On Jockeying in Queues

There are numerous queueing situations in which those awaiting service may be allowed to make choices which affect the time spent in the service system. In such cases, it should be possible to formulate a "strategy" for customers to follow in order to optimize a given parameter. A whole class of queue problems which involve "jockeying" can be regarded in this way, but has so far received little attention. Jockeying can be described as the movement of of a waiting customer from one queue to another (of shorter length or which appears to be moving faster, etc.) in anticipation of a shorter delay. We have so far considered steady state solutions for just a few of the various possible jockeying disciplines in two-server systems with heterogeneous "exponential" servers and Poisson inputs. As a basis for comparison, we first treat the heterogeneous server problems where arriving customers join the shorter of two independent waiting lines and are not permitted to jockey. The same problem is then considered allowing instantaneous jockeying from the longer to the shorter line when the difference in line lengths exceeds one. The results, in terms of expected line lengths and delays, are identical to those obtained by Gumbel (Gumbel, H. 1960. Waiting lines with heterogeneous servers. Oper. Res. 8 504.) for heterogeneous servers fed from a single queue. When the same problem, with customer preferences for a specific line, is considered the results are identical to those of Krishnamoorthi (Krishnamoorthi, B. 1963. On a poisson queue with two heterogeneous servers. Oper. Res. 11 321). It is important to note that in these two problems the slower server has a larger throughput than might be expected from the classical theory. In other words, the slower server acts as a trapping state. In the last problems treated here, customers join a preferred line and may jockey at a rate proportional to the line lengths or proportional to the difference in line lengths.