be unitless  Only variance is not enough  Use the coefficient of variation  CV=  / 

Interpreting the variability measures CV i = coefficient of variation of interarrival times i) constant or deterministic arrivals CV i = 0 ii) completely random or independent arrivals CV i =1 iii) scheduled or negatively correlated arrivals CV i < 1 iv) bursty or positively correlated arrivals CV i > 1

Little’s Law or WIP = THROUGHPUT RATE x FLOWTIME For a queue: N= W Inventory I [units]... Flow Time T [hrs]

A Queueing System c , CV s ArrivalDeparture, CV a n L t L n S t S

What to manage in such a process?  Inputs –Arrival rate / distribution –Service or processing time / distribution  System structure –Number of servers c –Number of queues –Maximum queue capacity/buffer capacity K  Operating control policies –Queue-service discipline

Performance Measures  Sales –Throughput –Abandoning rate  Cost –Capacity utilization –Queue length / total number in process  Customer service –Waiting time in queue / total time in process –Probability of blocking

The A/B/C notation  A: type of distribution for interarrival times  B: type of distribution for service times  C: the number of parallel servers M = exponential interarrival and service time distribution (same as Poisson arrival or service rate) D= deterministic interarrival or service time G= general distributions

Variation characteristics  distribution type M: CV a = CV s =1  distribution type D: CV a = CV s = 0  distribution type G: could be any value

Basic notation = mean arrival rate (units per time period)  = mean service rate (units per time period)  =  /  = utilization rate (traffic intensity) c = number of servers (sometimes also s) P 0 = probability that there are 0 customers in the system P n = probability that there are n customers in the system L s = mean number of customers in the system (N s ) L q = mean number of customers in the queue (N q ) W s = mean time in the system W q = mean time in the queue

Recall Little’s Law L q =  W q queue length = arrival rate * time in queue

The building block: M/M/1  An infinite or large population of customers arriving independently; no reservations  Poisson arrival rate (exponential interarrival times)  single server, single queue  no reneges or balking  no restrictions on queue length  first-come first-served (FCFS)  exponential service times

Facts for M/M/1  < 1 P 0 = 1-  P n = P 0  n L s = /(  ) W s = 1 / (  ) L q =  W q =  1 / (  )

For a general system with c servers W (or t S ) = average service time + W q (or t q ) Average wait = (scale effect) (utilization effect) (variability effect) W q = L q /  c  Note:

Generalized Throughput-Delay Curve Variability Increases Average Flow Time Ws Utilization (ρ)  100% 

In words:  in high utilization case: small decrease in utilization yields large improvement in response time  this marginal improvement decreases as the slack in the system increases

Levers to reduce waiting and increase QoS:  variability reduction + safety capacity  How to reduce system variability?  Safety Capacity = capacity carried in excess of expected demand to cover for system variability –it provides a safety net against higher than expected arrivals or services and reduces waiting time

Excel does it all!

Example: Secretarial Pool  4 Departments and 4 Departmental secretaries  Request rate for Operations, Accounting, and Finance is 2 requests/hour  Request rate for Marketing is 3 requests/hour  Secretaries can handle 4 requests per hour  Marketing department is complaining about the response time of the secretaries. They demand 30 min. response time.  College is considering two options: –Hire a new secretary –Reorganize the secretarial support

Current Situation Accounting Finance Marketing Operations 2 requests/hour 3 requests/hour 2 requests/hour 4 requests/hour

Current Situation: queueing notation Acc., Fin., Ops. Marketing = 2 requests/hour = 3 requests/hour  = 4 requests/hour C 2 [A] = 1 (totally random arrivals) C 2 [S] = 1 (assumption) C 2 [S] = 1 (assumption)

Current Situation: waiting times W = service time + Wq W = 0.25 hrs hrs = 30 minutes Accounting, Operations, Finance: Marketing: W = service time + Wq W = 0.25 hrs hrs = 60 minutes

Proposal: Secretarial Pool Accounting Finance Marketing Operations 16 requests/hour 9 requests/hour

Proposal: Secretarial Pool Wq = hrs. W= hrs hrs.= 17 minutes In the proposed system, faculty members in all departments get their requests back in 17 minutes on the average. (Around 50% improvement for Acc, Fin, and Ops and 75% improvement for Marketing)

The impact of task integration (pooling)  balances utilization...  reduces resource interference... ...therefore reduces the impact of temporary bottlenecks  there is more benefit from pooling in a high utilization and high variability process  pooling is beneficial as long as it does not introduce excessive variability in a low variability system the benefits exceed the task time reductions due to specialization

Examples of pooling in business  Consolidating back office work  Call centers  Single line versus separate queues

Capacity design using queueing models  Criteria for design waiting time probability of excessive waiting minimize probability of lost sales maximize revenues

Example: bank branch  48 customers arrive per hour, 50 % for teller service and 50 % for ATM service  On average, 5 minutes to service each request or 12 per hour.  Can model as two independent queues in parallel, each with mean arrival rate of =24 customers per hour  Want to find number of tellers and ATMs to ensure customers will find an available teller or ATM at least 95 % of the time

How many tellers and ATMs? P(delay) or P(wait) less than 5%: 6 Tellers and 6 ATMs

Example  A mail order company has one department for taking customer orders and another for handling complaints. Currently each has a separate phone number. Each department has 7 phone lines. Calls arrive at an average rate of 1 per minute and are served at 1.5 per minute. Management is thinking of combining the departments into a single one with a single phone number and 14 phone lines.  The proportion of callers getting a busy signal will….?  Average flow experienced by customers will….?

Example  A bank would like to improve its drive-in service by reducing waiting and transaction times. Average rate of customer arrivals is 30/hour. Customers form a single queue and are served by 4 windows in a FCFS manner. Each transaction is completed in 6 minutes on average. The bank is considering to lease a high speed information retrieval and communication equipment that would cost 30 YTL per hour. The facility would reduce each teller’s transaction time to 4 minutes per customer.  a. If our manager estimates customer cost of waiting in queue to be 20 YTL per customer per hour, can she justify leasing this equipment?  b. The competitor provides service in 8 minutes on average. If the bank wants to meet this standard, should it lease the new equipment?

Example Global airlines is revamping its check-in operations at its hub terminal. This is a single queue system where an available server takes the next passenger. Arrival rate is estimated to be 52 passengers per hour. During the check-in process, an agent confirms reservation, assigns a seat, issues a boarding pass, and weighs, labels, dispatches baggage. The entire process takes on average 3 minutes. Agents are paid 20 YTL an hour and it is estimated that Global loses 1 YTL for every minute a passenger spends waiting in line. How many agents should Global staff at its hub terminal? How many agents does it need to meet the industry norm of 3 minutes wait?

Capacity Management  First check if average capacity is enough: is there a perpetual queue? If not, increase capacity  Capacity may be enough on average but badly distributed over time periods experiencing demand fluctuations: check if there is a predictable queue, do proper scheduling; you may need more people to accommodate scheduling constraints  Find sources of variability and try to reduce them: these create the stochastic queue

Want to eliminate as much variability as possible from your processes: how?  specialization in tasks can reduce task time variability  standardization of offer can reduce job type variability  automation of certain tasks  IT support: templates, prompts, etc.  incentives

Tips for queueing problems  Make sure you use rates not times for  and   Use consistent units: minutes, hours, etc.  If the problem states “constant service times” or an “automated machine with practically constant times” this means: deterministic service so CV s =0  Check the objective: –Cost minimization? –Service level satisfaction at lowest cost? –Etc.  Read carefully to understand difference between “waiting”, “standing in line” (in queue) “in system” or “total flow time” or “providing service”