EASTERN MEDITERRANEAN UNIVERSITY DEPARTMENT OF INDUSTRIAL ENGINEERING IENG

EASTERN MEDITERRANEAN UNIVERSITY DEPARTMENT OF INDUSTRIAL ENGINEERING IENG 314 OPERATIONS RESEARCH II Ilkin Jafarov 148692

Modeling the Service Process The Kendall-Lee for Queuning Systems IENG 314: Bonus Homework

What is Queuing Theory? Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service. Queuing Model

Queue Discipline The queue discipline describes the method used to determine the order in which customers are served. • FCFS • LCFS • SIRO

Erlang Distribution 1

Modeling the Service Process

Example Lets suppose there are three automated cash register in a shop, all of them are busy, and a one customer is waiting in queue. And lets suppose before going to cash register all customers scan their product and add them into theirs online basket. In the cash register they only pay the total price of the basket. The manager of the shop ask us to find probability that the waiting customer will be the last of the four customers to complete service in similar cases. Lets assume the first customer was the first one that came to the cash register, so by FCFS discipline he/she will be the first to complete service. Then customer 4 will enter service. Customer 4’s service time has the same distribution as the remaining service times of customers 2 and 3. Thus, customers 4, 2, and 3 will have the same chance of being the last customer to complete service. That means, customer 4 has a 1/3 chance of being the last customer to complete service. Unfortunately, actual service times may not be consistent with the no-memory property. For this reason, we often assume that s(t) is an Erlang distribution with shape parameter k and rate parameter kµ.

The Kendall–Lee Notation for Queuing Systems Where do we use the Kendall–Lee Notation? The notation that we will discuss is used to describe a queuing system in which all arrivals wait in a single line until one of s identical parallel servers is free. 1 Example The customer in server 1 is the next customer to complete service, then (FCFS) the first customer in line would enter server 1. The next customer in line would enter service after the next service completion, and so on. 2

Characteristics of The Kendall–Lee Notation 1/2/3/4/5/6 or A/S/C/K/N/D A – Arrival process M - Poisson distribution; random arrival process D – Degenerate distribution; Deterministic or fixed arrival rate G – General distribution E – Erlang distribution S – Service time distribution M – Exponential service time D – Degenerate distribution; Deterministic or fixed service time E – Erlang distribution C – Number of servers Can range from 1 to many K – Capacity of the whole system, i. e. , maximum number of customers allowed in queue plus those being serviced. By default taken as infinity N – Size of the population from where customers are coming. By default taken as infinity D – Queue discipline, i. e. , the order in which the customers in queue are being served FCFS – First come first serve; Default LCFS – Last come first serve SIRO – Service in random order PS – Processor sharing PNPN – Priority service

Examples of Simple Form of Kendall-Lee Notation The simplest form of Kendall's notation consists of 3 letters: a/b/c Where, a represents the probability distribution of customers arrivals (M, E, D, G); b represents the probability distribution of service time (M, E, D, G); c is the number of servers; For example, queuing system M/M/1 (simple system) means Poisson distribution for arrival and Exponential distribution for service time and one server. Queuing system M/M/2 has customers arrive according to Poisson distribution with Exponential distribution for service time and the system has two servers. Queuing system M/M/3 indicates that the customers arrive according to Poisson distribution while the service time is constant with 3 servers. M/M/s – Airline Ticket Counter M/D/1 – Automated Car Wash M/G/1 – Auto Repair Shop