Topic V. Multiple-Channel Queuing Model Olga Marukhina, Associate Professor, Control System Optimization Department, Tomsk Polytechnic University Marukhina@tpu.ru

Multichannel, single-phase waiting lines are found in many banks today: A common line is formed, and the customer at the head of the line proceeds to the first free teller. Multiserver systems have parallel service providers offering the same service. Multiserver examples include grocery stores (multiple cashiers), drive-through banks (multiple drive-through windows), and gas stations (multiple gas pumps).

Most common multi channel system contains parallel stations serving a single queue on FIFO basis All service stations provide the same service The single queue may separate into shorter queues in front of respective service stations Also when advantageous, calling units can shift from one queue to another All servers are assumed to perform at the same rate.

Example 1 Tom Jones, the mechanic at Golden Muffler Shop, is able to install new mufflers at an average rate of 3 per hour (or about 1 every 20 minutes), according to an exponential distribution. Customers seeking this service arrive at the shop on the average of 2 per hour, following a Poisson distribution. They are served on a first-in, first-out basis and come from a very large (almost infinite) population of possible buyers. From this description, we are able to obtain the operating characteristics of Golden Muffler’s queuing system:

Each mechanic installs mufflers at the rate of about µ = 3 per hour. The Golden Muffler Shop has decided to open a second garage and hire a second mechanic to handle installations. Customers, who arrive at the rate of about λ = 2 per hour, will wait in a single line until 1st of the 2d mechanics is free. Each mechanic installs mufflers at the rate of about µ = 3 per hour. We need to compare both systems. VS ?

Utilization factor of the system probability of zero cars in the system average number of cars in the queue (waiting) average number of cars in the system 0.375 hours (22.5 minutes) average time a car spends in the system 0.042 hours (2.5 minutes) average time a car spends in the queue (waiting) Average number of services

We can summarize the characteristics of the 2- channel model and compare them to those of the single-channel model as follows: The increased service has a dramatic effect on almost all characteristics. For instance, note that the time spent waiting in line drops from 40 minutes to only 2.5 minutes.

GPSS model

GPSS World Simulation Report - 2mech.18.1 Title GPSS World Simulation Report - 2mech.18.1 Sunday, October 04, 2015 11:59:02 The title line of the standard report is taken from the name of the Model File that produced the report. The Date and Time of the running of the model is also included. General Information START TIME END TIME BLOCKS FACILITIES STORAGES 0.000 480.000 9 0 1·  START TIME. The absolute system clock at the beginning of the measurement period. Utilizations and space-time products are based on the START TIME. The START TIME is set equal to the absolute system clock by a RESET or CLEAR statement. END TIME. The absolute clock time that the termination count became 0. BLOCKS. The number of Block entities in the simulation at the end of the simulation. FACILITIES. The number of Facility entities in the simulation at the end of the simulation. STORAGES. The number of Storage entities in the simulation at the end of the simulation. NAME VALUE MECHANIC 10000.000 NAME. User assigned names used in your GPSS World model since the last Translation. VALUE. The numeric value assigned to the name. System assigned numbers start at 10000.

Blocks       LABEL. Alphanumeric name of this Block if given one. LOC. Numerical position of this Block in the model. "Location". BLOCK TYPE. The GPSS Block name. ENTRY COUNT. The number of Transactions to enter this Block since the last RESET or CLEAR statement or since the last Translation. CURRENT COUNT. The number of Transactions in this Block at the end of the simulation. RETRY. The number of Transactions waiting for a specific condition depending on the state of this Block entity.

Queues       QUEUE. Name or number of the Queue entity. MAX. The maximum content of the Queue entity during the measurement period. A measurement period begins with the Translation of a model or the issuing of a RESET or CLEAR command. CONT. The current content of the Queue entity at the end of the simulation period. ENTRY. Entry count. The total count of Queue entries during the measurement period. ENTRY(0). "Zero entry" count. The total count of Queue entries with a 0 residence time. AVE.CONT. Queue average length of the AVE.TIME. The average time per unit of Queue content utilized during the measurement period. AVE.(-0). The average time per unit of Queue content utilized during the measurement period, adjusted for "zero entries". The space-time product divided by (the total entry count less the zero entry count). RETRY. The number of Transactions waiting for a specific condition depending on the state of this Queue entity.

Storages STORAGE. Name or number of the Storage entity. CAP. The Storage capacity of the Storage entity defined in the STORAGE statement. REM. The number of unused Storage units at the end of the simulation. MIN. The minimum number of Storage units in use during the measurement period. A measurement period begins with the Translation of a model. MAX. The maximum number of Storage units in use during the measurement period. ENTRIES. The number of "entries" into the Storage entity during the measurement period. AVL. The availability state of the Storage entity at the end of the simulation. 1 means available, 0 means unavailable. AVE.C. The time weighted average of the Storage content during the measurement period. Average number of services. UTIL. The fraction of the total space-time product of the Storage entity utilized during the measurement period. RETRY. The number of Transactions waiting for a specific condition depending on the state of this Storage entity. DELAY. The number of Transactions waiting to enter ENTER blocks on behalf of this Storage entity.

Standard Reports http://www.minutemansoftware.com/refer ence/r11.htm

Arena model

a) The percentage of time that the machine is used. Example III. FOR YOU!!! There is only one copying machine in the student lounge of the business school. Students arrive at the rate of λ = 40 per hour (according to a Poisson distribution). Copying takes an average of 40 seconds, or μ = 90 per hour (according to an exponential distribution). Compute the following: a) The percentage of time that the machine is used. b) The average length of the queue. c) The average number of students in the system. d) The average time spent waiting in the queue. e) The average time in the system.

Thank you for attention!